IN

FORMAL LOGIC

INCLUDING A GENERALISATION OF LOGICAL PROCESSES IN THEIR APPLICATION TO COMPLEX INFERENCES

BY

JOHN NEVILLE KEYNES, M.A., Sc.D.

UNIVERSITY LECTURER IN MORAL SCIENCE AND FORMERLY FELLOW OF PEMBROKE
COLLEGE IN THE UNIVERSITY OF CAMBRIDGE

*FOURTH EDITION RE-WRITTEN AND ENLARGED*

𝕷𝖔𝖓𝖉𝖔𝖓

[*The Right of Translation and Reproduction is reserved*]

*First Edition* (*Crown* 8*vo.*) *printed* 1884.

*Second Edition* (*Crown* 8*vo.*) 1887.

*Third Edition* (*Demy* 8*vo.*) 1894.

*Fourth Edition* (*Demy* 8*vo.*) 1906.

IN this edition many of the sections have been re-written and a good deal of new matter has been introduced. The following are some of the more important modifications.

In Part I a new definition of “connotative name” is proposed, in the hope that some misunderstanding may thereby be avoided; and the treatment of negative names has been revised.

In Part II the problem of the import of judgments and propositions in its various aspects is dealt with in much more detail than before, and greater importance is attached to distinctions of modality. Partly in consequence of this, the treatment of conditional and hypothetical propositions has been modified. I have partially re-written the chapter on the existential import of propositions in order to meet some recent criticisms and to explain my position more clearly. Many other minor changes in Part II have been made.

Amongst the changes in Part III are a more systematic treatment of the process of the indirect reduction of syllogisms, and the introduction of a chapter on the characteristics of inference.

An appendix on the fundamental laws of thought has been added; and the treatment of complex propositions which previously constituted Part IV of the book has now been placed in an appendix.

The reader of this edition will perceive my indebtedness to
Sigwart’s *Logic*. I have received valuable help from
Professor J. S. Mackenzie and from my son, Mr J. M. Keynes; and I
cannot express too strongly the debt I once more owe to Mr W. E.
Johnson, who by his criticisms has enabled me to improve my exposition
in many parts of the book, and also to avoid some errors.

J. N. KEYNES.

6, HARVEY ROAD,

CAMBRIDGE,

4 *September* 1906.

^{1} With some omissions.

IN addition to a somewhat detailed exposition of certain portions of what may be called the book-work of formal logic, the following pages contain a number of problems worked out in detail and unsolved problems, by means of which the student may test his command over logical processes.

In the expository portions of Parts I, II, and III, dealing
respectively with terms, propositions, and syllogisms, the traditional
lines are in the main followed, though with certain modifications;
*e.g.*, in the systematisation of immediate inferences, and in
several points of detail in connexion with the syllogism. For purposes
of illustration Euler’s diagrams are employed to a greater
extent than is usual in English manuals.

In Part IV, which contains a generalisation of logical processes in their application to complex inferences, a somewhat new departure is taken. So far as I am aware this part constitutes the first systematic attempt that has been made to deal with formal reasonings of the most complicated character without the aid of mathematical or other symbols of operation, and without abandoning the ordinary non-equational or predicative form of proposition. This attempt has on the whole met with greater success than I had anticipated; and I believe that the methods formulated will be found to be both as easy and as effective as the symbolical methods of Boole and his followers. The book concludes with a general and sure method of solution of what Professor Jevons called the inverse problem, and which he himself seemed to regard as soluble only by a series of guesses.

The writers on logic to whom I have been chiefly indebted are De
Morgan, Jevons, and Venn. To Mr Venn I am peculiarly indebted, not
merely by reason of his published writings, vii especially his *Symbolic Logic*, but
also for most valuable suggestions and criticisms while this book was
in progress. I am glad to have this opportunity of expressing to him
my thanks for the ungrudging help he has afforded me. I am also under
great obligation to Miss Martin of Newnham College, and to Mr
Caldecott of St John’s College, for criticisms which I have
found extremely helpful.

CAMBRIDGE,

19 *January* 1884.

THIS edition has been carefully revised, and numerous sections have been almost entirely re-written.

In addition to the introduction of some brief prefatory sections, the following are among the more important modifications. In Part I an attempt has been made to differentiate the meanings of the three terms connotation, intension, comprehension, with the hope that such differentiation of meaning may help to remove an ambiguity which is the source of much of the current controversy on the subject of connotation. In Part II a distinction between conditional and hypothetical propositions is adopted for which I am indebted to Mr W. E. Johnson; and the treatment of the existential import of propositions has been both expanded and systematised. In Part IV particular propositions, which in the first edition were practically neglected, are treated in detail; and, while the number of mere exercises has been diminished, many points of theory have received considerable development. Throughout the book the unanswered exercises are now separated from the expository matter and placed together at the end of the several chapters in which they occur. An index has been added.

I have to thank several friends and correspondents, amongst whom I must especially mention Mr Henry Laurie of the University of Melbourne and Mr W. E. Johnson of King’s College, Cambridge, for suggestions and criticisms from which I have derived the greatest assistance. Mr Johnson has kindly read the proof sheets throughout; and I am particularly indebted to him for the generous manner in which he has placed at my disposal not only his time but also the results of his own work on various points of formal logic.

CAMBRIDGE,

22 *June* 1887.

THIS edition has been in great part re-written and the book is again considerably enlarged.

In Part I the mutual relations between the extension and the intension of names are examined from a new point of view, and the distinction between real and verbal propositions is treated more fully than in the two earlier editions. In Part II more attention is paid to tables of equivalent propositions, certain developments of Euler’s and Lambert’s diagrams are introduced, the interpretation of propositions in extension and intension is discussed in more detail, and a brief explanation is given of the nature of logical equations. The chapters on the existential import of propositions and on conditional, hypothetical, and disjunctive (or, as I now prefer to call them, alternative) propositions have also been expanded, and the position which I take on the various questions raised in these chapters is I hope more clearly explained. In Parts III and IV there is less absolutely new matter, but the minor modifications are numerous. An appendix is added containing a brief account of the doctrine of division.

In the preface to earlier editions I was glad to have the opportunity of acknowledging my indebtedness to Professor Caldecott, to Mr W. E. Johnson, to Professor Henry Laurie, to Dr Venn, and to Mrs Ward. In the present edition my indebtedness to Mr Johnson is again very great. Many new developments are due to his suggestion, and in every important discussion in the book I have been most materially helped by his criticism and advice.

CAMBRIDGE,

25 *July* 1894.

INTRODUCTION. | ||

SECTION | PAGE | |

1. | The General Character of Logic | 1 |

2. | Formal Logic | 1 |

3. | Logic and Language | 3 |

4. | Logic and Psychology | 5 |

5. | The Utility of Logic | 6 |

PART I. | ||

TERMS. | ||

CHAPTER I. | ||

THE LOGIC OF TERMS. | ||

6. | The Three Parts of Logical Doctrine | 8 |

7. | Names and Concepts | 10 |

8. | The Logic of Terms | 11 |

9. | General and Singular Names | 11 |

10. | Proper Names | 13 |

11. | Collective Names | 14 |

12. | Concrete and Abstract Names | 16 |

13. | Can Abstract Names be subdivided into General and Singular? | 19 |

14, 15. | Exercises | 21 |

CHAPTER II. | ||

EXTENSION AND INTENSION. | ||

16. | The Extension and the Intension of Names | 22 |

17. | Connotation, Subjective Intension, and Comprehension. | 23 |

18. | Sigwart’s distinction between Empirical, Metaphysical, and Logical Concepts | 27 |

xii | ||

19. | Connotation and Etymology | 28 |

20. | Fixity of Connotation | 28 |

21. | Extension and Denotation | 29 |

22. | Dependence of Extension and Intension upon one another | 31 |

23. | Inverse Variation of Extension and Intension | 35 |

24. | Connotative Names | 40 |

25. | Are proper names connotative? | 41 |

26 to 30. | Exercises | 47 |

CHAPTER III. | ||

REAL, VERBAL, AND FORMAL PROPOSITIONS. | ||

31. | Real, Verbal, and Formal Propositions | 49 |

32. | Nature of the Analysis involved in Analytic Propositions | 53 |

33 to 37. | Exercises | 56 |

CHAPTER IV. | ||

NEGATIVE NAMES AND RELATIVE NAMES. | ||

38. | Positive and Negative Names | 57 |

39. | Indefinite Character of Negative Names | 59 |

40. | Contradictory Terms | 61 |

41. | Contrary Terms | 62 |

42. | Relative Names | 63 |

43 to 45. | Exercises | 65 |

PART II. | ||

PROPOSITIONS. | ||

CHAPTER I. | ||

IMPORT OF JUDGMENTS AND PROPOSITIONS. | ||

46. | Judgments and Propositions | 66 |

47. | The Abstract Character of Logic | 68 |

48. | Nature of the Enquiry into the Import of Propositions | 70 |

49. | The Objective Reference in Judgments | 74 |

50. | The Universality of Judgments | 76 |

51. | The Necessity of Judgments | 77 |

52. | Exercise | 78 |

xiii | ||

CHAPTER II. | ||

KINDS OF JUDGMENTS AND PROPOSITIONS. | ||

53. | The Classification of Judgments | 79 |

54. | Kant’s Classification of Judgments | 81 |

55. | Simple Judgments and Compound Judgments | 82 |

56. | The Modality of Judgments | 84 |

57. | Modality in relation to Simple Judgments | 85 |

58. | Subjective Distinctions of Modality | 86 |

59. | Objective Distinctions of Modality | 87 |

60. | Modality in relation to Compound Judgments | 90 |

61. | The Quantity and the Quality of Propositions | 91 |

62. | The traditional Scheme of Propositions | 92 |

63. | The Distribution of Terms in a Proposition | 95 |

64. | The Distinction between Subject and Predicate in the traditional Scheme of Propositions | 96 |

65. | Universal Propositions | 97 |

66. | Particular Propositions | 100 |

67. | Singular Propositions | 102 |

68. | Plurative Propositions and Numerically Definite Propositions | 103 |

69. | Indefinite Propositions | 105 |

70. | Multiple Quantification | 105 |

71. | Infinite or Limitative Propositions | 106 |

72 to 78. | Exercises | 107 |

CHAPTER III. | ||

THE OPPOSITION OF PROPOSITIONS. | ||

79. | The Square of Opposition | 109 |

80. | Contradictory Opposition | 111 |

81. | Contrary Opposition | 114 |

82. | The Opposition of Singular Propositions | 115 |

83. | The Opposition of Modal Propositions | 116 |

84. | Extension of the Doctrine of Opposition | 117 |

85. | The Nature of Significant Denial | 119 |

86 to 95. | Exercises. | 124 |

xiv | ||

CHAPTER IV. | ||

IMMEDIATE INFERENCES. | ||

96. | The Conversion of Categorical Propositions | 126 |

97. | Simple Conversion and Conversion per accidens. | 128 |

98. | Inconvertibility of Particular Negative Propositions | 130 |

99. | Legitimacy of Conversion | 130 |

100. | Table of Propositions connecting any two terms | 132 |

101. | The Obversion of Categorical Propositions | 133 |

102. | The Contraposition of Categorical Propositions | 134 |

103. | The Inversion of Categorical Propositions | 137 |

104. | The Validity of Inversion | 139 |

105. | Summary of Results | 140 |

106. | Table of Propositions connecting any two terms and their contradictories | 141 |

107. | Mutual Relations of the non-equivalent Propositions connecting any two terms and their contradictories | 142 |

108. | The Elimination of Negative Terms | 144 |

109. | Other Immediate Inferences | 147 |

110. | Reduction of immediate inferences to the mediate form | 151 |

111 to 124. | Exercises | 153 |

CHAPTER V. | ||

THE DIAGRAMMATIC REPRESENTATION OF PROPOSITIONS. | ||

125. | The use of Diagrams in Logic | 156 |

126. | Euler’s Diagrams | 157 |

127. | Lambert’s Diagrams | 163 |

128. | Dr Venn’s Diagrams | 166 |

129. | Expression of the possible relations between any two classes by means of the propositional forms
A, E, I, O | 168 |

130. | Euler’s diagrams and the class-relations between S, not-S, P, not-P | 170 |

131. | Lambert’s diagrams and the class-relations between S, not-S, P, not-P | 174 |

132 to 134. | Exercises | 176 |

xv | ||

CHAPTER VI. | ||

PROPOSITIONS IN EXTENSION AND IN INTENSION. | ||

135. | Fourfold Implication of Propositions in Connotation and Denotation | 177 |

(1) Subject in denotation, predicate in connotation | 179 | |

(2) Subject in denotation, predicate in denotation | 181 | |

(3) Subject in connotation, predicate in connotation | 184 | |

(4) Subject in connotation, predicate in denotation | 186 | |

136. | The Reading of Propositions in Comprehension | 187 |

CHAPTER VII. | ||

LOGICAL EQUATIONS AND THE QUANTIFICATION OF THE PREDICATE. | ||

137. | The employment of the symbol of Equality in Logic | 189 |

138. | Types of Logical Equations | 191 |

139. | The expression of Propositions as Equations | 194 |

140. | The eight propositional forms resulting from the explicit Quantification of the Predicate | 195 |

141. | Sir William Hamilton’s fundamental Postulate of Logic | 195 |

142. | Advantages claimed for the Quantification of the Predicate | 196 |

143. | Objections urged against the Quantification of the Predicate | 197 |

144. | The meaning to be attached to the word some in the eight propositional forms recognised by Sir
William Hamilton | 199 |

145. | The use of some in the sense of some only | 202 |

146. | The interpretation of the eight Hamiltonian forms of proposition, some being used in its
ordinary logical sense | 203 |

147. | The propositions U and Y | 204 |

148. | The proposition η | 206 |

149. | The proposition ω | 206 |

150. | Sixfold Schedule of Propositions obtained by recognising Y and η, in addition
to A, E, I, O | 207 |

151, 152. | Exercises | 209 |

CHAPTER VIII. | ||

THE EXISTENTIAL IMPORT OF CATEGORICAL PROPOSITIONS. | ||

153. | Existence and the Universe of Discourse | 210 |

154. | Formal Logic and the Existential Import of Propositions | 215 |

155. | The Existential Formulation of Propositions | 218 |

156. | Various Suppositions concerning the Existential Import of Categorical Propositions | 218 |

xvi | ||

157. | Reduction of the traditional forms of proposition to the form of Existential Propositions | 221 |

158. | Immediate Inferences and the Existential Import of Propositions | 223 |

159. | The Doctrine of Opposition and the Existential Import of Propositions | 227 |

160. | The Opposition of Modal Propositions considered in connexion with their Existential Import | 231 |

161. | Jevons’s Criterion of Consistency | 232 |

162. | The Existential Import of the Propositions included in the Traditional Schedule | 234 |

163. | The Existential Import of Modal Propositions | 244 |

164 to 172. | Exercises | 245 |

CHAPTER IX. | ||

CONDITIONAL AND HYPOTHETICAL PROPOSITIONS. | ||

173. | The distinction between Conditional Propositions and Hypothetical Propositions | 249 |

174. | The Import of Conditional Propositions | 252 |

175. | Conditional Propositions and Categorical Propositions | 253 |

176. | The Opposition of Conditional Propositions | 256 |

177. | Immediate Inferences from Conditional Propositions | 259 |

178. | The Import of Hypothetical Propositions | 261 |

179. | The Opposition of Hypothetical Propositions | 264 |

180. | Immediate Inferences from Hypothetical Propositions | 268 |

181. | Hypothetical Propositions and Categorical Propositions | 270 |

182. | Alleged Reciprocal Character of Conditional and Hypothetical Judgments | 270 |

183 to 188. | Exercises | 273 |

CHAPTER X. | ||

DISJUNCTIVE (OR ALTERNATIVE) PROPOSITIONS. | ||

189. | The terms Disjunctive and Alternative as applied to Propositions | 275 |

190. | Two types of Alternative Propositions | 276 |

191. | The Import of Disjunctive (Alternative) Propositions | 277 |

192. | Scheme of Assertoric and Modal Propositions | 282 |

193. | The Relation of Disjunctive (Alternative) Propositions to Conditionals and Hypotheticals | 282 |

194 to 196. | Exercises | 284 |

xvii | ||

PART III. | ||

SYLLOGISMS. | ||

CHAPTER I. | ||

THE RULES OF THE SYLLOGISM. | ||

197. | The Terms of the Syllogism | 285 |

198. | The Propositions of the Syllogism | 287 |

199. | The Rules of the Syllogism | 287 |

200. | Corollaries from the Rules of the Syllogism | 289 |

201. | Restatement of the Rules of the Syllogism | 291 |

202. | Dependence of the Rules of the Syllogism upon one another | 291 |

203. | Statement of the independent Rules of the Syllogism | 293 |

204. | Proof of the Rule of Quality | 294 |

205. | Two negative premisses may yield a valid conclusion; but not syllogistically | 295 |

206. | Other apparent exceptions to the Rules of the Syllogism | 297 |

207. | Syllogisms with two singular premisses | 298 |

208. | Charge of incompleteness brought against the ordinary syllogistic conclusion | 300 |

209. | The connexion between the Dictum de omni et nullo and the ordinary Rules of the Syllogism | 301 |

210 to 242. | Exercises | 302 |

CHAPTER II. | ||

THE FIGURES AND MOODS OF THE SYLLOGISM. | ||

243. | Figure and Mood | 309 |

244. | The Special Rules of the Figures; and the Determination of the Legitimate Moods in each Figure | 309 |

245. | Weakened Conclusions and Subaltern Moods | 313 |

246. | Strengthened Syllogisms | 314 |

247. | The peculiarities and uses of each of the four figures of the syllogism | 315 |

248 to 255. | Exercises | 317 |

xviii | ||

CHAPTER III. | ||

THE REDUCTION OF SYLLOGISMS. | ||

256. | The Problem of Reduction | 318 |

257. | Indirect Reduction | 318 |

258. | The mnemonic lines Barbara, Celarent, &c. | 319 |

259. | The direct reduction of Baroco and Bocardo | 323 |

260. | Extension of the Doctrine of Reduction | 324 |

261. | Is Reduction an essential part of the Doctrine of the Syllogism? | 325 |

262. | The Fourth Figure | 328 |

263. | Indirect Moods | 329 |

264. | Further discussion of the process of Indirect Reduction | 331 |

265. | The Antilogism | 332 |

266. | Equivalence of the Moods of the first three Figures shewn by the Method of Indirect Reduction | 333 |

267. | The Moods of Figure 4 in their relation to one another | 334 |

268. | Equivalence of the Special Rules of the First Three Figures | 335 |

269. | Scheme of the Valid Moods of Figure 1 | 336 |

270. | Scheme of the Valid Moods of Figure 2 | 336 |

271. | Scheme of the Valid Moods of Figure 3 | 337 |

272. | Dictum for Figure 4 | 338 |

273 to 287. | Exercises | 339 |

CHAPTER IV. | ||

THE DIAGRAMMATIC REPRESENTATION OF SYLLOGISMS. | ||

288. | Euler’s diagrams and syllogistic reasonings | 341 |

289. | Lambert’s diagrams and syllogistic reasonings | 344 |

290. | Dr Venn’s diagrams and syllogistic reasonings | 345 |

291 to 300. | Exercises | 347 |

CHAPTER V. | ||

CONDITIONAL AND HYPOTHETICAL SYLLOGISMS. | ||

301. | The Conditional Syllogism, the Hypothetical Syllogism, and the Hypothetico-Categorical Syllogism | 348 |

302. | Distinctions of Mood and Figure in the case of Conditional and Hypothetical Syllogisms | 349 |

303. | Fallacies in Hypothetical Syllogisms | 350 |

304. | The Reduction of Conditional and Hypothetical Syllogisms | 351 |

xix | ||

305. | The Moods of the Mixed Hypothetical Syllogism | 352 |

306. | Fallacies in Mixed Hypothetical Syllogisms | 353 |

307. | The Reduction of Mixed Hypothetical Syllogisms | 354 |

308. | Is the reasoning contained in the mixed hypothetical syllogism mediate or immediate? | 354 |

309 to 315. | Exercises | 358 |

CHAPTER VI. | ||

DISJUNCTIVE SYLLOGISMS. | ||

316. | The Disjunctive Syllogism | 359 |

317. | The modus ponendo tollens | 361 |

318. | The Dilemma | 363 |

319 to 321. | Exercises | 366 |

CHAPTER VII. | ||

IRREGULAR AND COMPOUND SYLLOGISMS. | ||

322. | The Enthymeme | 367 |

323. | The Polysyllogism and the Epicheirema | 368 |

324. | The Sorites | 370 |

325. | The Special Rules of the Sorites | 372 |

326. | The possibility of a Sorites in a Figure other than the First | 373 |

327. | Ultra-total Distribution of the Middle Term | 376 |

328. | The Quantification of the Predicate and the Syllogism | 378 |

329. | Table of valid moods resulting from the recognition of Y and η in addition to A, E, I, O | 381 |

330. | Formal Inferences not reducible to ordinary Syllogisms | 384 |

331 to 341. | Exercises | 388 |

CHAPTER VIII. | ||

PROBLEMS ON THE SYLLOGISM. | ||

342. | Bearing of the existential interpretation of propositions upon the validity of syllogistic reasonings | 390 |

343. | Connexion between the truth and falsity of premisses and conclusion in a valid syllogism | 394 |

344. | Arguments from the truth of one premiss and the falsity of the other premiss in a valid syllogism, or from the falsity of one premiss to the truth of the conclusion, or from the truth of one premiss to the falsity of the conclusion | 396 |

345. | Numerical Moods of the Syllogism | 400 |

346 to 375. | Exercises | 403 |

xx | ||

CHAPTER IX. | ||

THE CHARACTERISTICS OF INFERENCE. | ||

376. | The Nature of Logical Inference | 413 |

377. | The Paradox of Inference | 414 |

378. | The nature of the difference that there must be between premisses and conclusion in an inference | 415 |

379. | The Direct Import and the Implications of a Proposition | 420 |

380. | Syllogisms and Immediate Inferences | 423 |

381. | The charge of petitio principii brought against Syllogistic Reasoning | 424 |

CHAPTER X. | ||

EXAMPLES OF ARGUMENTS AND FALLACIES. | ||

382 to 408. | Exercises | 431 |

APPENDIX A. | ||

THE DOCTRINE OF DIVISION. | ||

409. | Logical Division | 441 |

410. | Physical Division, Metaphysical Division, and Verbal Division | 442 |

411. | Rules of Logical Division | 443 |

412. | Division by Dichotomy | 445 |

413. | The place of the Doctrine of Division in Logic | 446 |

APPENDIX B. | ||

THE FUNDAMENTAL LAWS OF THOUGHT. | ||

414. | The Three Laws of Thought | 450 |

415. | The Law of Identity | 451 |

416. | The Law of Contradiction | 454 |

417. | The Sophism of “The Liar” | 457 |

418. | The Law of Excluded Middle | 458 |

419. | Grounds on which the absolute universality and necessity of the law of excluded middle have been denied | 460 |

420. | Are the Laws of Thought also Laws of Things? | 463 |

421. | Mutual Relations of the three Laws of Thought | 464 |

422. | The Laws of Thought in relation to Immediate Inferences | 464 |

423. | The Laws of Thought and Formal Mediate Inferences | 466 |

xxi | ||

APPENDIX C. | ||

A GENERALISATION OF LOGICAL PROCESSES
IN THEIR APPLICATION TO COMPLEX PROPOSITIONS. | ||

CHAPTER I. | ||

THE COMBINATION OF TERMS. | ||

424. | Complex Terms | 468 |

425. | Order of Combination in Complex Terms | 469 |

426. | The Opposition of Complex Terms | 470 |

427. | Duality of Formal Equivalences in the case of Complex Terms | 472 |

428. | Laws of Distribution | 472 |

429. | Laws of Tautology | 473 |

430. | Laws of Development and Reduction | 474 |

431. | Laws of Absorption | 475 |

432. | Laws of Exclusion and Inclusion | 475 |

433. | Summary of Formal Equivalences of Complex Terms | 475 |

434. | The Conjunctive Combination of Alternative Terms | 478 |

435 to 439. | Exercises | 477 |

CHAPTER II. | ||

COMPLEX PROPOSITIONS AND COMPOUND PROPOSITIONS. | ||

440. | Complex Propositions | 478 |

441. | The Opposition of Complex Propositions | 478 |

442. | Compound Propositions | 478 |

443. | The Opposition of Compound Propositions | 480 |

444. | Formal Equivalences of Compound Propositions | 480 |

445. | The Simplification of Complex Propositions | 481 |

446. | The Resolution of Universal Complex Propositions into Equivalent Compound Propositions | 483 |

447. | The Resolution of Particular Complex Propositions into Equivalent Compound Propositions | 484 |

448. | The Omission of Terms from Complex Propositions | 485 |

449. | The Introduction of Terms into Complex Propositions | 485 |

450. | Interpretation of Anomalous Forms | 486 |

451 to 453. | Exercises | 487 |

xxii | ||

CHAPTER III. | ||

IMMEDIATE INFERENCES FROM COMPLEX PROPOSITIONS. | ||

454. | The Obversion of Complex Propositions | 488 |

455. | The Conversion of Complex Propositions | 489 |

456. | The Contraposition of Complex Propositions | 490 |

457. | Summary of the results obtainable by Obversion, Conversion, and Contraposition | 493 |

458 to 473. | Exercises | 494 |

CHAPTER IV. | ||

THE COMBINATION OF COMPLEX PROPOSITIONS. | ||

474. | The Problem of combining Complex Propositions | 498 |

475. | The Conjunctive Combination of Universal Affirmatives | 498 |

476. | The Conjunctive Combination of Universal Negatives | 499 |

477. | The Conjunctive Combination of Universals with Particulars of the same Quality | 500 |

478. | The Conjunctive Combination of Affirmatives with Negatives | 501 |

479. | The Conjunctive Combination of Particulars with Particulars | 501 |

480. | The Alternative Combination of Universal Propositions | 502 |

481. | The Alternative Combination of Particular Propositions | 502 |

482. | The Alternative Combination of Particulars with Universals | 502 |

483 to 486. | Exercises | 503 |

CHAPTER V. | ||

INFERENCES FROM COMBINATIONS OF COMPLEX PROPOSITIONS. | ||

487. | Conditions under which a universal proposition affords information in regard to any given term | 504 |

488. | Information jointly afforded by a series of universal propositions with regard to any given term | 506 |

489. | The Problem of Elimination | 508 |

490. | Elimination from Universal Affirmatives | 509 |

491. | Elimination from Universal Negatives | 510 |

492. | Elimination from Particular Affirmatives | 511 |

493. | Elimination from Particular Negatives | 511 |

494. | Order of procedure in the process of elimination | 511 |

495 to 533. | Exercises | 512 |

xxiii | ||

CHAPTER VI. | ||

THE INVERSE PROBLEM. | ||

534. | Nature of the Inverse Problem | 525 |

535. | A General Solution of the Inverse Problem | 527 |

536. | Another Method of Solution of the Inverse Problem | 530 |

537. | A Third Method of Solution of the Inverse Problem | 531 |

538. | Mr Johnson’s Notation for the Solution of Logical Problems | 533 |

539. | The Inverse Problem and Schröder’s Law of Reciprocal Equivalences | 534 |

540 to 550. | Exercises | 535 |

INDEX | 539 | |

REFERENCE LIST OF INITIAL LETTERS SHEWING THE AUTHORSHIP OR SOURCE OF QUESTIONS AND PROBLEMS.

- B = Professor J. I. Beare, Trinity College, Dublin;
- C = University of Cambridge;
- J = Mr W. E. Johnson, King’s College, Cambridge;
- K = Dr J. N. Keynes, Pembroke College, Cambridge;
- L = University of London;
- M = University of Melbourne;
- N = Professor J. S. Nicholson, University of Edinburgh;
- O = University of Oxford;
- O’S = Mr C. A. O’Sullivan, Trinity College, Dublin;
- R = the late Professor G. Croom Robertson;
- RR = Mr R. A. P. Rogers, Trinity College, Dublin;
- T = Dr F. A. Tarleton, Trinity College, Dublin;
- V = Dr J. Venn, Gonville and Caius College, Cambridge;
- W = Professor J. Ward, Trinity College, Cambridge.

*Note.* A few problems have been selected from the published
writings of Boole, De Morgan, Jevons, Solly, Venn, and Whately, from
the Port Royal Logic, and from the Johns Hopkins Studies in Logic. In
these cases the source of the problem is appended in full.

STUDIES AND EXERCISES IN FORMAL LOGIC.

1. *The General Character of
Logic*.—Logic may be defined as the science which
investigates the general principles of valid thought. Its object is to
discuss the characteristics of judgments, regarded not as
psychological phenomena but as expressing our knowledge and beliefs;
and, in particular, it seeks to determine the conditions under which
we are justified in passing from given judgments to other judgments
that follow from them.

As thus defined, logic has in view an ideal; it is concerned fundamentally with how we ought to think, and only indirectly and as a means to an end with how we actually think. It may accordingly be described as a normative or regulative science. This character it possesses in common with ethics and aesthetics. These three branches of knowledge—all of them based on psychology—form a unique trio, to be distinguished from positive sciences on the one hand, and from practical arts on the other. It may be said roughly that they are concerned with the ideal in the domains of thought, action, and feeling respectively. Logic seeks to determine the general principles of valid thought, ethics the general principles of right conduct, aesthetics the general principles of correct taste.

2. *Formal
Logic*.—As regards the scope of logic, one of the principal
questions ordinarily raised is whether the science is *formal* or
*material*, subjective or objective, concerned with 2 thoughts or with things. It
is usual to say that logic is *formal*, in so far as it is
concerned merely with the form of thought, that is, with our manner of
thinking irrespective of the particular objects about which we are
thinking; and that it is *material*, in so far as it regards as
fundamental the objective reference of our thought, and recognises as
of essential importance the differences existing in the objects
themselves about which we think.

Logic is certainly formal, or at any rate non-material, in the sense that it cannot guarantee the actual objective or material truth of any particular conclusions. Moreover any valid reasoning whatsoever must conform to some definite type, or—in other words—must be reducible to some determinate form; and one of the main objects of logic is by abstraction to discover what are the various types or forms to which all valid reasoning may be reduced.

But, on the other hand, it is essential that logic should recognise an objective reference in every judgment, that is, a reference outside the state of mind which constitutes the judgment itself: apart from this, as we shall endeavour to shew in more detail later on, the true nature of judgment cannot be understood. It is, moreover, possible for logic to examine and formulate certain general conditions which must be satisfied if our thoughts and judgments are to have objective validity; and the science may recognise and discuss certain general presuppositions relating to external nature which are involved in passing from the particular facts of observation to general laws.

Logic fully treated has then both a formal and a material side. The question may indeed be raised whether the distinction between form and matter is not a relative, rather than an absolute, distinction. All sciences are in a sense formal, since they abstract to some extent from the matter of thought. Thus physics abstracts in the main from the chemical properties of bodies, while geometry abstracts also from their physical properties, considering their figure only. In this way we become more and more formal as we become more and more general; and logic may be said to be more abstract, more 3 general, more formal, than any other science, except perhaps pure mathematics.

It is to be added that, within
the domain of logic itself, the answer to the question whether two
given propositions have or have not the same form may depend upon the
particular system of propositions in connexion with which they are
considered. Thus, if we carry our analysis no further than is usual in
ordinary formal logic, the two propositions, *Every angle in a
semi-circle is equal to a right angle*, *Any two sides of a
triangle are together greater than the third side*, may be
considered to be identical in form. Each is universal, and each is
affirmative; they differ only in matter. But it will be found that in
the logic of relatives, to which further reference will subsequently
be made, the two propositions (one expressing an equality and the
other an inequality) may be regarded as differing in form as well as
in matter; and, moreover, that the difference between them in form is
capable of being symbolically expressed.

The difficulty of
assigning a distinctive scope to formal logic *par excellence* is
increased by the fact that certain problems falling naturally into the
domain of material logic—for example, the inductive
methods—admit up to a certain point of a purely formal
treatment.

It is not possible then to draw a hard and fast line and to say that a certain determinate portion of logic is formal, and that the rest is not formal. We must content ourselves with the statement that when we speak of formal logic in a distinctive sense we mean the most abstract parts of the science, in which no presuppositions are made relating to external nature, and in which—beyond the recognition of the necessary objective reference contained in all judgments—there is an abstraction from the matter of thought. Because they are so abstract, the problems of formal logic as thus conceived admit usually of symbolic treatment; and it is with problems admitting of such treatment that we shall more particularly concern ourselves in the following pages.

3. *Logic and
Language*.—Some logicians, in their treatment of the problems
of formal logic, endeavour to abstract not 4 merely from the matter of thought but also
from the language which is the instrument of thought. This method of
treatment is not adopted in the following pages. In order to justify
the adoption of the alternative method, it is not necessary to
maintain that thought is altogether impossible without language. It is
enough that all thought-processes of any degree of complexity are as a
matter of fact carried on by the aid of language, and that
thought-products are normally expressed in language. That language is
in this sense the universal instrument of thought will not be denied;
and it seems a fair corollary that the principles by which valid
thought is regulated, and more especially the application of these
principles to the criticism of thought-products, cannot be adequately
discussed, unless account is taken of the way in which this instrument
actually performs its functions.

Language is full of
ambiguities, and it is impossible to proceed far with the problems
with which logic is concerned until a precise interpretation has been
placed upon certain forms of words as representing thought. In
ordinary discourse, to take a simple example, the word *some* may
or may not be used in a sense in which it is exclusive of *all* ;
it may be understood to mean *not-all* as well as
*not-none*, or its full meaning may be taken to be
*not-none*. The logician must decide in which of these senses the
word is to be understood in any given scheme of propositional forms.
Now, if thought were considered exclusively in itself, such a question
as this could not arise; it has to do with the expression of thought
in language. The fact that such questions do arise and cannot help
arising shews that actually to eliminate all consideration of language
from logic is an impossibility. A not infrequent result of attempting
to rise above mere considerations of language is needless prolixity
and dogmatism in regard to what are really verbal questions, though
they are not recognised as such.

The method of treating logic
here advocated is sometimes called *nominalist*, and the opposed
method *conceptualist*. A word or two of explanation is, however,
desirable in order that this use of terms may not prove misleading.
Nominalism and conceptualism usually denote certain doctrines
concerning the 5 nature
of general notions. Nominalism is understood to involve the assertion
that generality belongs to language alone and that there is nothing
general in thought. But a so-called nominalist treatment of logic does
not involve this. It involves no more than a clear recognition of the
importance of language as the instrument of thought; and this is a
circumstance upon which modern advocates of conceptualism have
themselves insisted.

It is perhaps necessary to add that on the view here taken logic in no way becomes a mere branch of grammar, nor does it cease to have a place amongst the mental sciences. Whatever may be the aid derived from language, it remains true that the validity of formal reasonings depends ultimately on laws of thought. Formal logic is, therefore, still concerned primarily with thought, and only secondarily with language as the instrument of thought.

In our subsequent discussion of the relation of *terms* to
*concepts*, and of *propositions* to *judgments*, we
shall return to a consideration of the question raised in this
section.2

4. *Logic and
Psychology*.—Since processes of reasoning are mental
processes depending upon the constitution of our minds, they fall
within the cognizance of psychology as well as of logic. But laws of
reasoning are regarded from different points of view by these two
sciences. Psychology deals with such laws in the sense of
uniformities, that is, as laws in accordance with which men are found
by experience normally to think and reason. Logic, on the other hand,
deals with laws of reasoning as regulative and authoritative, as
affording criteria by the aid of which valid and invalid reasonings
may be discriminated, and as determining the formal relations in which
different products of thought stand to one another.

Looking at the relations between logic and psychology from a slightly different standpoint, we observe that while the latter is concerned with the actual, the former is concerned with the ideal. Logic does not, like psychology, treat of all the ways in which men actually reach conclusions, or of all the various modes in which, through the association of ideas or otherwise, one belief actually generates another. It is concerned with 6 reasonings only as regards their cogency, and with the dependence of one judgment upon another only in so far as it is a dependence in respect of proof.

There are various other ways in which the contrast between the two sciences may be expressed. We may, for example, say that psychology is concerned with thought-processes, logic with thought-products; or that psychology is concerned with the origin of our beliefs, logic with their validity.

Logic has thus a unique character of its own, and is not a mere branch of psychology. Psychological and logical discussions are no doubt apt to overlap one another at certain points, in connexion, for example, with theories of conception and judgment. In the following pages, however, the psychological side of logic is comparatively little touched upon. The metaphysical questions also to which logic tends to give rise are as far as possible avoided.

5. *The Utility of Logic*.—We have seen
that logic has in view an ideal and treats of what ought to be. Its
object is, however, to investigate general principles, and it puts
forward no claim to be a practical art. Its utility is accordingly not
to be measured by any direct help that it may afford towards the
attainment of particular scientific truths. No doubt the procedure in
all sciences is subject to the general principles formulated by logic;
but, in details, the weighing of evidence will often be better
performed by the judgment of the expert than by any formal or
systematic observance of logical rules.

It is important to bear in mind that, in the study of logic, our immediate aim is the scientific investigation of general principles recognised as authoritative in relation to thought-products, not the formulation of a system of rules and precepts. It may be said that the art of dealing with particular concrete arguments, with the object of determining their validity, is related to the science of logic in the same way as the art of casuistry (that is, the art of deciding what it is right to do in particular concrete circumstances) is related to the science of ethics. Moreover, just as in the art of casuistry we meet with problems which are elusive and difficult to decide because in the concrete they cannot be brought exactly under the abstract 7 formulae of ethical science, so in the art of detecting fallacies we meet with arguments which cannot easily be brought under the abstract formulae of logical science. As it would be a mistake to subordinate ethics to the treatment of casuistical questions, so it would be a mistake to mould the science of logic with constant reference to concrete arguments which, either because of the ambiguity of the terms employed, or because of the uncertain bearing of the context in which they occur, elude any attempt to reduce them to a form to which general principles are directly applicable.

Wherein then consists the utility of logic? In answer to this question, it may be observed primarily that if logic determines truly the principles of valid thought, then its study is of value simply in that it adds to our knowledge. To justify the study of logic it is, as Mansel has observed, sufficient to shew that what it teaches is true, and that by its aid we advance in the knowledge of ourselves and of our capacities.

To this it must be added (in qualification of what has been said previously) that, while logic is not to be regarded as an art of attaining truth, it still does possess utility as propaedeutic to other studies and independently of the addition that it makes to our knowledge. Fallacious arguments can no doubt usually be recognised as such by an acute intellect apart from any logical study; and, as we have seen, it is not the primary function of logic to deal with particular concrete arguments. At the same time, it is only by the aid of logic that we can analyse a reasoning, explain precisely why a fallacious argument is faulty, and give the fallacy a name. In other words, while logic is not to be identified with the criticism of particular concrete arguments, such criticism when systematically undertaken must be based on logic.

Greater, however, than the indirect value of logic in its subsequent application to the examination of particular reasonings is its value as a general intellectual discipline. The study of logic cultivates the power of abstract thought; and it is not too much to say that, when undertaken with thoroughness, it affords a unique mental training.

6. *The Three Parts of
Logical Doctrine*—It has been usual to divide logical
doctrine into three parts, dealing with terms (or concepts),
propositions (or judgments), and reasonings respectively; and it will
be convenient to adopt this arrangement in the present treatise. At
the same time, we may in passing touch upon certain objections that
have been raised to this mode of treating the subject.

Mr Bosanquet treats of logic in two parts, not in three, giving no
separate discussion of names (or concepts). His main ground for taking
up this position is that “the name or concept has no reality in
living language or living thought, except when referred to its place
in a proposition or judgment” (*Essentials of Logic*, p.
87). He urges that “we ought not to think of propositions as
built up by putting words or names together, but of words or names as
distinguished though not separable elements in propositions.”
There is undoubted force in this argument, and attention should be
called to the points raised by Mr Bosanquet, even though we may not be
led to quite the same conclusion.

Logic is essentially concerned with truth and falsity as characteristics of thought, and truth and falsity are embodied in judgments and in judgments only. Hence the judgment 9 (or the proposition as expressing the judgment) may be regarded as fundamentally the logical unit. It would, moreover, now be generally agreed that the concept is not by itself a complete mental state, but is realised only as occurring in a context. Correspondingly the name does not by itself express any mental state. If a mere name is pronounced it leaves us in a state of expectancy, except in so far as it is the abbreviated expression of a proposition, as it may be when spoken in answer to a question or when the special circumstances or manner of its utterance connect it with a context that gives it predicative force.

At the same time, in ideal analysis the developed judgment yields the concept as at any rate a distinguishable element of which it is composed, while the proposition similarly yields the term; and in order that the import of judgments and propositions may be properly understood some discussion of concepts and terms is necessary.

The question as to the proper order of treatment remains. In
dealing with this question we need not trouble ourselves with the
enquiry as to whether the concept or the judgment has psychological
priority, that is to say, as to whether in the first instance the
process of forming judgments requires that concepts should have been
already formed, or whether on the other hand the process of forming
conceptions itself involves the formation of judgments, or whether the
two processes go on *pari passu*. It is enough that the developed
judgment and the proposition, as we are concerned with them in logic,
yield respectively the concept, and the term as elements out of which
they are constituted.

We shall then give a separate discussion of terms, and shall enter upon this part of the subject before discussing propositions. But in doing this we shall endeavour constantly to bear in mind that the proposition is the true logical unit, and that the logical import of terms cannot be properly understood except with reference to their employment in propositions.3

^{3} In this connexion attention may
be called to Mill’s well known dictum that “names
are names of things, not of our ideas,” Apart from its context,
the force of this antithesis may easily be misunderstood. It is clear
that every name that is employed in an intelligible sense must have
some mental equivalent, must call up some idea or other to our minds,
and must therefore in this sense be the name of an idea. It is not,
however, Mill’s intention to deny this. Nor, on the other hand,
does he intend to assert that things actually exist corresponding to
all the names we employ. His dictum really has reference to
*predication*. What he means is that when any name appears as the
subject of a proposition, an assertion is made not about the
corresponding idea, but about something which is distinct both from
the name and the idea, though both are related to it. He is in fact
affirming the objective reference that is essential to the conception
of truth or falsity. The discussion may, therefore, be said to be
properly part of the discussion of the import of propositions rather
than of names, and it would certainly be less puzzling if it were
introduced in that connexion. Our special object, however, in
referring to the matter here is not to criticise Mill, but to
illustrate the difficulty of discussing names logically apart from the
use that may be made of them for purposes of predication.

10 7. *Names and Concepts*.—We
have in the preceding section spoken more or less indiscriminately of
*names* (or terms) and of *concepts*, and this has been
intentional. We have already expressed our disagreement with those who
would exclude from logic all consideration of language. Our judgments
cannot have certainty and universal validity unless the ideas which
enter into them are fixed and determined; and, apart from the aid that
we derive from language, our ideas cannot be thus fixed and
determined.

It is, therefore, a mistake to treat of concepts to the exclusion of names. But, on the other hand, we must not forget that the logician is concerned with names only as representive of ideas. His real aim is to treat of ideas, though he may think it wiser to do so not directly, but indirectly by considering the names by which ideas are represented. For this reason it is well, now and then at any rate, to refer explicitly to the concept.

The so-called conceptualist school of logicians are apt in their treatment of the first part of logical doctrine to discuss problems of a markedly psychological character, as, for example, the mode of formation of concepts and the controversy between conceptualism and nominalism. Apart, however, from the fact that the conceptualist logicians do not draw so clear a line of distinction as do the nominalists between logic and psychology, the difference between the two schools is to a large extent 11 a mere difference of phraseology. Practically the same points, for example, are raised whether we discuss the extension and intension of concepts or the denotation and connotation of names. At the same time, it must be said that the attempt to deal with the intension of concepts to the entire exclusion of any consideration of the connotation of names appears to be responsible for a good deal of confusion.

8. *The Logic of
Terms*.—Attention has already been called to the relation of
dependence that exists between the logic of terms and the logic of
propositions. It will be found that we cannot in general fully
determine the logical characteristics of a given name without explicit
reference to its employment as a constituent of a proposition. We
cannot again properly discuss or understand the import of so-called
negative names without reference to negative judgments.

It must be added that in dealing with distinctions between names,
it is particularly difficult for the logician who follows at all on
the traditional lines to avoid discussing problems that belong more
appropriately to psychology, metaphysics, or grammar; and to some of
the questions which arise it may hardly be possible to give a
completely satisfactory answer from the purely logical point of view.
This remark applies especially to the distinction between
*abstract* and *concrete* terms, a distinction, moreover,
which is of little further logical utility or significance. It is
introduced in the following pages in accordance with custom; but
adequately to discriminate between things and their attributes is the
function of metaphysics rather than of logic. The portion of the logic
of terms (or concepts) to which by far the greatest importance
attaches is that which is concerned with the distinction between
extension and intension.

9. *General and Singular
Names*.—A *general* name is a name which is actually or
potentially predicable in the same sense of each of an indefinite
number of units; a *singular* or *individual* name is a name
which is understood in the particular circumstances in which it is
employed to denote some one determinate unit only.

The nature and logical importance of this distinction may 12 be illustrated by
considering names as the subjects of propositions. A general name is
the name of a divisible class, and predication is possible in respect
of the whole or a part of the class; a singular name is the name of a
unit indivisible. Hence we may take as the test or criterion of a
general name, the possibility of prefixing *all* or *some*
to it with any meaning.

Thus, *prime minister of England* is a general name, since it
is applicable to more than one individual, and statements may be made
which are true of all prime ministers of England or only of some. The
name *God* is singular to a monotheist as the name of the Deity,
general to a polytheist, or as the name of any object of worship.
*Universe* is general in so far as we distinguish different kinds
of universes, *e.g.*, the material universe, the terrestrial
universe, &c.; it is singular if we mean the totality of all
things. *Space* is general if we mean any portion of space,
singular if we mean space as a whole. *Water* is general.
Professor Bain takes a different view here; he says, “Names of
material—earth, atone, salt, mercury, water, flame—are
singular. They each denote the entire collection of one species of
material” (*Logic*, *Deduction*, pp. 48, 49). But when
we predicate anything of these terms it is generally of *any portion*
(or of some particular portion) of the material in question, and not
of the entire collection of it *considered as one aggregate* ;
thus, if we say, “Water is composed of oxygen and
hydrogen,” we mean any and every particle of water, and the name
has all the distinctive characters of the general name. Again, we can
distinguish *this* water from *that* water, and we can say,
“*some* water is not fit to drink”; but the word
*some* cannot, as we have seen above, be attached to a really
singular name. Similarly with regard to the other terms in question.
It is also to be observed that we distinguish between different kinds
of stone, salt, &c.4

^{4} Terms of the kind here under
discussion are called by Jevons *substantial terms*. (See
*Principles of Science*, 2, § 4.) Their peculiarity is that,
although they are concrete, the things denoted by them possess a
peculiar homogeneity or uniformity of structure; also we do not as a
rule use the indefinite article with them as we do with other general
names.

A name is to be regarded as general if it is *potentially*
13 predicable of more
than one object, although as a matter of fact it happens that it can
be truly affirmed of only one, *e.g.*, *an English sovereign
six times married*. A really singular name is not even potentially
applicable to more than one individual; *e.g.*, *the last of
the Mohicans*, *the eldest son of King Edward the First*. This
may be differently expressed by saying that a really singular name
implies in its signification the uniqueness of the corresponding
object. We may take as examples *the summum bonum*, *the centre
of gravity of the material universe*. It is not easy to find such
names except in cases where uniqueness results from some explicit or
implicit limitation in time or space or from some relation to an
object denoted by a proper name. Even in such a case as *the centre
of gravity of the material universe* some limitation in time
appears to be necessary, for the centre of gravity of the universe may
be differently situated at different periods.

Any general name may be transformed into a singular name by means
of an individualising prefix, such as a demonstrative pronoun
(*e.g.*, *this book*), or by the use of the definite
article, which usually indicates a restriction to some one determinate
person or thing (*e.g.*, *the Queen*, *the pole star*).
Such restriction by means of the definite article may sometimes need
to be interpreted by the context, *e.g.*, *the garden*,
*the river* ; in other cases some limitation of place or time or
circumstance is introduced which unequivocally defines the individual
reference, *e.g.*, *the first man*, *the present Lord
Chancellor*, *the author of Paradise Lost*.

On the other hand, propositions with singular names as subjects may sometimes admit of subdivision into universal and particular. This is the case when, with reference to different times or different conditions, a distinction is made or implied in regard to the manner of existence, actual or potential, of the object denoted by the name: for example, “Homer sometimes nods,” “The present Pope always dwells in the Vatican,” “This country is sometimes subject to earthquakes.”5

10. *Proper
Names*.—A *proper* name is a name assigned as a mark to
distinguish an individual person or thing from others, 14 without implying in its
signification the possession by the individual in question of any
specific attributes. Such names are given to objects which possess
interest in respect of their individuality and independently of their
specific nature. For the most part they are confined to persons and
places; but they are also given to domestic animals, and sometimes to
inanimate objects to which affection-value is attached, as, for
example, by children to their dolls. Proper names form a sub-class of
singular names, being distinguished from the singular names of which
examples were given in the preceding section in that they denote
individual objects without at the same time necessarily conveying any
information as to particular properties belonging to those objects.6

Many proper names, *e.g.*, *John*, *Victoria*, are
as a matter of fact assigned to more than one individual; but they are
not therefore general names, since on each particular occasion of
their use, with the exception noted below, there is an understood
reference to some one determinate individual only. There is, moreover,
no implication that different individuals who may happen to be called
by the same proper name have this name assigned to them on account of
properties which they possess in common.7 The exception above
referred to occurs when we speak of the class composed of those who
bear the name, and who are constituted a distinct class by this common
feature alone: *e.g.*, “All Victorias are honoured in their
name,” “Some Johns are not of Anglo-Saxon origin, but are
negroes.” The subjects of such propositions as these must,
however, be regarded as elliptical; written out more fully, they
become *all persons called Victoria*, *some individuals named
John*.

^{7} Professor
Bain brings out this distinction in his definition of a general name:
“A general name is applicable to a number of things in virtue of
their being similar, or having something in common.”

11. *Collective
Names*.—A *collective* name is one which is applied to a
group of similar things regarded as constituting a single whole;
*e.g.*, *regiment*, *nation*, *army*. A
*non-collective* name, *e.g.*, *stone*, may also be the
name of something which is 15 composed of a number of precisely similar
parts, but this is not in the same way present to the mind in the use
of the name.8

^{8} To *collective* name as
above defined there is no distinctive antithetical term in ordinary
use. The antithesis between the *collective* and the
*distributive* use of names arises, as we shall see, in connexion
with predication only.

A collective name may be singular or general. It is the name of a
group or collection of things, and so far as it is capable of being
correctly affirmed in the same sense of only one such group, it is
singular; *e.g.*, *the* 29*th regiment of foot*, *the
English nation*, *the Bodleian Library*, But if it is capable
of being correctly affirmed in the same sense of each of several such
groups it is to be regarded as general; *e.g.*, *regiment*,
*nation*, *library*.9

^{9} It is pointed out by Dr Venn
that certain proper names may be regarded as collective, though such
names are not common. “One instance of them is exhibited in the
case of geographical groups. For instance, the Seychelles, and the
Pyrenees, are distinctly, in their present usage, proper names,
denoting respectively two groups of things. They simply denote these
groups, and give us no information whatever about any of their
characteristics” (*Empirical Logic*, p. 172).

Some logicians imply an antithesis between collective and general names, either regarding collectives as a sub-class of singulars, or else recognising a threefold division into singular, collective, and general. There is, properly speaking, no such antithesis; and both the above alternatives must be regarded as misleading, if not actually erroneous; for, as we have just seen, the class of collective names overlaps each of the other classes.

The correct and really important logical antithesis is between the
*collective* and the *distributive* use of names. A
collective name such as *nation*, or any name in the plural
number, is the name of a collection or group of similar things. These
we may regard as one whole, and something may be predicated of them
that is true of them only as a whole; in this case the name is used
*collectively*. On the other hand, the group may be regarded as a
series of units, and something may be predicated of these which is
true of them taken individually; in this case the name is used
*distributively*.10

^{10} It is held by Dr Venn
(*Empirical Logic*, p. 170) that *substantial terms* are
always used collectively when they appear as subjects of general
propositions. If, however, we take such a proposition as “Oil is
lighter than water” it seems clear that the subject is used not
collectively, but distributively; for the assertion is made of each
and every portion of oil, whereas if we used the term collectively our
assertion would apply only to all the portions taken together. The
same is clearly true in other instances; for example, in the
propositions, “Water is composed of oxygen and hydrogen,”
“Ice melts when the temperature rises above 32°
Fahr.”

16 The above distinction may be illustrated by the propositions, “All the angles of a triangle are equal to two right angles,” “All the angles of a triangle are less than two right angles.” In the first case the predication is true only of the angles all taken together, while in the second it is true only of each of them taken separately; in the first case, therefore, the term is used collectively, in the second distributively. Compare again the propositions, “The people filled the church,” “The people all fell on their knees.”11

^{11} When in an argument we pass
from the collective to the distributive use of a term, or *vice
versâ*, we have what is technically called a *fallacy of
division* or *of composition* as the case may be. The
following are examples: The people who attended Great St Mary’s
contributed more than those who attended Little St Mary’s,
therefore, *A* (who attended the former) gave more than *B*
(who attended the latter); All the angles of a triangle are less than
two right angles, therefore *A*, *B*, and *C*, which
are all the angles of a triangle, are together less than two right
angles. The point of the old riddle, “Why do white sheep eat
more than black?” consists in the unexpected use of terms
collectively instead of distributively.

12. *Concrete and Abstract
Names*.—The distinction between concrete and abstract names,
as ordinarily recognised, may be most briefly expressed by saying that
a *concrete* name is the name of a *thing*, whilst an
*abstract* name is the name of an *attribute*. The question,
however, at once arises as to what is meant by a *thing* as
distinguished from an *attribute* ; and the only answer to be
given is that by a thing we mean whatever is regarded as possessing
attributes. It would appear, therefore, that our definitions may be
made more explicit by saying that a *concrete* name is the name
of anything which is regarded as possessing attributes, *i.e.*,
as a *subject of attributes* ; while an *abstract* name is
the name of anything which is regarded as an attribute of something
else, *i.e.*, as an *attribute of subjects*.12

^{12} The distinction is sometimes
expressed by saying that an abstract name is the name of an attribute,
a concrete name the name of a *substance*. If by *substance*
is merely meant whatever possesses attributes, then this distinction
is equivalent to that given in the text; but if, as would ordinarily
be the case, a fuller meaning is given to the term, then the division
of names into abstract and concrete is no longer an exhaustive one.
Take such names as *astronomy*, *proposition*,
*triangle*: these names certainly do not denote attributes; but,
on the other hand, it seems paradoxical to regard them as names of
substances. On the whole, therefore, it is best to avoid the term
*substance* in this connexion.

17 This distinction is in most cases easy of application; for example,
*plane triangle* is the name of all figures that possess the
attribute of being bounded by three straight lines, and is a concrete
name; *triangularity* is the name of this distinctive attribute
of triangles, and is an abstract name. Similarly, *man*,
*living being*, *generous* are concretes; *humanity*,
*life*, *generosity* are the corresponding abstracts.13

^{13} It will be observed that,
according to the above definitions, a name is not called abstract,
simply because the corresponding idea is the result of abstraction,
*i.e.*, attending to some qualities of a thing or class of things
to the exclusion as far as possible of others. In this sense all
general names, such as *man*, *living being*, &c., would
be abstract.

Abstract and concrete names usually go in pairs as in the above
illustrations. A concrete general name is the name of a class of
things grouped together in virtue of some quality or set of qualities
which they possess in common; the name given to the quality or
qualities themselves apart from the individuals to which they belong
is the corresponding abstract.14 Using the terms
*connote* and *denote* in their technical senses, as defined
in the following chapter, an abstract name *denotes* the
qualities which are *connoted* by the corresponding concrete
name. This relation between concretes and the corresponding abstracts
is the one point in connexion with abstract and concrete names that is
of real logical importance, and it may be observed that it does not in
itself give rise to the somewhat fruitless subtleties with which the
distinction is apt to be 18 associated. For when two names are given
which are thus related, there will never be any difficulty in
determining which is concrete and which is abstract in relation to the
other.

^{14} Thus, in
the case of every general concrete name there is or may be constructed
a corresponding abstract. But this is not true of proper names or
other singular names regarded strictly as such. We may indeed have
such abstracts as *Caesarism* and *Bismarckism*. These
names, however, do not denote all the differentiating attributes of
Caesar and Bismarck respectively, but only certain qualities supposed
to be specially characteristic of these individuals. In forming the
above abstracts we generalise, and contemplate a certain type of
character and conduct that may possibly be common to a whole class.
Compare page 45.

But whilst the distinction is absolute and unmistakeable when names are thus given in pairs, the application of our definitions is by no means always easy when we consider names in themselves and not in this definite relation to other names. We shall find indeed that if we adopt the definitions given above, then the division of names into abstract and concrete is not an exclusive one in the sense that every name can once and for all be assigned exclusively to one or other of the two categories.

We are at any rate driven to this if we once admit that attributes
may themselves be the subjects of attributes, and it is difficult to
see how this admission can be avoided. If, for example, we say that
“unpunctuality is irritating,” we ascribe the attribute of
being irritating to unpunctuality, which is itself an attribute.
*Unpunctuality*, therefore, although primarily an abstract name,
can also be used in such a way that it is, according to our
definition, concrete.

Similarly when we consider that an attribute may appear in different forms or in different degrees, we must regard it as something which can itself be modified by the addition of a further attribute; as, for example, when we distinguish physical courage from moral courage, or the whiteness of snow from the whiteness of smoke, or when we observe that the beauty of a diamond differs in its characteristics from the beauty of a landscape.

Hence, if the definitions under discussion are adopted, we arrive at the conclusion that while some names are concrete and never anything but concrete, names which are primarily formed as abstracts and continue to be used as such are apt also to be used as concretes, that is to say, they are names of attributes which can themselves be regarded as possessing attributes. They are abstract names when viewed in one relation, concrete when viewed in another.15

^{15} The use of the same term as
both abstract and concrete in the manner above described must be
distinguished from the not unfrequent case of quite another kind in
which a name originally abstract changes its meaning and comes to be
used in the sense of the corresponding concrete; as, for example, when
we talk of *the Deity* meaning thereby God, not the qualities of
God. Compare Jevons, *Elementary Lessons in Logic*, pp. 21,
22.

19 It must be admitted that this result is paradoxical. As yielding a division of names that is non-exclusive, it is also unscientific. There are two ways of avoiding this difficulty.

In the first place, we may further modify our definitions and say
that an *abstract* name is the name of anything which *can* be
regarded as an attribute of something else (whether it is or is not
itself a subject of attributes), while a *concrete* name is the
name of that which *cannot* be regarded as an attribute of something
else. This distinction is simple and easy of application, it is in
accordance with popular usage, and it satisfies the condition that the
members of a division shall be mutually exclusive. But it may be
doubted whether it has any logical value.

A second way of avoiding the difficulty is to give up for logical purposes the distinction between concrete and abstract names, and to substitute for it a distinction between the concrete and the abstract use of names. A name is then used in a concrete sense when the thing called by the name is contemplated as a subject of attributes, and in an abstract sense when the thing called by the name is contemplated as an attribute of subjects. It follows from what has been already said that some names can be used as concrete only, while others can be used either as abstract or as concrete. This solution is satisfactory from the logical point of view, since logic is concerned not with names as such, but with the use of names in propositions. It may be added that as logicians we have very little to do with the abstract use of names, A consideration of the import of propositions will shew that when a name appears either as the subject or as the predicate of a non-verbal proposition its use is always concrete.

13. *Can Abstract Names be
subdivided into General and Singular?*—The question whether
any abstract names can be considered general has given rise to much
difference of opinion amongst logicians. On the one hand, it is argued
that all 20 abstract
names must necessarily be singular, since an attribute considered
purely as such and apart from its concrete manifestations is one and
indivisible, and cannot admit of numerical distinction.16 On
the other hand, it is urged that some abstracts must certainly be
considered general since they are names of attributes of which there
are various kinds or subdivisions; and in confirmation of this view it
is pointed out that we frequently write abstracts in the plural
number, as when we say, “Redness and yellowness are
*colours*,” “Patience and meekness are
*virtues*.”17

^{16} This represents the view
taken by Jevons. See *Principles of Science*, 2, § 3.

^{17} Compare
Mill, *Logic*, i. 2, § 4.

The solution of the question really depends upon our use of the
term *abstract*.

If we adopt the definition given in the last paragraph but one of
the preceding section, and include under abstract names the names of
attributes which are themselves the subjects of attributes, these
latter attributes possibly varying in different instances, then there
can be no doubt that some abstracts are general; for they are the
names of a class of things which, while having something in common,
are also distinguishable *inter se*.

So far, however, as the question is raised in regard to the abstract (as distinguished from the concrete) use of names in the manner indicated in the last paragraph of the preceding section, we are led to the conclusion that it is only when names are used in a concrete sense that they can be considered general. For it is clear that the name of an attribute can be described as general only in so far as the attribute is regarded as exhibiting characteristics which vary in different instances, only in so far, that is to say, as it is itself a subject of attributes; and when the attribute is so regarded, the name is used in a concrete, not an abstract, sense.

Take the propositions, “Some colours are painfully
vivid,” “All yellows are agreeable,” “Some courage is the
result of ignorance,” “Some cruelty is the result of
fear,” “All cruelty is detestable.” The subjects of
these propositions are certainly 21 general. According to the definition given
in the last paragraph but one of the preceding section they are also
abstract. If, however, in place of distinguishing between abstract and
concrete names *per se*, we distinguish between the abstract and
the concrete use of names as proposed in the last paragraph of the
preceding section, then the terms in question are all used in a
concrete, not an abstract, sense.

14. Discuss Mill’s
statement that “names are names of things, not of our
ideas,” with special reference to the following names:
*dodo*, *mermaid*, *chimaera*, *toothache*,
*jealousy*, *idea*. [C.]

15. Discuss the logical characteristics of adjectives. [K.]

16. *The Extension and the
Intension of Names*.18—Every concrete general name is the name
of a real or imaginary class of objects which possess in common
certain attributes; and there are, therefore, two aspects under which
it may be regarded. We may consider the name (i) in relation to the
objects which are called by it; or (ii) in relation to the qualities
belonging to those objects. It is desirable to have terms by which to
refer to this broad distinction without regard to further refinements
of meaning; and the terms *extension* and *intension* will
accordingly be employed to express in the most general way these two
aspects of names respectively.19

^{18} We may speak also of the
extension and the intension of concepts. In the discussion, however,
of questions concerning extension and intension, it is essential to
recognise the part played by language as the instrument of thought.
Hence it seems better to start from names rather than from concepts.
Neglect to consider names explicitly in this connexion has been
responsible for much confusion.

^{19} It is usual to employ the
terms *comprehension* and *connotation* as simply synonymous
with *intension*, and *denotation* as synonymous with
*extension*. We shall, however, presently find it convenient to
differentiate the meanings of these terms. The force of the terms
*extension* and *intension* in the most general sense might
perhaps also be expressed by the pair of terms *application* and
*implication*.

The *extension* of a name then
consists of objects of which the name can be predicated; its
*intension* consists of properties which can be predicated of it.
For example, by the extension of *plane triangle* we mean a
certain class of geometrical figures, and by its intension certain
qualities belonging to such figures. 23 Similarly, by the extension of *man* is
meant a certain class of material objects, and by its intension the
qualities of rationality, animality, &c., belonging to these
objects.

17. *Connotation,
Subjective Intension, and Comprehension*.—The term
*intension* has been used in the preceding section to express in
the most general way that aspect of general names under which we
consider not the objects called by the names but the qualities
belonging to those objects. Taking any general name, however, there
are at least three different points of view from which the qualities
of the corresponding class may be regarded; and it is to a want of
discrimination between these points of view that we may attribute many
of the controversies and misunderstandings to which the problem of the
connotation of names has given rise.

(1) There are those qualities which are essential to the class in
the sense that the name implies them in its definition. Were any of
this set of qualities absent the name would not be applicable; and any
individual thing lacking them would accordingly not be regarded as a
member of the class. The standpoint here taken may be said to be
*conventional*, since we are concerned with the set of
characteristics which are supposed to have been conventionally agreed
upon as determining the application of the name.

(2) There are those qualities which in the mind of any given
individual are associated with the name in such a way that they are
normally called up in idea when the name is used. These qualities will
include the marks by which the individual in question usually
recognises or identifies an object as belonging to the class. They may
not exhaust the essential qualities of the class in the sense
indicated in the preceding paragraph, but on the other hand they will
probably include some that are not essential to it. The standpoint
here taken is *subjective* and relative. Even when there is
agreement as to the actual meaning of a name, the qualities that we
naturally think of in connexion with it may vary both from individual
to individual, and, in the case of any given individual, from time to
time.

We may consider as a special case under this head the 24 complete group of
attributes *known* at any given time to belong to the class. All
these attributes can be called up in idea by any person whose
knowledge of the class is fully up to date; and this group may,
therefore, be regarded as constituting the most scientific form of
intension from the subjective point of view.

(3) There is the sum-total of qualities actually possessed in
common by all members of the class. These will include all the
qualities included under the two preceding heads,20 and usually many
others in addition. The standpoint here taken is *objective*.21

^{20} It is here assumed, as
regards the qualities mentally associated with the name, that our
knowledge of the class, so far as it extends, is
correct.

^{21} When the
objective standpoint is taken, there is an implied reference to some
particular universe of discourse, within which the class denoted by
the name is supposed to be included. The force of this remark will be
made clearer at a subsequent stage.

In seeking to give a
precise meaning to *connotation*, we may start from the above
classification. It suggests three distinct senses in which the term
might possibly be used, and as a matter of fact all three of these
senses have been selected by different logicians, sometimes without
any clear recognition of divergence from the usage of other writers.
It is desirable that we should be quite clear in our own minds in
which sense we intend to employ the term.

(i) According to Mill’s usage, which is that adopted in the
following pages, the conventional standpoint is taken when we speak of
the *connotation* of a name. On this view, we do not mean by the
connotation of a class-name all the qualities possessed in common by
the class; nor do we necessarily mean those particular qualities which
may be mentally associated with the name; but we mean just those
qualities on account of the possession of which any individual is
placed in the class and called by the name. In other words, we include
in the connotation of a class-name only those attributes upon which
the classification is based, and in the absence of any of which the
name would not be regarded as applicable. For example, although all
equilateral triangles are equiangular, equiangularity is not on this
view included in the connotation of equilateral 25 triangle, since it is not
a property upon which the classification of triangles into equilateral
and non-equilateral is based; although all kangaroos may happen to be
*Australian* kangaroos, this is not part of what is necessarily
implied by the use of the name, for an animal subsequently found in
the interior of New Guinea, but otherwise possessing all the
properties of kangaroos, would not have the name kangaroo denied to
it; although all ruminant animals are cloven-hoofed, we cannot regard
cloven-hoofed as part of the meaning of ruminant, and (as Mill
observes) if an animal were to be discovered which chewed the cud, but
had its feet undivided, it would certainly still be called ruminant.

(ii) Some writers who regard proper names as connotative appear to include in the connotation of a name all those attributes which the name suggests to the mind, whether or not they are actually implied by it. And it is to be observed in this connexion that a name may in the mind of any given individual be closely associated with properties which even the same individual would in no way regard as implied in the meaning of the name, as, for instance, “Trinity undergraduate” with a blue gown. This interpretation of connotation is, therefore, clearly to be distinguished from that given in the preceding paragraph.

We may further distinguish the view, apparently taken by some
writers, according to which the connotation of a class-name at any
given time would include all the properties *known* at that time to
belong to the class.

(iii) Other writers use the term in still another sense and would
include in the connotation of a class-name all the properties, known
and unknown, which are possessed in common by all members of the
class. Thus, Mr E. C. Benecke writes,—“Just as the word
‘man’ denotes every creature, or class of creatures,
having the attributes of humanity, whether we know him or not, so does
the word properly connote the *whole* of the properties common to the
class, whether we know them or not. Many of the facts, known to
physiologists and anatomists about the constitution of man’s
brain, for example, are not involved in most men’s idea of the
brain; the possession 26 of a brain precisely so constituted does
not, therefore, form any part of their meaning of the word
‘man.’ Yet surely this is properly connoted by the word….
We have thus the denotation of the concrete name on the one side and
its connotation on the other, occupying perfectly analogous positions.
Given the connotation,—the denotation is all the objects that
possess the whole of the properties so connoted. Given the
denotation,—the connotation is the whole of the properties
possessed in common by all the objects so denoted” (*Mind*,
1881, p. 532). Jevons uses the term in the same sense. “A term
taken in intent (connotation) has for its meaning the whole infinite
series of qualities and circumstances which a thing possesses. Of
these qualities or circumstances some may be known and form the
description or definition of the meaning; the infinite remainder are
unknown” (*Pure Logic*, p. 6).22

^{22} Bain appears to use the term
in an intermediate sense, including in the connotation of a class-name
not *all* the attributes common to the class but all the
*independent* attributes, that is, all that cannot be derived or
inferred from others.

While rejecting the use of the term
*connotation* in any but the first of the above mentioned senses,
we shall find it convenient to have distinctive terms which can be
used with the other meanings that have been indicated. The three terms
connotation, intension, and comprehension are commonly employed almost
synonymously, and there will certainly be a gain in endeavouring to
differentiate their meanings. *Intension*, as already suggested,
may be used to indicate in the most general way what may be called the
implicational aspect of names; the complex terms *conventional
intension*, *subjective intension*, and *objective
intension* will then explain themselves. *Connotation* may be
used as equivalent to *conventional intension* ; and
*comprehension* as equivalent to *objective intension*.
*Subjective intension* is less important from the logical
standpoint, and we need not seek to invent a single term to be used as
its equivalent.23

^{23} For anyone who is given the
meaning of a name but knows nothing of the objects denoted by the
name, subjective intension coincides with connotation. Were the ideal
of knowledge to be reached, subjective intension would coincide with
comprehension.

27 *Conventional intension* or
*connotation* will then include only those attributes which
constitute the meaning of a name;24 *subjective
intension* will include those that are mentally associated with it,
whether or not they are actually signified by it; *objective
intension* or *comprehension* will include all the attributes
possessed in common by all members of the class denoted by the name.
We might perhaps speak more strictly of the *connotation* of the
*name* itself, the *subjective intension* of the
*notion* which is the mental equivalent of the name, and the
*comprehension* of the *class* which is denoted by the
name.25

^{24} It is to be observed that in
speaking of the connotation of a name we may have in view either the
signification that the name bears in common acceptation, or some
special meaning assigned to it by explicit definition for some
scientific or other specific purpose.

^{25} The distinctions of meaning
indicated in this section will be found essential for clearness of
view in discussing certain questions to which we shall pass on
immediately; in particular, the questions whether connotation and
denotation necessarily vary inversely, and whether proper names are
connotative.

18.
*Sigwart’s distinction between Empirical, Metaphysical, and
Logical Concepts*.—Sigwart observes that in speaking of
concepts we ought to distinguish between three meanings of the word.
These three meanings of “concept” he describes as follows.26

^{26} *Logic*, I. p. 245. This and future references to Sigwart
are to the English translation of his work by Mrs
Bosanquet.

(1) By a concept may be meant a natural psychological production,—the general idea which has been developed in the natural course of thought. Such ideas are different for different people, and are continually in process of formation; even for the individual himself they change, so that a word does not always keep the same meaning even for the same person. Strictly speaking, it is only by a fiction which neglects individual peculiarities that we can speak of the concepts corresponding to the terms used in ordinary language.

(2) In contrast with this empirical meaning the concept may be
viewed as an ideal; it is then the mark at which we aim in our
endeavour to attain knowledge, for we seek to find in it an
*adequate copy of the essence of things*. 28

(3) Between these two meanings of the word, which may be called the
*empirical* and the *metaphysical*, there lies the
*logical*. This has its origin in the logical demand for
certainty and universal validity in our judgments. All that is
required is that our ideas should be absolutely fixed and determined,
and that all who make use of the same system of denotation should have
the same ideas.

This threefold distinction may be usefully compared with that drawn in the preceding section. Sigwart is approaching the question from a different point of view, but it will be observed that his three “meanings of concept” correspond broadly with subjective intension, objective intension, and conventional intension respectively.

It may be added that Mr Bosanquet’s distinction
(*Logic*, I. pp. 41 to 46) between the
objective reference of a name (its *logical meaning*) and its
content for the individual mind (the *psychical idea*) appears to
some extent to correspond to the distinction between connotation and
subjective intension.

19. *Connotation and
Etymology*.—The connotation of a name must not be confused
with its etymology. In dealing with names from the etymological or
historical point of view we consider the circumstances in which they
were first imposed and the reasons for their adoption; also the
successive changes, if any, in their meaning that have subsequently
occurred. In making precise the connotation to be attached to a name
we may be helped by considering its etymology. But we must clearly
distinguish between the two; in finally deciding upon the connotation
to be assigned to a name for any particular scientific purpose, we may
indeed find it necessary to depart not merely from its original
meaning, but also from its current meaning in everyday discourse.

20. *Fixity of
Connotation*.—It has been already pointed out that subjective
intension is variable. A given name will almost certainly call up in
the minds of different persons different ideas; and even in the case
of the same person it will probably do so at different times. The
question may be raised how far the same is true of connotation. It has
been implied in the preceding section that the scientific use of a
name may differ 29
from its use in everyday discourse; and there can be no doubt that as
a matter of fact different people may by the same name intend to
signify different things, that is to say, they would include different
attributes in the connotation of the name. It is, moreover, not
unfrequently the case that some of us may be unable to say precisely
what is the meaning that we ourselves attach to the words we use.

At the same time a clear distinction ought to be drawn between
subjective intension and connotation in respect of their variability.
Subjective intension is necessarily variable; it can never be
otherwise. Connotation, on the other hand, is only variable by
accident; and in so far as there is variation language fails of its
purpose. “Identical reference,” as Mr Bosanquet puts it,
“is the root and essence of the system of signs which we call
language” (*Logic*, I. p. 16).
It is only by some conventional agreement which shall make language
fixed that scientific discussions can be satisfactorily carried on;
and there would be no variation in the connotation of names in the
case of an ideal language properly employed. In dealing with
reasonings from the point of view of logical doctrine, it is,
therefore, no unreasonable assumption to make that in any given
argument the connotation of the names employed is fixed and definite;
in other words, that every name employed is either used in its
ordinary sense and that this is precisely determined, or else that,
the name being used with a special meaning, such meaning is adhered to
consistently and without equivocation.

21. *Extension and
Denotation*.—The terms *extension* and *denotation*
are usually employed as synonymous, but there will be some advantage
in drawing a certain distinction between them. We shall find that when
names are regarded as the subjects of propositions there is an implied
reference to some *universe of discourse*, which may be more or less
limited. For example, we should naturally understand such propositions
as *all men are mortal*, *no men are perfect*, to refer to
all men who have actually existed on the earth, or are now existing,
or will exist hereafter, but we should not understand them to refer to
all fictitious persons or all beings possessing the essential
characteristics of men whom we are able to conceive or imagine. 30 The meaning of
*universe of discourse* will be further illustrated subsequently.
The only reason for introducing the conception at this point is that
we propose to use the term *denotation* or *objective
extension* rather than the term extension simply when there is an
explicit or implicit limitation to the objects actually to be found in
some restricted universe. By the *subjective extension* of a
general name, on the other hand, we shall understand the whole range
of objects real or imaginary to which the name can be correctly
applied, the only limitation being that of logical conceivability.
Every name, therefore, which can be used in an intelligible sense will
have a positive subjective extension, but its denotation in a universe
which is in some way restricted by time, place, or circumstance may be
zero.27

^{27} The value of the above
distinction may be illustrated by reference to the divergence of view
indicated in the following quotation from Mr Monck, who uses the terms
*denotation* and *extension* as synonymous: “It is a
matter of accident whether a general name will have any extension (or
denotation) or not. *Unicorn*, *griffin*, and *dragon*
are general names because they have a meaning, and we can suppose
another world in which such beings exist; but the terms have no
extension, because there are no such animals in this world. Some
logicians speak of these terms as having an extension, because we can
*suppose* individuals corresponding to them. In this way every
general term would have an extension which might be either real or
imaginary. It is, however, more convenient to use the word
*extension* for a real extension (past, present, or future)
only” (*Introduction to Logic*, p. 10). It should be added,
in order to prevent possible misapprehension, that by *universe of
discourse*, as used in the text, we do not necessarily mean the
material universe; we may, for example, mean the universe of
fairy-land, or of heraldry, and in such a case, *unicorn*,
*griffin*, and *dragon* may have denotation (in our special
sense), as well as subjective extension, greater than zero. What is
the particular universe of reference in any given proposition will
generally be determined by the context. For logical purposes we may
assume that it is conventionally understood and agreed upon, and that
it remains the same throughout the course of any given argument. As Dr
Venn remarks, “We might include amongst the assumptions of logic
that the speaker and hearer should be in agreement, not only as to the
meaning of the words they use, but also as to the conventional
limitations under which they apply them in the circumstances of the
case” (*Empirical Logic*, p. 180).

In the
sense here indicated, *denotation* is in certain respects the
correlative of *comprehension* rather than of *connotation*.
For in speaking of denotation we are, as in the case of comprehension,
taking an objective standpoint; and there is, moreover, in the case of
comprehension, as in that of denotation, a 31 tacit reference to some particular universe
of discourse. Since, however, denotation is generally speaking
determined by connotation, there is one very important respect in
which connotation and denotation are still correlatives.

22. *Dependence of
Extension and Intension upon one another*.28—Taking any
class-name *X*, let us first suppose that there has been a
conventional agreement to use it wherever a certain selected set of
properties *P*_{1},
*P*_{2}, … *P _{m}*, are present. This
set of properties will constitute the

^{28} This section may be omitted
on a first reading.

^{29} It will be assumed in the
remainder of this section that we are throughout speaking with
reference to a given universe of discourse.

These
properties may not, and almost certainly will not, exhaust the
properties common to *Q*_{1},
*Q*_{2}, … *Q _{y}*. Let all the common
properties be

Now it will always be possible in one or more ways to make out of
*Q*_{1},
*Q*_{2}, … *Q _{y}*, a selection

^{30} It may chance to be necessary
to make *Q*_{1},
*Q*_{2}, … *Q _{n}* coincide with

^{31} Mr Johnson points out to me
that in pursuing this line of argument certain restrictions of a
somewhat subtle kind are necessary in regard to what may be called our
“universe of attributes.” The “universe of
objects” which is what we mean by the “universe of
discourse,” implies *individuality of object* and
*limitation of range of objects* ; and if we are to work out a
thoroughgoing reciprocity between attributes and objects, we must
recognise in our “universe of attributes” restrictions
analogous to the above, namely, *simplicity of attribute* and
*limitation of range of attributes*. By “simplicity of
attribute” is meant that the universe of attributes must not
contain any attribute which is a *logical function* of any other
attribute or set of attributes. Thus, if *A*, *B* are two
attributes recognised in our universe, we must not admit such
attributes as *X* (= *A and B*), or *Y* (= *A or
B*), or *Z* (= *not-A*). We may indeed have a negatively
defined attribute, but it must not be the formal contradictory of
another or formally involve the contradictory of another. The
following example will shew the necessity of this restriction. Let
*P*_{1}, *P*_{2}, *P*_{3}, be
selected as typical of the whole class *P*_{1},
*P*_{2}, *P*_{3} *P*_{4},
*P*_{5}, *P*_{6}; and let
*A*_{1} be an attribute possessed by
*P*_{1} alone, *A*_{2} an attribute
possessed by *P*_{2} alone, and so on. Then if we
recognise *A*_{1} *or A*_{2} *or
A*_{3} as a distinct attribute, it is at once clear that
*P*_{1}, *P*_{2}, *P*_{3} will
no longer be typical of the whole class; and the same result follows
if *not-A*_{4} is recognised as a distinct attribute.
Similarly, without the restriction in question *any* selection
(short of the whole) would necessarily fail to be typical of the whole
class. As a concrete example, suppose that we select from the class of
*professional men* a set of examples that have in common no
attribute except those that are common to the whole class. It may turn
out that our examples are all *barristers* or *doctors*, but
none of them *solicitors*. Now if we recognise as a distinct
attribute being “either a barrister or a doctor,” our
selected group will thereby have an extra common attribute not
possessed by every professional man. The same result will follow if we
recognise the attribute “non-solicitor.” Not much need be
added as regards the necessity of some limitation in the range of
attributes which are recognised. The mere fact of our having selected
a certain group would indeed constitute an additional attribute, which
would at once cause the selection to fail in its purpose, unless this
were excluded as inessential. Similarly, such attributes as position
in space or in time &c. must in general be regarded as
inessential. For example, I might draw on a sheet of paper a number of
triangles sufficiently typical of the whole class of triangles, but
for this it would be necessary to reject as inessential the common
property which they would possess of all being drawn on a particular
sheet of paper.

We have then, with reference to *X*,

(1) *Connotation*:
*P*_{1} … *P _{m}* ;

(2)

(3)

(4)

Of these, either the connotation or the exemplification will suffice to mark out or completely identify the class, although they do not exhaust either all its common properties or all the individuals contained in it. In other words, whether we start from the connotation or from the exemplification, the denotation and the comprehension will be the same.32

^{32} It will be observed that
connotation and exemplification are distinguished from comprehension
and denotation in that they are *selective*, while the latter
pair are *exhaustive*. In making our selection our aim will
usually be to find the *minimum* list which will suffice for our
purpose.

33 For a concrete illustration of the above,
the term *metal* may be taken. From the chemical point of view a
metal may be defined as an element which can replace hydrogen in an
acid and thus form a salt. This then is the *connotation* of the
name. Its *denotation* consists of the complete list of elements
fulfilling the above condition now known to chemists, and possibly of
others not yet discovered.33 The members of the
whole class thus constituted are, however, found to possess other
properties in common besides those contained in the definition of the
name, for example, fusibility, the characteristic lustre termed
metallic, a high degree of opacity, and the property of being good
conductors of heat and electricity. The complete list of these
properties forms the *comprehension* of the name. Now a chemist
would no doubt be able from the full denotation of metal to make a
selection of a limited number of metals which would be precisely
typical of the whole class;34 that is to say, his
selected list would possess in common only such properties as are
common to the whole class. This selected class would constitute the
*exemplification* of the name.

^{33} It is necessary to
distinguish between the *known* extension of a term and its full
denotation, just as we distinguish between the *known* intension
of a term and its full comprehension.

^{34} He would take metals as
different from one another as possible, such as aluminium, antimony,
copper, gold, iron, mercury, sodium, zinc.

We have so far
assumed that (1) connotation or *intensive definition* has first
been arbitrarily fixed, and that this has successively determined (2)
denotation, (3) comprehension, and—with a certain range of
choice—(4) exemplification. But it is clear that theoretically
we might start by arbitrarily fixing (i) the exemplification or
*extensive definition* ; and that this would successively
determine (ii) comprehension, (iii) denotation, and then—again
with a certain range of choice35—(iv)
connotation.

^{35} It is ordinarily said that
“of the denotation and connotation of a term one may, both
cannot, be arbitrary,” and this is broadly true. It is possible,
however, to make the statement rather more exact. Given either
intensive or extensive definition, then both denotation and
comprehension are, with reference to any assigned universe of
discourse, absolutely fixed. But different intensive definitions, and
also different extensive definitions, may sometimes yield the same
results; and it is therefore possible that, everything else being
given, connotation or exemplification may still be *within certain
limits indeterminate*. For example, given the class of *parallel
straight lines*, the connotation may be determined in two or three
different ways; or, given the class of *equilateral equiangular
triangles*, we may select as connotation either having three equal
sides or having three equal angles. Again, given the connotation of
*metal*, it would no doubt be possible to select in more ways
than one a limited number of metals not possessing in common any
attributes which are not also possessed by the remaining members of
the class.

34 It is interesting from a theoretical point
of view to note the possibility of this second order of procedure; and
this order may, moreover, be said to represent what actually
occurs—at any rate in the first instance—in certain cases,
as, for example, in the case of natural groups in the animal,
vegetable, and mineral kingdoms. Men form classes out of vaguely
recognised resemblances long before they are able to give an intensive
definition of the class-name, and in such a case if they are asked to
explain their use of the name, their reply will be to enumerate
typical examples of the class. This would no doubt ordinarily be done
in an unscientific manner, but it would be possible to work it out
scientifically. The extensive definition of a name will take the form:
*X is the name of the class of which Q*_{1},
*Q*_{2}, … *Q _{n} are typical*. This
primitive form of definition may also be called

^{36} It is not of course meant
that when we start from an extensive definition, we are classing
things together at random without any guiding principle of selection.
No doubt we shall be guided by a resemblance between the objects which
we place in the same class, and in this sense intension may be said
always to have the priority. But the resemblance may be unanalysed, so
that we may be far more familiar with the application of the
class-name than with its implication; and even when a connotation has
been assigned to the name, it may be extensively controlled, and
constantly subject to modification, just because we are much more
concerned to keep the denotation fixed than the
connotation.

In this connexion the names of simple
feelings which are incapable of analysis may be specially considered.
For the names of ultimate elements, there is, says Sigwart,37 no
definition; we must assume that everyone attaches the same meaning to
them. To such names we may indeed be able to assign a proximate genus,
as when we say “red is a colour”; but we 35 cannot add a specific
difference. It is, however, only an *intensive definition* that
is wanting in these cases; and the deficiency is supplied by means of
an *extensive definition*. The way in which we make clear to
others our use of such a term as “red” is by pointing out
or otherwise indicating various objects which give rise in us to the
feeling. Thus “red” is the colour of field poppies, hips
and haws, ordinary sealing-wax, bricks made from certain kinds of
clay, &c. This is nothing more or less than an extensive
definition as above defined.

^{37} Logic, I. p. 289.

In the case of most names, however, where formal definition is attempted, it is more usual, as well as really simpler, to start from an intensive definition, and this in general corresponds with the ultimate procedure of science. For logical purposes, it is accordingly best to assume this order of procedure, unless an explicit statement is made to the contrary.38

^{38} It is worth noticing that in
practice an intensive definition is often followed by an enumeration
of typical examples, which, if well selected, may themselves almost
amount to an extensive definition. In this case, we may be said to
have the two kinds of definition supplementing one
another.

23.
*Inverse Variation of Extension and Intension*.39—In general, as intension is increased
or diminished, extension is diminished or increased accordingly, and
*vice versâ*. If, for example, *rational* is added to the
connotation of animal, the denotation is diminished, since all
non-rational animals are now excluded, whereas they were previously
included. On the other hand, if the denotation of *animal* is to
be extended so as to include the vegetable kingdom, it can only be by
omitting *sensitive* from the connotation. Hence the following
law has been formulated: “In a series of common terms standing
to one another in a relation of subordination40 *the extension and
the intension vary inversely*.” Is this law to be accepted?
It must be observed at the outset that the notion of inverse variation
is at any rate not to be interpreted in any strict mathematical or
numerical sense. It is certainly not true that whenever the number of
36 attributes included
in the intension is altered in any manner, then the number of
individuals included in the extension will be altered in some assigned
numerical proportion. If, for example, to the connotation of a given
name different single attributes are added, the denotation will be
affected in very different degrees in different cases. Thus, the
addition of *resident* to the connotation of *member of the
Senate of the University of Cambridge* will reduce its denotation
in a much greater degree than the addition of *non-resident*.
There is in short no *regular* law of variation. The statement
must not then be understood to mean more than that any increase or
diminution of the intension of a name will necessarily be accompanied
by *some* diminution or increase of the extension as the case may be,
and *vice versâ*.41 We will discuss the alleged law in this form,
considering, first, connotation and denotation, exemplification and
comprehension; and, secondly, denotation and comprehension.42

^{39} This section may be omitted
on a first reading.

^{40} As in the *Tree of
Porphyry*: Substance, Corporeal Substance (Body), Animate Body
(Living Being), Sensitive Living Being (Animal), Rational Animal
(Man). In this series of terms the intension is at each step
increased, and the extension diminished.

^{41} It has been said that while
the extension of a term is capable of quantitative measurement, the
same is not equally true of intension. “The parts of extension
may be counted, but it is inept to count the parts in intension. For
they are not external to each other, and they form a whole such as
cannot be divided into units except by the most arbitrary
dilaceration. And if it were so divided, all its parts would vary in
value, and there would be no reason to expect that ten of them (that
is, ten attributes) should have twice the amount or value of
five” (Bosanquet, *Logic*, I. p.
59). There is some force in this, and it is decisive against
interpreting *inverse variation* in the present connexion in any strict
numerical sense. But, at the same time, no error is committed and no
difficulty of interpretation arises, if we content ourselves with
speaking merely of the enlargement or restriction of the intension of
a term. There can be no doubt that intension is increased when we pass
from animal to man, or from man to negro; or again when we pass from
triangle to isosceles triangle, or from isosceles triangle to
right-angled isosceles triangle.

^{42} The
discussion is purposely made as formal and exact as possible. If
indeed the doctrine of inverse variation cannot be treated with
precision, it is better not to attempt to deal with it at
all.

**A.** (1) Let connotation be supposed arbitrarily
fixed, and used to determine denotation in some assigned universe of
discourse. Then it will not be true that connotation and denotation
will necessarily vary inversely. For suppose the connotation of a
name, *i.e.*, the attributes signified by it, to be *a*,
*b*, *c*. It may happen that in fact wherever the attributes
*a* and *b* are present, the attributes *c* and
*d* are also present. 37 In this case, if *c* is dropped from
the connotation, or *d* added to it, the denotation of the name
will remain unaffected. We have concrete examples of this, if we
suppose *equiangularity* added to the connotation of
*equilateral triangle*, or *cloven-hoofed* to that of
*ruminant*, or *having jaws opening up and down* to that of
*vertebrate*, or if we suppose *invalid* dropped from the
connotation of *invalid syllogism with undistributed middle*. It
is clear, however, that *if* any alteration in denotation takes
place when connotation is altered, it must necessarily be in the
opposite direction. Some individuals possessing the attributes
*a* and *b* may lack the attribute *c* or the attribute
*d* ; but no individuals possessing the attributes *a*,
*b*, *c*, or *a*, *b*, *c*, *d* can fail
to possess the attributes *a*, *b*, or *a*, *b*,
*c*. For example, if to the connotation of *metal* we add
*fusible*, it makes no difference to the denotation; but if we
add *having great weight*, we exclude potassium, sodium, &c.

The *law of variation of denotation with connotation* may then
be stated as follows:—If the connotation of a term is
arbitrarily enlarged or restricted, the denotation in an assigned
universe of discourse will either remain unaltered or will change in
the opposite direction.43

^{43} Since reference is here made
to the actual denotation of a term in some assigned universe of
discourse, the above law may be said to turn partly on material, and
not on purely formal, considerations. It should, therefore, be added
that although an alteration in the connotation of a term will not
always alter its actual denotation in an assigned universe of
discourse, it will always affect potentially its subjective extension.
If, for example, the connotation of a term *X* is *a*,
*b*, *c*, and we add *d* ; then the (real or imaginary)
class of *X*’s that are not *d* is necessarily
excluded from, while it was previously included in, the subjective
extension of the term *X*. Hence, if the connotation of a term is
arbitrarily enlarged or restricted, the *subjective extension*
will be potentially restricted or enlarged accordingly. Cf. Jevons,
*Principles of Science*, 30, § 13.

(2) Let
exemplification be supposed arbitrarily fixed, and used to determine
comprehension. It is unnecessary to shew in detail that a
corresponding *law of variation of comprehension with exemplification*
will hold good, namely:—If the exemplification (extensive
definition) of a term is arbitrarily enlarged or restricted, the
comprehension in an assigned universe of discourse will either remain
unaltered or will change in the opposite direction. 38

**B.** We may now consider the relation between the
*comprehension* and the *denotation* of a term. Let
*P*_{1}, *P*_{2}, … *P _{x}* be
the totality of attributes possessed by the class

^{44} What may be arbitrary is the
intensive definition (*P*_{1},
*P*_{2}, … *P _{m}*) or the extensive
definition (

We cannot suppose any direct arbitrary
alteration either in comprehension or in denotation. We can, however,
establish the following law of inverse variation, namely, that *any
arbitrary alteration in either intensive definition or extensive
definition which results in an alteration of either denotation or
comprehension will also result in an alteration in the opposite
direction of the other*.

Let *X* and *Y* be two terms which are so related that
the definition (either intensive or extensive, as the case may be) of
*Y* includes all that is included in the definition of *X*
and more besides. We have to shew that either the denotations and
comprehensions of *X* and *Y* will be identical or if the
denotation of one includes more than the denotation of the other then
its comprehension will include less, and *vice versâ*.

(*a*) Let *X* and *Y* be determined by connotation
or intensive definition. Thus, let *X* be determined by the set
of properties *P*_{1} … *P _{m}* and

Then

If the former, no object included in the denotation of

If the latter, then the denotation of

(*b*) Let *X* and *Y* be determined by
exemplification or extensive definition. Thus, let *X* be
determined by the set of examples
*Q*_{1} … *Q _{n}*, and

Then

If the former, no property included in the comprehension of

If the latter, then the comprehension of

All cases have now been considered, and it has been shewn that the law above formulated holds good universally. This law and the two laws given on page 37 must together be substituted for the law of inverse relation between extension and intension in its usual form if full precision of statement is desired.

It should be observed that in speaking of variations in
comprehension or denotation, no reference is intended to changes in
things or in our knowledge of them. The variation is always supposed
to have originated in some arbitrary alteration in the intensive or
extensive definition of a given term, or in passing from the
consideration of one term to that of another with a different
extensive or intensive definition. Thus fresh things may be discovered
to belong to a class, and the comprehension of the class-name may not
thereby be affected. But in this case the denotation has not itself
varied; only our knowledge of it has varied. Or we may discover fresh
attributes previously overlooked; in which case similar remarks will
apply. Again, new things may be brought into existence which come
under the denotation of the name, and still its comprehension may
remain unchanged. Or possibly new qualities may be developed by 40 the whole of the class.
In these cases, however, there is no *arbitrary* alteration in
the application or implication of the name, and hence no real
exception to what has been laid down above.

24. *Connotative
Names*.—Mill’s use of the word *connotative*,
which is that generally adopted in modern works on logic, is as
follows: “A non-connotative term is one which signifies a
subject only, or an attribute only. A connotative term is one which
denotes a subject, and implies an attribute” (*Logic*, I.
2, § 5). According to this definition, a connotative name must not
only possess extension, but must also have a conventional intension
assigned to it.

Mill considers that the following kinds of names are connotative in
the above sense:—(1) All concrete general names. (2) Some
singular names. For example, *city* is a general name, and as
such no one would deny it to be connotative. Now if we say *the
largest city in the world*, we have individualised the name, but it
does not thereby cease to be connotative. Proper names are, however,
according to Mill, non-connotative, since they merely denote a subject
and do not imply any attributes. To this point, which is a subject of
controversy, we shall return in the following section. (3) While
admitting that most abstract names are non-connotative, since they
merely signify an attribute and do not denote a subject, Mill
maintains that some abstracts may justly be “considered as
connotative; for attributes themselves may have attributes ascribed to
them; and a word which denotes attributes may connote an attribute of
those attributes” (*Logic*, I. 2, § 5).

The wording of Mill’s definition is unfortunate and is probably responsible for a good deal of the controversy that has centred round the question as to whether certain classes of names are or are not connotative.

All names that we are able to use in an intelligible sense must have subjective intension for us. For we must know to what objects or what kinds of objects the names are applicable, and we cannot but associate some properties with these objects and therefore with the names.

Moreover all names that have denotation in any given 41 universe of discourse must have comprehension also; for no object can exist without possessing properties of some kind.

If then any name can properly be described as non-connotative, it cannot be in the sense that it has no subjective intension or no comprehension. This is at least obscured when Mill speaks of non-connotative names as not implying any attributes; and if misunderstanding is to be avoided, his definitions must be amended, so as to make it quite clear that in a non-connotative name it is connotation only that is lacking, and not either subjective intension or comprehension.

A *connotative name* may be defined as a name whose
application is determined by connotation or *intensive
definition*, that is, by a conventionally assigned attribute or set
of attributes. A *non-connotative name* is an *exemplificative
name*, a name whose application is determined by exemplification or
*extensive definition* in the sense explained in section 22; in other words, it is a name whose application
is determined by pointing out or indicating, by means of a description
or otherwise, the particular individual (if the name is singular), or
typical individuals (if the name is general), to which the name is
attached.

If it is allowed that the application of any names can be determined in the latter way, as distinguished from the former, then it must be allowed that some names are non-connotative.

25. *Are proper names
connotative?*—To this question absolutely contradictory
answers are given by ordinarily clear thinkers as being obviously
correct. To some extent, however, the divergence is merely verbal, the
terms “connotation” and “connotative name”
being used in different senses.

It is necessary at the outset to guard against a misconception
which quite obscures the real point at issue. Thus, with reference to
Mill, Jevons says, “Logicians have erroneously asserted, as it
seems to me, that singular terms are devoid of meaning in intension,
the fact being that they exceed all other terms in that kind of
meaning” (*Principles of Science*, 2, § 2, with a reference
to Mill in a foot-note). But Mill distinctly states that some singular
names are connotative, *e.g.*, *the* 42 *sun*,45
*the first emperor of Rome* (*Logic*, I. 2, § 5). We may
certainly narrow down the extension of a term till it becomes
individualised without destroying its connotation; “the present
Professor of Pure Mathematics in University College, London” is
a singular term—its extension cannot be further
diminished—but it is certainly connotative.

^{45} The question has been asked
on what grounds *the sun* can be regarded as connotative, while
*John* is considered non-connotative; compare T. H. Green,
*Philosophical Works*, ii. p. 204. The answer is that *sun*
is a general name with a definite signification which determines its
application, and that it does not lose its connotation when
individualised by the prefix *the* ; while *John*, on the
other hand, is a name given to an object merely as a mark for purposes
of future reference, and without signifying the possession by that
object of any conventionally selected attributes.

It must then be understood that only one class of singular names, namely,
*proper names*, are affirmed to be non-connotative; and that no
more is meant by this than that their application is not determined by
a conventionally assigned set of attributes.46 The ground may be
further cleared by our explicitly recognising that, although proper
names have no connotation, they nevertheless have both subjective
intension and comprehension. An individual object can be recognised
only through its attributes; and a proper name when understood by me
to be a mark of a certain individual undoubtedly suggests to my mind
certain qualities.47 The qualities thus suggested by the name
constitute its subjective intension. The comprehension of the name
will include a good deal more than its subjective intension, namely,
43 the whole of the
properties that belong to the individual denoted.

^{46} The treatment of the question
adopted in this work has been criticised on the ground that it is
question-begging, since in section 10 proper names have really been
*defined* as non-connotative. This criticism cannot, however, be
pressed unless it is at the same time maintained that the definition
given in section 10 yields a denotation different from that ordinarily
understood to belong to proper names.

^{47} A proper name may have
suggestive force even for those who are not actually acquainted with
the person or thing denoted by it. Thus *William Stanley Jevons*
may suggest any or all of the following to one who never heard the
name before: an organised being, a human being, a male, an
Anglo-Saxon, having some relative named Stanley, having parents named
Jevons. But at the same time, the name cannot be said necessarily to
signify any of these things, in the sense that if they were wanting it
would be misapplied. Consider, for example, such a name as *Victoria
Nyanza*. Some further remarks bearing on this point will be found
later on in this section.

It will be found that most
writers who regard proper names as possessing connotation really mean
thereby either subjective intension or comprehension. Thus Jevons puts
his case as follows:—“Any proper name such as John Smith,
is almost without meaning until we know the John Smith in question. It
is true that the name alone connotes the fact that he is a Teuton, and
is a male; but, so soon as we know the exact individual it denotes the
name surely implies, also, the peculiar features, form, and character,
of that individual. In fact, as it is only by the peculiar qualities,
features, or circumstances of a thing, that we can ever recognise it,
no name could have any fixed meaning unless we attached to it,
mentally at least, such a definition of the kind of thing denoted by
it, that we should know whether any given thing was denoted by it or
not. If the name John Smith does not suggest to my mind the qualities
of John Smith, how shall I know him when I meet him? For he certainly
does not bear his name written upon his brow” (*Elementary
Lessons in Logic*, p. 43). A wrong criterion of connotation in
Mill’s sense is here taken. The connotation of a name is not the
quality or qualities by which I or any one else may happen to
recognise the class which it denotes. For example, I may recognise an
Englishman abroad by the cut of his clothes, or a Frenchman by his
pronunciation, or a proctor by his bands, or a barrister by his wig;
but I do not mean any of these things by these names, nor do they (in
Mill’s sense) form any part of the connotation of the names.
Compare two such names as *Henry Montagu Butler* and *the
Master of Trinity College, Cambridge*. At the present time they
denote the same person; but the names are not equivalent,—the
one is given to a certain individual as a mark to distinguish him from
others, and has no further signification; the other is given because
of the performance of certain functions, on the cessation of which the
name would cease to apply. Surely there is a distinction here, and one
which it is important that we should not overlook.

It may indeed fairly be said that many, if not most, proper 44 names do signify
something, in the sense that they were chosen in the first instance
for a special reason. For example, *Strongi’th’arm*,
*Smith*, *Jungfrau*. But such names even if in a certain
sense connotative when first imposed soon cease to be so, since their
subsequent application to the persons or things designated is not
dependent on the continuance of the attribute with reference to which
they were originally given. As Mill puts it, *the name once given is
independent of the reason*. In other words, we ought carefully to
distinguish between the *connotation* of a name and its
*history*. Thus, a man may in his youth have been strong, but we
should not continue to calling strong in his dotage; whilst the name
*Strongi’th’arm* once given would not be taken from
him. Again, the name *Smith* may in the first instance have been
given because a man plied a certain handicraft, but he would still be
called by the same name if he changed his trade, and his descendants
continue to be called Smith whatever their occupations may be.48

^{48} It cannot, however, be said
that the name necessarily implies ancestors of the same name. As Dr
Venn remarks, “he who changes his family name may grossly
deceive genealogists, but he does not tell a falsehood”
(*Empirical Logic*, p. 185).

It has been argued that
proper names must be connotative because the use of a proper name
conveys more information than the use of a general name. “Few
persons,” says Mr Benecke,49 “will deny
that if I say *the principal speaker was Mr Gladstone*, I am
giving not less but more information than if, instead of *Mr
Gladstone*, I say either *a member of Parliament*, or *an
eminent man*, or *a statesman*, or *a Liberal leader*. It
will be admitted that the predicate *Mr Gladstone* tells us all
that is told us by all these other connotative predicates put
together, and more; and, if so, I cannot see how it can be denied that
it also connotes more.” It is clear, however, that the
information given when a thing is called by any name depends not on
the connotation of the name, but on its intension for the person
addressed. To anyone who knows that Mr Gladstone was Prime Minister in
1892 the same information is afforded whether a speaker is referred to
as *Mr Gladstone* or as *Prime Minister of* 45 *Great Britain and
Ireland in* 1892. But it certainly cannot be maintained that the
connotation of these two names is identical.

^{49} In a paper on *the
Connotation of Proper Names* read before the Aristotelian
Society.

In criticism of the position that the
application of a proper name such as *Gladstone* is determined by
some attribute or set of attributes, we may naturally ask, *what*
attribute or set of attributes? The answer cannot be that the
connotation consists of the complete group of attributes possessed by
the individual designated; for it is absurd to require any such
enumeration as this in order to determine the application of the name.
It is, however, impossible to select some particular attributes of the
individual in question, and point to them as a group that would be
accepted as constituting the definition of the name; and if it is said
that the application of the name is determined by *any* set of
attributes that will suffice for identification, the case is given up.
For this amounts to identifying the individual by a description (that
is, practically by exemplification), not by a particular set of
attributes conventionally attached to the name as such. The truth is
that no one would ever propose to give an *intensive definition*
of a proper name. All names, however, that are connotative must
necessarily admit of intensive definition.50

^{50} Mr Bosanquet arrives at the
conclusion that “a proper name has a connotation, but not a
fixed general connotation. It is attached to a unique individual, and
connotes whatever may be involved in his identity, or is instrumental
in bringing it before the mind” (*Essentials of Logic*, p.
93). So far as I can understand this statement, it amounts to saying
that proper names have comprehension and subjective intension, but not
connotation, in the senses in which I have defined these
terms.

Proper names of course become connotative when they are used to designate a certain type of person; for example, a Diogenes, a Thomas, a Don Quixote, a Paul Pry, a Benedick, a Socrates. But, when so used, such names have really ceased to be proper names at all; they have come to possess all the characteristics of general names.51

^{51} Compare Gray’s
lines,—

“Some village Hampden, that, with dauntless breast,

The little tyrant of his fields withstood,

Some mute inglorious Milton here may rest,

Some Cromwell guiltless of his country’s blood.”

Attention
may be called to a class of singular names, such as 46 *Miss Smith*,
*Captain Jones*, *President Roosevelt*, *the Lake of
Lucerne*, *the Falls of Niagara*, which may be said to be
partially but only partially connotative. Their peculiarity is that
they are partly made up of elements that have a general and permanent
signification, and that consequently some change in the object denoted
might render them no longer applicable, as, for example, if Captain
Jones received promotion and were made a major; while, at the same
time, such connotation as they possess is by itself insufficient to
determine completely their application. It may be said that their
application is *limited*, but not *determined*, by reference
to specific assignable attributes. They occupy an intermediate
position, therefore, between connotative singular names, such as
*the first man*, and strictly proper names.

We may in this connexion touch upon Jevons’s argument that such a name as “John Smith” connotes at any rate “Teuton” and “male.” This is not strictly the case, since “John Smith” might be a dahlia, or a racehorse, or a negro, or the pseudonym of a woman, as in the case of George Eliot. In none of these cases could the name be said to be misapplied as it would be if a dahlia or a horse were called a man, or a negro a Teuton, or a woman a male. At the same time, it cannot be denied that certain proper names are in practice so much limited to certain classes of objects, that some incongruity would be felt if they were applied to objects belonging to any other class. It is, for example, unlikely that a parent would deliberately have his daughter christened “John Richard.” So far as this is the case, the names in question may be said to be partially connotative in the same way as the names referred to in the preceding paragraph, though to a less extent; that is to say, their application is limited, though not determined, by reference to specific attributes. We should have a still clearer case of a similar kind if the right to bear a certain name carried with it specific legal or social privileges.52

^{52} Compare Bosanquet,
*Logic*, i. p. 53.

The position has been taken that every proper name is at least partially connotative inasmuch as it necessarily implies individuality and the property of being called by the name in question. If we refer to anything by any name whatsoever, it 47 must at any rate have the quality of being called by that name. If we call a man John when he really passes by the name of James, we make a mistake; we attribute to him a quality which he does not possess,—that of passing by the name of John. This argument, although it does not appear to establish the conclusion that proper names are in any degree connotative, nevertheless calls attention to a distinctive peculiarity of proper names that is worthy of notice. The denotation of connotative names may, and usually does, vary from time to time; and this is true of connotative singular names as well as of general names. But it is clearly essential in the case of a proper name that (in any given use) the name shall be consistently affixed to the same individual object. It is, however, one thing to say that the identity of the object called by the name with that to which the name has previously been assigned is a condition essential to the correct use of a proper name, and another thing to say that this is connoted by a proper name. If indeed by connotation we mean the attributes by reason of the possession of which by any object the name is applicable to that object, it seems a case of ὕστερον πρότερον to include in the connotation the property of being called by the name.

26. Are such concepts as
“equilateral triangle” and “equiangular
triangle” identical or different? [K.]

[This question should be considered with reference to the
discussion in sections 17 and 18.]

27. Let *X*_{1}, *X*_{2},
*X*_{3}, *X*_{4}, and *X*_{5}
constitute the whole of a certain universe of discourse: also let
*a*, *b*, *c*, *d*, *e*, *f* exhaust the
properties of *X*_{1}; *a*, *b*, *c*,
*d*, *e*, *g*, those of *X*_{2}; *b*,
*c*, *d*, *f*, *g*, those of *X*_{3};
*a*, *b*, *d*, *e*, *f*, those of
*X*_{4}; and *a*, *c*, *e*, *f*,
*g* those of *X*_{5}.

(i) Given that, under
these conditions, a term has the connotation *a*, *b*, find
its denotation and its comprehension, and determine an exemplification
that would yield the same result.

(ii) Given that, under the
same conditions, a term has the exemplification *X*_{4},
*X*_{5}, find its comprehension and its denotation, and
determine a connotation that would yield the same result. [K.]

48 28. On what grounds may it be held that names may
possess (*a*) denotation without connotation, (*b*)
connotation without denotation?

Give illustrations shewing that
the denotation of a term of which the connotation is known must be
regarded as relative to the proposition in which it is used as subject
and to the context in which the proposition occurs. [J.]

29. What do you consider
to be the question really at issue when it is asked whether proper
names are connotative?

Enquire whether the following names are
respectively connotative or non-connotative: *Caesar*,
*Czar*, *Lord Beaconsfield*, *the highest mountain in
Europe*, *Mont Blanc*, *the Weisshorn*, *Greenland*,
*the Claimant*, *the pole star*, *Homer*, *a Daniel
come to judgment*. [K.]

30. Bring out any special points that arise in the
discussion of the extensional and intensional aspects of the following
terms respectively: *the Rosaceae*, *equilateral triangle*,
*colour*, *giant*. [C.]

31. *Real*
(*Synthetic*), *Verbal* (*Analytic or Synonymous*),
*and Formal Propositions*.—(1) A *real proposition* is
one which gives information of something more than the meaning or
application of the term which constitutes its subject; as when a
proposition predicates of a connotative subject some attribute not
included in its connotation, or when a connotative term is predicated
of a non-connotative subject. For example, *All bodies have
weight*, *The angles of any triangle are together equal to two
right angles*, *Negative propositions distribute their
predicates*, *Wordsworth is a great poet*.

Real propositions are also described as *synthetic*,
*ampliative*, *accidental*.

(2) A *verbal proposition* is one which gives information only in
regard to the meaning or application of the term which constitutes its
subject.53

^{53} Although verbal propositions
may be distinguished from real propositions in accordance with the
above definitions, it may be argued that every verbal proposition
implies a real proposition of a certain sort behind it. For the
question as to what meaning is attached to a given term in ordinary
discourse, or by a given individual, is a question of matter of fact,
and a statement respecting it may be true or false. Thus, *X means
abc* is a verbal proposition; but such propositions as *The
meaning commonly attached to the term X is abc*, *The meaning
attached in this work to the term X is abc*, *The meaning with
which it would be most convenient to employ the term X is abc*, are
real. Looked at from this point of view the distinction between verbal
and real propositions may perhaps be thought to be a rather subtle
one. It remains true, however, that the proposition *X means abc*
is verbal relatively to its subject *X*. Out of the given
material we cannot by any manipulation obtain a real predication about
*X*, that is, about *the thing signified by the term X*, but
only about *the meaning of the term X*. The real proposition
involved can thus only be obtained by substituting for the original
subject another subject.

50 Two classes of
verbal propositions are to be distinguished, which may be called
respectively *analytic* and *synonymous*. In the former the
predicate gives a partial or complete analysis of the connotation of
the subject; *e.g.*, *Bodies are extended*, *An
equilateral triangle is a triangle having three equal sides*, *A
negative proposition has a negative copula*.54 *Definitions*
are included under this division of verbal propositions; and the
importance of definitions is so great, that it is clearly erroneous to
speak of verbal propositions as being in all cases trivial. In general
they are trivial only in so far as their true nature is misunderstood;
when, for example, people waste time in pretending to prove what has
been already assumed in the meaning assigned to the terms employed.55

^{54} Since we do not here really
advance beyond an analysis of the subject-notion, Dr Bain describes
the verbal proposition as the “notion under the guise of the
proposition.” Hence the appropriateness of treating verbal
propositions under the general head of Terms.

^{55} By a *verbal dispute* is
meant a dispute that turns on the meaning of words. Dr Venn observes
that purely verbal disputes are very rare, since “a different
usage of words almost necessarily entails different convictions as to
facts” (*Empirical Logic*, p. 296). This is true and
important; it ought indeed always to be borne in mind that the problem
of scientific definition is not a mere question of words, but a
question of things. At the same time, disputes which are *partly*
verbal are exceedingly common, and it is also very common for their
true character in this respect to be unrecognised. When this is the
case, the controversy is more likely than not to be fruitless. The
questions whether proper names are connotative, and whether every
syllogism involves a *petitio principii*, may be taken as
examples. We certainly go a long way towards the solution of these
questions by clearly differentiating between different meanings which
may be attached to the terms employed.

Besides propositions giving a more or less complete analysis of the
connotation of names, the following—which we may speak of as
*synonymous* propositions—are to be included under the head
of verbal propositions: (*a*) where the subject and predicate are
both proper names, *e.g.*, *Tully is Cicero* ; (*b*)
where they are dictionary synonyms, *e.g.*, *Wealth is
riches*, *A story is a tale*, *Charity is love*. In these
cases information is given only in regard to the application or
meaning of the terms which appear as the subjects of the propositions.

Analytic propositions are also described as *explicative* and
as *essential*. Very nearly the same distinction, therefore, as
51 that between
*verbal* and *real* propositions is expressed by the pairs
of terms—*analytic* and *synthetic*,
*explicative* and *ampliative*, *essential* and
*accidental*. These terms do not, however, cover quite the same
ground as verbal and real, since they leave out of account
*synonymous* propositions, which cannot, for example, be properly
described as either analytic or synthetic.56

^{56} Thus, Mansel calls attention
to “a class of propositions which are not, in the strict sense
of the word, analytical, *viz.*, those in which the predicate is
a single term synonymous with the subject” (Mansel’s
*Aldrich*, p. 170).

The distinction between real and verbal propositions as above given
assumes that the use of terms is fixed by their connotation and that
this connotation is determinate.57 Whether any given
proposition is as a matter of fact verbal or real will depend on the
meaning attached to the terms which it contains; and it is clear that
logic cannot lay down any rule for determining under which category
any given proposition should be placed.58 Still, while we
cannot with certainty distinguish a real proposition by its form, it
may be observed that the attachment of a sign of quantity, such as
*all*, *every*, *some*, &c., to the subject of a
proposition may in general be regarded as an indication that in the
view of the person laying down the 52 proposition a fact is being stated and not
merely a term explained. Verbal propositions, on the other hand, are
usually unquantified or indesignate (see section 69). For example, in
order to give a partially correct idea of the meaning of such a name
as *square*, we should not say “all squares are four-sided
figures,” or “every square is a four-sided figure,”
but “a square is a four-sided figure.”59

^{57} We can, however, adapt the
distinction to the case in which the use of terms is fixed by
extensive definition. We may say that whilst a proposition (expressed
affirmatively and with a copula of inclusion) is *intensively
verbal* when the connotation of the predicate is a part or the
whole of the connotation of the subject, it is *extensively
verbal* when the subject taken in extension is a part or the whole
of the extensive definition of the predicate. Thus, if the use of the
term *metal* is fixed by an extensive definition, that is to say,
by the enumeration of certain typical metals, of which we may suppose
*iron* to be one, then it is a verbal proposition to say that
*iron is a metal*. If, however, *tin* is not included
amongst the typical metals, then it is a real proposition to say that
*tin is a metal*.

^{58} It does not follow from this
that the distinction between verbal and real propositions is of no
logical importance. Although the logician cannot *quâ* logician
determine in doubtful cases to which category a given proposition
belongs, he can point out what are the conditions upon which this
depends, and he can shew that in any discussion or argument no
progress is possible until it is clearly understood by all who are
taking part whether the propositions laid down are to be interpreted
as being real or merely verbal. To refer to an analogous case, it
will not be said that the distinction between truth and falsity is of
no logical importance because the logician cannot *quâ* logician
determine whether a given proposition is true or false.

^{59} It should be added that we
may formally distinguish a full definition from a real proposition by
connecting the subject and the predicate by the word
“means” instead of the word “is.”

(3) There are propositions usually classed as verbal which ought
rather to be placed in a class by themselves, namely, those which are
valid whatever may be the meaning of the terms involved; *e.g.*,
*All A is A*, *No A is **not-A*, *All Z is either B or
not-B*, *If all A is B then no not-B is A*, *If all A is B
and all B is C then all A is C*. These may be called *formal
propositions*, since their validity is determined by their bare
form.60

^{60} Propositions which are in
appearance purely tautologous have sometimes an epigrammatic force and
are used for rhetorical purposes, *e.g.*, *A man’s a
man* (*for a’ that*). In such cases, however, there is
usually an implication which gives the proposition the character of a
real proposition; thus, in the above instance the true force of the
proposition is that *Every man is as such entitled to respect*.
“In the proposition, *Children are children*, the
subject-term means only the age characteristic of childhood; the
predicate-term the other characteristics which are connected with it.
By the proposition, *War is war*, we mean to say that when once a
state of warfare has arisen, we need not be surprised that all the
consequences usually connected with it appear also. Thus the predicate
adds new determinations to the meaning in which the subject was first
taken” (Sigwart, *Logic*, I. p. 86).

Formal propositions are the only propositions whose validity is examined and guaranteed by logic itself irrespective of other sources of knowledge, and many of the results reached in formal logic may be summed up in such propositions; for any formally valid reasoning can be expressed by a formal hypothetical proposition as in the last two of the examples given above.

A formal proposition as here defined must not be confused with a
proposition expressed in symbols. A formal proposition need not indeed
be expressed in symbols at all. Thus, the proposition *An animal is
an animal* is a formal proposition; 53 *All S is P* is not. Strictly speaking,
a symbolic expression, such as *All S is P*, is to be regarded as
a *propositional form*, rather than as a proposition *per
se*. For it cannot be described as in itself either true or false.
What we are largely concerned with in logic are relations between
propositional forms; because these involve corresponding relations
between all propositions falling into the forms in question.

We have then three classes of propositions—*formal*,
*verbal*, and *real*—the validity or invalidity of
which is determined respectively by their bare form, by the mere
meaning or application of the terms involved, by questions of fact
concerning the things denoted by these terms.61

^{61} Real propositions are divided
into true and false according as they do or do not accurately
correspond with facts. By verbal and formal propositions we usually
mean propositions which from the point of view taken are valid. A
proposition which from either of these points of view is invalid is
spoken of as a *contradiction in terms*. Properly speaking we
ought to distinguish between a *verbal* contradiction in terms
and a *formal* contradiction in terms, the contradiction
depending in the first case upon the force of the terms employed and
in the second case upon the mere form of the proposition; *e.g.*,
*Some men are not animals*, *A is not-A*. Any purely formal
fallacy may be said to resolve itself into a formal contradiction in
terms. It should be added that a mere term, if it is complex, may
involve a contradiction in terms; *e.g.*, *Roman Catholic*
(if the separate terms are interpreted literally), *A
not-A*.

32. *Nature of the Analysis
involved in Analytic Propositions*.—Confusion is not
unfrequently introduced into discussions relating to analytic
propositions by a want of agreement as to the nature of the analysis
involved. If identified, as above, with a division of the verbal
proposition, an analytic proposition gives an analysis, partial or
complete, of the *connotation* of the subject-term. Some writers,
however, appear to have in view an analysis of the *subjective
intension* of the subject-term. There is of course nothing
absolutely incorrect in this interpretation, if consistently adhered
to, but it makes the distinction between analytic and synthetic
propositions logically valueless and for all practical purposes
nugatory. “Both intension and extension,” says Mr Bradley,
“are relative to our knowledge. And the perception of this truth
is fatal to a well-known Kantian distinction. A judgment is not fixed
as ‘synthetic’ or ‘analytic’: its character
varies with the knowledge 54 possessed by various persons and at
different times. If the meaning of a word were confined to that
attribute or group of attributes with which it set out, we could
distinguish those judgments which assert within the whole one part of
its contents from those which add an element from outside; and the
distinction thus made would remain valid for ever. But in actual
practice the meaning itself is enlarged by synthesis. What is added
to-day is implied to-morrow. We may even say that a synthetic
judgment, so soon as it is made, is at once analytic.”62

^{62} *Principles of Logic*,
p. 172. Professor Veitch expresses himself somewhat similarly.
“Logically all judgments are analytic, for judgment is an
assertion by the person judging of what he knows of the subject spoken
of. To the person addressed, real or imaginary, the judgment may
contain a predicate new—a new knowledge. But the person making
the judgment speaks analytically, and analytically only; for he sets
forth a part of what he knows belongs to the subject spoken of. In
fact, it is impossible anyone can judge otherwise. We must judge by
our real or supposed knowledge of the thing already in the mind”
(*Institutes of Logic*, p. 237).

If by intension is meant subjective intension, and by an analytic
judgment one which analyses the intension of the subject, the above
statements are unimpeachable. It is indeed so obviously true that in
this sense synthetic judgments are only analytic judgments in the
making, that to dwell upon the distinction itself at any length would
be only waste of time. It is, however, misleading to identify
subjective intension with *meaning* ;63 and this is
especially the case in the present connexion, since it may be
maintained with a certain degree of plausibility that *some*
synthetic judgments are only analytic judgments in the making, even
when by an analytic judgment is meant one which analyses the
*connotation* of the subject. For undoubtedly the connotation of
names is not in practice unalterably fixed. As our knowledge
progresses, many of our 55 definitions are modified, and hence a form
of words which is synthetic at one period may become analytic at
another.

^{63} Compare the following
criticism of Mill’s distinction between real and verbal
propositions: “If every proposition is merely verbal which
asserts something of a thing under a name that already presupposes
what is about to be asserted, then every statement by a scientific man
is *for him* merely verbal” (T. H. Green, *Works*, ii.
p. 233). This criticism seems to lose its force if we bear in mind the
distinction between connotation and subjective intension.

But, in the first place, it is very far indeed from being a
universal rule that newly-discovered properties of a class are taken
ultimately into the connotation or intensive definition of the
class-name. Dr Bain (*Logic*, *Deduction*, pp. 69 to 73)
seems to imply the contrary; but his doctrine on this point is not
defensible on the ground either of logical expediency or of actual
practice. As to logical expediency, it is a generally recognised
principle of definition that we ought to aim at including in a
definition the minimum number of properties necessary for
identification rather than the maximum which it is possible to
include.64 And as to what actually occurs, it is easy to
find cases where we are able to say with confidence that certain
common properties of a class never will as a matter of fact be
included in the definition of the class-name; for example,
*equiangularity* will never be included in the definition of
*equilateral triangle*, or *having cloven hoofs* in the
definition of *ruminant animal*.

^{64} If we include in the
definition of a class-name all the common properties of the class, how
are we to make any universal statement of fact about the class at all?
Given that the property *P* belongs to the whole of the class
*S*, then by hypothesis *P* becomes part of the meaning of
*S*, and the proposition *All S is P* merely makes this
verbal statement, and is no assertion of any matter of fact at all. We
are, therefore, involved in a kind of vicious circle.

In the second place, even when freshly discovered properties of things come ultimately to be included in the connotation of their names, the process is at any rate gradual, and it would, therefore, be incorrect to say—in the sense in which we are now using the terms—that a synthetic judgment becomes in the very process of its formation analytic. On the other hand, it may reasonably be assumed that in any given discussion the meaning of our terms is fixed, and the distinction between analytic and synthetic propositions then becomes highly significant and important. It may be added that when a name changes its meaning, any proposition in which it occurs does not strictly speaking remain the same proposition as before. We ought 56 rather to say that the same form of words now expresses a different proposition.65

^{65} This point is brought out by
Mr Monck in the admirable discussion of the above question contained
in his *Introduction to Logic*, pp. 130 to 134.

33. State which of the following propositions you consider real, and which verbal, giving your reasons in each case:

(i) | All proper names are singular; |

(ii) | A syllogism contains three and only three terms; |

(iii) | Men are vertebrates; |

(iv) | All is not gold that glitters; |

(v) | The dodo is an extinct bird; |

(vi) | Logic is the science of reasoning; |

(vii) | Two and two are four; |

(viii) | All equilateral triangles are equiangular; |

(ix) | Between any two points one, and only one, straight line can be drawn; |

(x) | Any two sides of a triangle are together greater than the third side. |

[C.]

34. Enquire whether the
following propositions are real or verbal: (*a*) Homer wrote the
*Iliad*, (*b*) Milton wrote *Paradise Lost*. [C.]

35. How would you
characterise a proposition which is *formally* inferred from the
conjunction of a *verbal* proposition with a *real material*
proposition? Explain your view by the aid of an illustration. [J.]

36. If all *x* is
*y*, and some *x* is *z*, and *p* is the name of
those *z*’s which are *x* ; is it a verbal proposition
to say that all *p* is *y*? [V.]

37. Is it possible to make
any term whatever the subject (*a*) of a verbal proposition,
(*b*) of a real proposition? [J.]

38. *Positive and Negative
Names*.—A pair of names of the forms *A* and
*not-A* are commonly described as positive and negative
respectively. The true import of the negative name *not-A*,
including the question whether it really has any signification at all,
has, however, given rise to much discussion.

Strictly speaking neither affirmation nor negation has any meaning except in reference to judgments or propositions. A concept or a term cannot be itself either affirmed or denied. If I affirm, it must be a judgment or a proposition that I affirm; if I deny, it must be a judgment or a proposition that I deny.

Starting from this position, Sigwart is led to the conclusion that,
“taken literally, the formula *not-A*, where *A*
denotes any idea, has no meaning whatever” (*Logic*, I. p.
134). Apart from the fact that the mere absence of an idea is not
itself an idea, *not-A* cannot be interpreted to mean the
*absence* of *A* in thought; for, on the contrary, it
implies the *presence* of *A* in thought. We cannot, for
instance, think of *not-white* except by thinking of
*white*. Nor again can we interpret *not-A* as denoting
whatever does not necessarily accompany *A* in thought. For, if
so, *A* and *not-A* would not as a rule be exclusive or incompatible. For example, *square*, *solid*, do not
necessarily accompany *white* in thought; but there is no
opposition between these ideas and the idea of *white*. In order
to interpret *not-A* as a real negation we must, says Sigwart, tacitly introduce a judgment or rather a series of judgments, 58 meaning by *not-A* “whatever is not *A*,” that is, everything whatsoever of which *A* must be denied. “I must review in thought all
possible things in order to deny *A* of them, and these would be
the positive objects denoted by *not-A*. But even if there were any use in this, it would be an impossible task” (p. 135).

Whilst agreeing with much that Sigwart says in this connexion, I cannot altogether accept his conclusion. We shall return to the question from the more controversial point of view in the following section. In the meantime we may indicate the result to which Sigwart’s general argument really seems to lead us.

We must agree that *not-A* cannot be regarded as representing
any independent concept; that is to say, we cannot form any idea of
*not-A* that negates the notion *A*. It is, therefore, true
that, taken literally (that is, as representing an idea which is the
pure negation of the idea *A*), the formula *not-A* is
unintelligible. Regarding *not-A*, however, as equivalent to *whatever is not A*, we may say that its justification and
explanation is to be found primarily by reference to the
*extension* of the name. The thinking of anything as *A*
involves its being distinguished from that which is not *A*. Thus
on the extensive side every concept divides the universe with
reference to which it is thought (whatever that may be) into two
mutually exclusive subdivisions, namely, a portion of which *A*
can be predicated and a portion of which *A* cannot be
predicated. These we designate *A* and *not-A* respectively.
While it may be said that *A* and *not-A* involve
*intensively* only one concept, they are *extensively*
mutually exclusive.

Confining ourselves to connotative names, we may express the
distinction between positive and negative names somewhat differently
by saying that a *positive* name implies the *presence* in
the things called by the name of a certain specified attribute or set
of attributes, while a *negative* name implies the *absence*
of one or other of certain specified attributes. A negative name,
therefore, has its denotation determined indirectly. The class denoted
by the positive name is determined positively, and then the negative
name denotes what is left.

59 39. *Indefinite Character
of Negative Names*.—*Infinite* and *indefinite* are
designations that have been applied to negative names when interpreted
in such a way as not to involve restriction to a limited universe of
discourse. For without such restriction (explicit or implicit) a
negative name, for example, *not-white*, must be understood to
denote the whole infinite or indefinite class of things of which
*white* cannot truly be affirmed, including such entities as
virtue, a dream, time, a soliloquy, New Guinea, the Seven Ages of Man.

Many logicians hold that no significant term can be really infinite
or indefinite in this way.66 They say that if a
term like *not-white* is to have any meaning at all, it must be
understood as denoting, not all things whatsoever except white things,
but only things that are black, red, green, yellow, etc., that is, all
*coloured* things except such as are white. In other words, the universe
of discourse which any pair of contradictory terms *A* and
*not-A* between them exhaust is considered to be necessarily
limited to the proximate genus of which *A* is a species; as, for
example, in the case of *white* and *not-white*, the
universe of colour.

^{66} This is at the root of
Sigwart’s final difficulty with regard to negative names, as
indicated in the preceding section. Later on he points out that in
division we are justified in including negative characteristics of the
form *not-A* in a concept, although we cannot regard *not-A*
itself as an independent concept. Thus we may divide the concept
*organic being* into *feeling* and *not-feeling*, a
specific difference being here constituted by the absence of a
characteristic which is compatible with the remaining characteristics,
but is not necessarily connected with them (*Logic*, I. p. 278). Compare also Lotze, *Logic*, § 40.

It is doubtless the case that we seldom or never make use of
negative names except with reference to some proximate genus. For
instance, in speaking of *non-voters* we are probably referring
to the inhabitants of some town or locality whom we subdivide into
those who have votes and those who have not. In a similar way we
ordinarily deny *red* only of things that are coloured,
*squareness* only of things that have some figure, etc., so that
there is an implicit limitation of sphere. It may be granted further
that a proposition containing a negative name interpreted as infinite
can have little or no practical value. But it does not follow that
some limitation 60 of
sphere is necessary in order that a negative term may have
*meaning*. The argument is used that it is an utterly impossible
feat to hold together in any one idea a chaotic mass of the most
different things. But the answer to this argument is that we do not
profess to hold together the things denoted by a negative name by
reference to any positive elements which they may have in common: they
are held together simply by the fact that they all lack some one or
other of certain determinate elements. In other words, the argument
only shews that a *negative* name has no *positive* concept
corresponding to it.67 It may be added that if this argument had
force, it would apply also to the subdivision of a genus with
reference to the presence or absence of a certain quality. If we
divide coloured objects into *red* and *not-red*, we may say
equally that we cannot hold together coloured objects other than red
by any positive element that they have in common: the fact that they
are all coloured is obviously insufficient for the purpose.

^{67} For a good statement of the
counter-argument, compare Mrs Ladd Franklin in *Mind*, January,
1892, pp. 130, 1.

A somewhat different argument is implied by Sigwart when he says,
“If *A* = *mortal*, where will justice, virtue, law,
order, distance find a place? They are neither *mortal beings*,
nor yet *not-mortal* beings, for they are not beings at
all.” The answer seems clear. They are *not*-(*mortal
beings*), and therefore *not-A*. As a rule, it is needless to exclude explicitly from a species what does not even belong to some higher genus. But the fact of the exclusion remains.

Granting then that in practice we rarely, if ever, employ a
negative name except with reference to some proximate genus, we
nevertheless hold that *not-A* is perfectly intelligible whatever
the universe of discourse may be and however wide it may be. For it
denotes in that universe whatever is not denoted by the corresponding
positive name. Moreover in formal processes we should be unnecessarily
hampered if not allowed to pass unreservedly from *X is not A* to
*X is not-A*.68

^{68} Writers who take the view
which we are here criticising must in consistency deny the universal
validity of the process of immediate inference called obversion. Thus
Lotze, rightly on his own view, will not allow us to pass from
*spirit is not matter* to *spirit is not-matter* ; in fact he
rejects altogether the form of judgment *S is not-P*
(*Logic*, § 40). Some writers, who follow Lotze on the general
question here raised, appear to go a good deal further than he does,
not merely disallowing such a proposition as *virtue is not-blue*
but also such a proposition as *virtue is not blue*, on the
ground that if we say “virtue is not blue,” there is no
real predication, since the notion of colour is absolutely foreign to
an unextended and abstract concept such as “virtue.”
Lotze, however, expressly draws a distinction between the two forms
*S is non-Q* and *S is not Q*, and tells us that
“everything which it is wished to secure by the affirmative
predicate *non-Q* is secured by the intelligible negation of
*Q*” (*Logic*, § 72; cf. § 40). On the more extreme
view it is wrong to say that *Virtue is either blue or it is not
blue* ; but Lotze himself does not thus deny the universality of the
law of excluded middle.

61 From this point of view attention may be called to the difference
in ordinary use between such forms as *unholy*, *immoral*,
*discourteous* and such forms as *non-holy*,
*non-moral*, *non-courteous*. The latter *may* be used with
reference to any universe of discourse, however extensive. But not so
the former; in their case there is undoubtedly a restriction to some
universe of discourse that is more or less limited in its range. We
can, for example, speak of a *table* as *non-moral*,
although we cannot speak of it as *immoral*. A want of
recognition of this distinction may be partly responsible for the
denial that any terms can properly be described as infinite or
indefinite.69

^{69} It should be added that in
the ordinary use of language the negative prefix does not always make
a term negative as here defined. Thus, as Mill points out, “the
word *unpleasant*, notwithstanding its negative form, does not
connote the mere absence of pleasantness, but a less degree of what is
signified by the word *painful*, which, it is hardly necessary to
say, is positive.” On the other hand, some names positive in
form may, with reference to a limited universe of discourse, be
negative in force; *e.g.*, *alien*, *foreign*. Another
example is the term *Turanian*, as employed in the science of
language. This term has been used to denote groups lying outside the
Aryan and Semitic groups, but not distinguished by any positive
characteristics which they possess in common.

40. *Contradictory
Terms*.—A positive name and the corresponding negative are
spoken of as *contradictory*. We may define contradictory terms
as a pair of terms so related that between them they exhaust the
entire universe to which reference is made, whilst in that universe
there is no individual of which both can be affirmed at the same time.
It is desirable to repeat here that contradiction can exist primarily
between 62 judgments
or propositions only, so that as applied to terms or ideas the notion
of contradiction must be interpreted with reference to predication.
*A* and *not-A* are spoken of as contradictory because they
cannot without contradiction be predicated together of the same
subject. Thus it is in their exclusive character that they are termed
contradictory; as between them exhausting the universe of discourse
they might rather be called *complementary*.70

^{70} Dr Venn (*Empirical
Logic*, p. 191) distinguishes between *formal contradictories*
and *material contradictories*, according as the relation in
which the pair of terms stand to one another is or is not apparent
from their mere form. Thus *A* and *not-A* are formal
contradictories; so are *human* and *non-human*. Material
contradictories, on the other hand, are not constructed “for the
express purpose of indicating their mutual relation.” No formal
contradiction, for example, is apparent between *British* and
*Foreign*, or between *British* and *Alien* ; and yet
“within their range of appropriate application—which in
the latter case includes persons only, and in the former case is
extended to produce of most kinds—these two pairs of terms
fulfil tolerably well the conditions of mutual exclusion and
collective exhaustion.”

41. *Contrary
Terms*.—Two terms are usually spoken of as *contrary*71 to
one another when they denote things which can be regarded as standing
at opposite ends of some definite scale in the universe to which
reference is made; *e.g.*, *first* and *last*,
*black* and *white*, *wise* and *foolish*,
*pleasant* and *painful.*72 Contraries differ
from contradictories in that they admit of a mean, and therefore do
not between them exhaust the entire universe of discourse. It follows
that, although two contraries cannot both be true of the same thing at
the same time, they may both be false. Thus, a colour may be neither
black nor 63 white,
but blue; a feeling may be neither pleasant nor painful, but
indifferent.

^{71} De Morgan uses the terms
contrary and contradictory as equivalent, his definition of them
corresponding to that given in the preceding section.

^{72} It has been already pointed
out that the negative prefix does not always make a term really
negative in force. Thus *pleasant* and *unpleasant* are not
contradictories, for they admit of a mean; when we say that anything
is unpleasant, we intend something more than the mere denial that it
is *pleasant*. It should be added that a pair of terms of this kind may
also fail to be contraries as above defined, since while admitting of
a mean they may at the same time not denote extremes. *Unpleasant*, for
example, denotes only that which is mildly painful: unless intended
ironically, it would be a misuse of terms to speak of the tortures of
the Inquisition as merely unpleasant. Compare Carveth Read,
*Logic*, p. 49.

It will be observed that not every term has a contrary as above
defined, for the thing denoted by a term may not be capable of being
regarded as representing the extreme in any definite scale. Thus
*blue* can hardly be said to have a contrary in the universe of
colour, or *indifferent* in the universe of feeling.

By some writers, the term *contrary* is used in a wider sense
than the above, contrariety being identified with simple
incompatibility (a mean between the two incompatibles being possible);
thus, *blue* and *yellow* equally with *black*, would
in this sense be called *contraries* of *white*.73 Other writers use
the term *repugnant* to express the mere relation of
incompatibility; thus *red*, *blue*, *yellow* are in
this sense *repugnant* to one another.74

^{73} There is much to be said in
favour of this wider use of the term *contrary*. Compare the
discussion of contrary propositions in section 81.

^{74} So long as we are confined to
simple terms the relations of contrariety and repugnancy cannot be
expressed *formally* or in mere symbols. But it is otherwise when
we pass on to the consideration of complex terms. Thus, while
*XY* and *not-X or not-Y* are formal contradictories,
*XY* and *X not-Y* may be said to be formal repugnants,
*XY* and *not-X not-Y* formal contraries (in the narrower of
the two senses indicated above).

42. *Relative
Names*.—A name is said to be *relative*, when, over and
above the object that it denotes, it implies in its signification
another object, to which in explaining its meaning reference must be
made. The name of this other object is called the *correlative*
of the first. Non-relative names are sometimes called *absolute*.

Jevons considers that in certain respects all names are relative.
“The fact is that everything must really have relations to
something else, the water to the elements of which it is composed, the
gas to the coal from which it is manufactured, the tree to the soil in
which it is rooted “ (*Elementary Lessons in Logic*, p.
26). Again, by the law of relativity, consciousness is possible only
in circumstances of change. We cannot think of any object except as
distinguished from something else. Every term, therefore, implies its
negative as an object 64 of thought. Take the term *man*. It is
an ambiguous term, and in many of its meanings is clearly
relative,—for example, as opposed to master, to officer, to
wife. If in any sense it is absolute it is when opposed to not-man;
but even in this case it may be said to be relative to not-man. To
avoid this difficulty, Jevons remarks, “Logicians have been
content to consider as relative terms those only which imply some
peculiar and striking kind of relation arising from position in time
or space, from connexion of cause and effect, &c.; and it is in
this special sense, therefore, that the student must use the
distinction.”

A more satisfactory solution of the difficulty may be found by
calling attention to the distinction already drawn between the point
of view of connotation (which has to do with the signification of
names) and the subjective and objective points of view respectively.
From the subjective point of view all notions are relative by the law
of relativity above referred to. Again, from the objective point of
view all things, at any rate in the phenomenal world, are relative in
the sense that they could not exist without the existence of something
else; *e.g.*, man without oxygen, or a tree without soil. But
when we say that a *name* is relative, we do not mean that what
it denotes cannot exist or be thought about without something else
also existing or being thought about; we mean that its signification
cannot be explained without reference to something else which is
called by a correlative name, *e.g.*, *husband*,
*parent*. It cannot be said that in this sense all names are
relative.

The fact or facts constituting the ground of both correlative names
is called the *fundamentum relationis*. For example, in the case
of partner, the fact of partnership; in the case of husband and wife,
the facts which constitute the marriage tie; in the case of ruler and
subject, the control which the former exercises over the latter.

Sometimes the relation which each correlative bears to the other is the same; for example, in the case of partner, where the correlative name is the same name over again. Sometimes it is not the same; for example, father and son, slave-owner and slave. 65

The consideration of relative names is not of importance except in connexion with the logic of relatives, to which further reference will be made subsequently.

43. Give one example of each of the following,—(i) a collective general name, (ii) a singular abstract name, (iii) a connotative singular name, (iv) a connotative abstract name. Add reasons justifying your example in each case. [K.]

44. Discuss the logical
characteristics of the following names:—*beauty*,
*fault*, *Mrs Grundy*, *immortal*, *nobility*,
*slave*, *sovereign*, *the Times*, *truth*,
*ungenerous*. [K.]

[In discussing the character of any name it is necessary first of
all to determine whether it is *univocal*, that is, used in one
definite sense only, or *equivocal* (or *ambiguous*), that
is, used in more senses than one. In the latter case, its logical
characteristics may vary according to the sense in which it is used.]

45. It has been maintained that the doctrine of terms is extra-logical. Justify or controvert this position. [J.]

46. *Judgments and
Propositions*.—In passing to the next division of our subject
we are confronted, first of all, with a question which is partly, but
not entirely, a question of phraseology. Shall we speak of the
*judgment* or of the *proposition*? The usage of logicians
differs widely. Some treat almost exclusively of judgments; others
almost exclusively of propositions. It will be found that for the most
part the former are those who tend to emphasise the psychological or
the metaphysical aspects of logic, while the latter are those who are
more inclined to develop the symbolic or the material aspects.

To a certain extent it is a matter of little importance which of
the alternatives is ostensively adopted. Those who deal with judgments
from the logical standpoint must when pressed admit that they can deal
with them only as expressed in language, and all their illustrations
necessarily consist of judgments expressed in language. But a
*judgment expressed in language* is precisely what is meant by a
*proposition*. Hence in treating of judgments it is impossible
not to treat also of propositions. 67

On the other hand, so far as we treat of propositions in logic, we
treat of them not as grammatical sentences, but as assertions, as
verbal expressions of judgments. The logical proposition is the
proposition as understood; and a *proposition as understood* is a
*judgment*. Hence in treating of propositions in logic we
necessarily treat also of judgments.

In a large degree, then, the problem does resolve itself into a merely verbal question. At the same time, reasons and counter-reasons may be adduced in favour of the one alternative and in favour of the other.

On the one side, it is said that the use of the term
*proposition* tends to confuse the sentence as a grammatical
combination of words with the proposition as apprehended and
intellectually affirmed; and it is urged that in treating of
propositions the logician tends to become a mere grammarian.

On the other side, it is submitted that the logician is primarily
concerned, not with the *process* of judgment, the discussion of
which belongs to the sphere of psychology, but with judgment as a
*product*, and moreover that he is concerned with this product only in
so far as it assumes a fixed and definite form, which it cannot do
until it receives verbal expression; and it is urged that if we
concentrate our attention on judgments without explicit regard to
their expression in language, our treatment tends to become too
psychological.

It has been said above that logically we can deal with judgments only as expressed in propositions; and no doubt all judgments can with more or less effort be so expressed. But as a matter of fact we constantly judge in a vague sort of way without the precision that is necessary even in loose modes of expression, and we find that to give expression to our judgments may sometimes require very considerable effort. It must be remembered that logic has in view an ideal. Its object is to determine the conditions to which valid judgments must conform, and it is concerned with the characteristics of actual judgments only in subordination to this end. From this point of view it is specially important that we should deal with judgments in the only form in which it is possible for them to attain precision; and this consideration appears to be conclusive in favour of our 68 treating explicitly of propositions in some part at any rate of a logical course.

No doubt in dealing with propositions we have to raise certain questions that relate to the usage of language. Unfortunately the same propositional form may be understood as expressing very different judgments. It is therefore requisite that in any scientific treatment of logic we should discuss the interpretation of the propositional forms that we recognise. This problem is akin to the problem of definition which has to be faced sooner or later in every science; and, as is also true of a definition, the solution in any particular case is largely of the nature of a convention. But this does not detract from its importance as conducing to clearness of thought.

The question of the interpretation of propositional forms is as a matter of fact one that cannot be altogether avoided on any treatment of logic; and it is of importance to recognise explicitly that in discussing this question we are not dealing with judgments pure and simple. Words are like mathematical symbols, and the meaning of a given form of words is not something inherent either in the words themselves or in the thoughts that they may represent, but is dependent on a convention established by those who employ the words. In the force of a given judgment, however, there can be nothing that is dependent on convention. This distinction is not always remembered by those who confine their attention mainly to judgments, and they are consequently sometimes led to express themselves with an appearance of dogmatism on questions that do not really admit of dogmatic treatment.

But while in certain aspects of logical enquiry it is requisite to
deal explicitly with *propositions*, it must never be forgotten that as
logicians we are concerned with propositions only as the expressions
of judgments; and there are numerous occasions when we have to go
behind propositional forms and ask what are the fundamental
characteristics of the judgments that they express.

47. *The Abstract Character
of Logic*.—Reference has been made in the preceding section
to the necessity for logical purposes of making our judgments precise.
For only if they 69
are precise is it possible to determine with accuracy what are their
logical implications considered either individually or in conjunction
with one another. It has also been pointed out that we can make our
judgments precise only by expressing them in propositional forms, the
interpretation of which has been agreed upon.

But this is not without its disadvantages. Sometimes the full force of an actual judgment hardly admits of being expressed in words, and even the force of a proposition as understood may not be found exclusively in the words of which it composed, but may depend partly on the context in which it is placed. Hence the isolated proposition must frequently be regarded as in a sense an abstraction, leaving behind it some portion of the actual judgment for which it stands.

This is indeed much less true of the propositions of science than of those of everyday life; and the more fully a statement is independent of context the more fully may it be regarded as fulfilling its purpose from the scientific standpoint. Still the abstract character of logic must be frankly recognised. “Just as thought is abstract in its dealings with reality, so logic is abstract in its dealings with ordinary thought.”75

^{75} Hobhouse, *Theory of
Knowledge*, p. 7.

That they are in some degree abstractions is true not only of propositions, but also of inferences, as we have to deal with them in logic. Much of the reasoning of everyday life does not admit of expression in the form of definite premisses and conclusions such as would satisfy the canons of logic. The grounds upon which our conclusions are based are often so complex, and the influence which some of them exert upon our beliefs is so subtle and delicate, that they cannot be completely set forth. This will be realised at once if an attempt is made to apply the rules of logic to any ordinary inference; and an explanation is herein found why the illustrations given in logical text-books frequently appear so artificial and unreal.

It must be admitted that the abstract character of logic detracts to some extent from its utility as an art, though the extent of this drawback may easily be exaggerated. Regarded as a science, however, the value, of logic remains unimpaired. 70 Other sciences besides logic have to proceed by abstractions and separations that do not fully correspond to the complexities of nature; and this often becomes the more true the higher the stage that the science has reached. Its necessary abstractness does not prevent logic from analysing successfully the characteristics of the developed judgment or from determining the principles of valid reasoning. If we were to seek to treat logical problems without abstraction we should be in danger of destroying the scientific character of logic without achieving any valuable result even from the purely utilitarian point of view. It is of little value to criticise received systems without providing any new constructive system in their place.

48. *Nature of the Enquiry
into the Import of Propositions*.—Under the general head of
the *import of propositions* it is usual to include problems that
are really very different in character.76

^{76} Compare Mr W. E. Johnson in
*Mind*, April, 1895, p. 242.

(1) There is, in the first place, the fundamental problem or series
of problems as to what are the *essential characteristics of
judgments*, and therefore of propositions as expressing judgments.
The discussion of questions of this character must be based directly
on psychological or philosophical considerations, and in the solutions
nothing arbitrary or conventional can find a place.

Under this head are to be included such problems as the following: Do all judgments contain a reference to reality? In what sense, if any, can all judgments claim to possess universality or necessity? What is the nature of significant denial? Are distinctions of modality subjective or objective?

(2) In the *interpretation of propositional forms* we have an
enquiry of a very different character, an enquiry which relates
distinctively to propositions, and not to judgments considered apart
from their expression. The problem is indeed to determine what is the
precise judgment that a given proposition shall be understood to
express; and, in consequence of the uncertainty and ambiguity of
ordinary language, the solution of the problem includes an optional or
selective element.

71 As a simple illustration of the kind of problem that we here have
in view, we may note that in the traditional scheme of propositions,
*All S is P*, *No S is P*, *Some S is P*, *Some S is
not P*, the signs of quantity have to be interpreted. The
existential and modal import of these propositions is also partly a
question of interpretation.

In connexion with the interpretation of propositions, the
distinction between *meaning* and *implication* has to be
considered. What we do in interpreting propositions is to assign to
them a meaning; and when the meaning has once been fixed, the
implications are determined in accordance with logical principles.

The dividing line between meaning and implication is not in
practice always easy to draw, and some writers seek to ignore it by
including within the scope of meaning all the implications of a
proposition. But this is a fatal error. The assignment of meaning is
within certain limits arbitrary and selective. But if element *a*
necessarily involves element *b*, then *a* having been
assigned as part of the meaning of a given propositional form, it is
no question of meaning as to whether the form in question does or does
not imply *b*, and there is nothing arbitrary or selective in the
solution of this question.

Sometimes the elements *a* and *b* mutually involve one
another. It may then be a question of interpretation whether *a*
shall be included in meaning, *b* thus becoming an implication,
or whether *b* shall be included in meaning, *a* becoming an
implication.

A failure to recognise what is really the point at issue in a case
like this has sometimes caused discussions to take a wrong turn. Thus
the question is raised whether the import of the proposition *All S
is P* is that the class *S* is included in the class *P*,
or that the set of attributes *S* is invariably accompanied by
the set of attributes *P* ; and these are regarded as antagonistic
theories. If the implications of a proposition are regarded as part of
its import, then the proposition may be said to import both these
things. But if by the import of a proposition we intend to signify its
meaning only, then we may adopt an interpretation that will make
either of them (but not both) part of its import, or our
interpretation may be such 72 that the proposition imports neither of
them. The question here raised is dealt with in more detail later on.

(3) A third problem, distinct from both those described above, arises in connexion with the expression of judgments in propositional form.

In ordinary discourse we meet with an infinite variety of forms of statement. To recognise and deal separately with all these forms in our treatment of logical problems would, however, be impracticable. We have, therefore, in some at any rate of our discussions, to limit ourselves to a certain number of selected forms; and in such discussions we have to assume that the judgments with which we are dealing are at the outset expressed in one or other or a combination of these selected forms.

This reduction of a statement to some canonical form has been
called by Mr Johnson its *formulation*.

A given statement, since it involves many different relations which
mutually implicate one another, may be formulated in a number of
different ways; and it is needless to say that there is no one scheme
of formulating propositions that we are bound to accept to the
exclusion of others. Different schemes are useful for different
purposes, and several schedules of propositions (for example,
equational and existential schedules) will presently be considered in
addition to the traditional fourfold schedule. It should be added that
a given scheme may profess to cover part only of the field. Thus the
traditional schedule (*All S is P*, etc.) professes to be a
scheme for categorical judgments only, and (as traditionally
interpreted) for assertoric judgments only.

With reference to the reduction of a statement to a form in which it belongs to a given schedule two points call for notice.

(*a*) There is danger lest some part of the force of the
original statement may be lost.

To a certain extent this is inevitable, especially if the original statement contains suggestion or innuendo in addition to what it definitely affirms; and this must be taken in connexion with what has already been said about the abstract character of logic. If, however, there is any substantial loss of 73 import, the scheme stands condemned so far as it professes to be a complete scheme of formulation. It may, as we have seen, not profess to be a complete scheme, but only to formulate statements falling within a certain category, for example, assertoric statements or categorical statements.

It is to be added that a statement which does not admit of being
translated into any one of the simple forms included in a given scheme
may still be capable of being expressed by a conjunctive or
disjunctive combination of such simple forms. Thus, if the statement
*Some S is P* is made with an emphasis on *some*, implying
*not all*, then the statement cannot be expressed in any one of
the forms of the traditional schedule of propositions, but it is
equivalent to *Some S is P and some S is not P*.

(*b*) In the reduction of a statement to a form in which it
belongs to a given schedule there may be involved what must be
admitted to be *inference*. As, for instance, if statements are
given in the ordinary predicative form and have to be expressed in an
equational scheme.

It may perhaps be urged that this is legitimate, simply on the
ground that one of the postulates of logic is that we be allowed to
substitute for any given form of words the technical form (and in an
equational system this will be an equation) which is equivalent to it.
Have we not, however, in reality a vicious circle if a process which
involves inference is to be regarded as a *postulate* of logic?

The difficulty here raised is a serious one only if we suppose ourselves rigidly limited in logic to a single scheme of formulation; and the solution is to be found in our not confining ourselves to any one scheme, but in our recognising several and investigating the logical relations between them. We can then refuse to regard any substitution of one set of words for another as pre-logical except in so far as it consists of a merely verbal transformation: and our postulate will merely be that we are free to make verbal changes as we please; it will not by itself authorise any change of an inferential character. For a change of this kind, appeal must be made to logical principles.

74 We have then in this section distinguished between three problems
any or all of which may be involved in discussions concerning the
import of propositions. We have

(1) the discussion of the essential nature of judgments and of the
fundamental distinctions between judgments;

(2) the interpretation of
propositional forms;

(3) the discussion and comparison of logical
schedules or schemes of propositions, drawn up with a view to the
expression of judgments in a limited number of propositional forms.

These problems are inter-related and do not admit of being discussed in complete isolation. It is clear, for instance, that the drawing up of a schedule of propositions needs to be supplemented by the exact interpretation of the different forms which it is proposed to recognise; and both in the drawing up of the schedule and in the interpretation we shall be guided and controlled by a consideration of fundamental distinctions between judgments.

The problems are, however, in themselves distinct; and some misunderstanding may be avoided if we can make it clear what is the actual problem that we are discussing at any given point.

In particular, it is important to recognise that in the formulation and interpretation of propositions there is an arbitrary and selective element which is absent from the more fundamental problem. Systems of formulation and interpretation, therefore, if only they are intelligible and self-consistent, can hardly be condemned as radically wrong, though they may be rejected as inconvenient or unsuitable. When, however, we are dealing with the fundamental import of judgments, the questions raised do become questions of absolute right or wrong.

It should be added that in the present treatise, since it is concerned with logic in its more formal aspects, questions of interpretation and formulation occupy a position of greater relative importance than they would in a treatment of the science more fully developed on the philosophical side.

49. *The Objective
Reference in Judgments*.—A judgment can be formed or
understood only through the occurrence of certain psychical events in
the minds of those who form or 75 understand it; and in this sense it may be
included amongst subjective states. It is, however, distinguished from
all other subjective states by the fact that it *claims to be
true*.

This claim to be true implies an objective reference. For a merely subjective state is not, as such, either true or false; it is simply an occurrence. Thus, the distinction between truth and falsity is inapplicable to an emotion or a volition. An emotion may be pleasurable or painful; it may be strong or weak; it may or may not impel to action; but we cannot describe it as true or false.

And the same applies to a judgment regarded as no more than a
subjective connexion of ideas. The claim to truth necessarily involves
more than this, namely, a reference to something external to the
psychical occurrence involved in the formation of the judgment. Every
judgment implies, therefore, on the part of the judging mind, the
recognition of an objective system of reality of some sort. The
validity that is claimed for judgment is an *objective validity*.

The word “objective” is always a dangerous word to use,
and some further explanation may be given of the meaning to be
attached to it here. When we say that a judgment refers to an
*objective* system, we mean a system that subsists independently
of the act of judgment itself, and that is not dependent on the
passing fancy of the person who forms the judgment. An objective
system of reality in this sense may, however, include subjective
states, that is, states of consciousness. A body of psychological
doctrine consists of judgments relating to states of mind. But such
judgments have an external reference (that is, external to the
judgments themselves) just as much as a body of judgments relating to
material phenomena. Indeed the doctrine of judgment here laid down is
not inconsistent with the theory of subjective idealism that resolves
all phenomena into states of consciousness.

Even when a judgment relates to purely fictitious objects there is still an external reference,—in this case, to the world of convention.

The particular aspect or portion of the total system of reality
referred to in any judgment may sometimes be 76 conveniently spoken of as the *universe of
discourse*. The limits, if any, intended to be placed upon the
universe of discourse in any given proposition are usually not
explicitly stated; but they must be considered to be implicit in the
judgment which the proposition is meant to express, and to be capable
of being themselves expressed should there be any danger of
misunderstanding. At the same time, it is only fair to add that
attempts to define the universe of discourse are likely to raise
metaphysical difficulties as to the ultimate nature of reality. What
is of main importance from the logical standpoint is the recognition
that there is a reference to *some* system of reality which is to
be distinguished from the uncontrolled course of our own ideas. And so
far as a distinction can be drawn between different systems of
reality, there is need of the assumption that, when we combine
judgments or view them in their mutual relations, the universe of
discourse is the same throughout.

50. *The Universality of
Judgments*.—The fundamental characteristic then of judgments
is their objective reference, their claim to objective validity. It
follows that all judgments claim *universality*, that is to say,
they claim to be acknowledged as true not for a given person only, or
for a limited number of persons, but for everyone; and again, not for
a given time only, or for a limited time, but for all time. In other
words, the import of a judgment is not merely to express some
connexion of ideas in my own mind; but to express something that
claims to be *true*. And truth is not relative to the individual,
nor is it when fully set forth limited by considerations of time.

We shall have subsequently to deal with the ordinary distinction
between universal and particular propositions; but it will be clear
that the claim to universality which we are now considering is one
that must be made on behalf of so-called particular, as well as of
so-called universal, propositions. The judgment that *some men are
six feet in height* claims universal acceptance just as much as the
judgment that *all men are mortal*.

Some judgments again contain an explicit or implicit reference to
time. But this is really part of the judgment. As 77 soon as the judgment is
fully stated it becomes independent of time. It may perhaps be said
that the judgment *France is under Bourbon rule* was true two
centuries ago, but is not true now. But the judgment as it stands,
without context, is incompletely stated. That France is (or was) under
Bourbon rule in the year 1906 A.D. is for
all time false; that France is (or was) under Bourbon rule in the year
1706 A.D. is for all time true.

In regard to the nature and significance of the reference to time
in judgments, Mr Bosanquet draws a useful distinction between the time
*of* predication and the time *in* predication.77 By
the time *of* predication is meant the time at which some
thinking being makes the judgment; and this in no way affects the
truth of the judgment. But, as Sigwart points out, everything which
exists as a particular thing occupies a definite position in time.
Hence all judgments relating to particular things, including singular
judgments and so-called narrative judgments, relate to some definite
time, past, present, or future, with reference to which alone the
statements made are valid. This is the time *in* predication, and
the reference to it must be regarded as an intrinsic part of the
judgment itself, although it is not always explicitly mentioned.

^{77} *Logic*, I. p. 215.
Compare Sigwart, *Logic*, § 15.

It will be seen that the recognition of the universality of all judgments in the sense here indicated is but the recognition in another aspect of their objective character.

51. *The Necessity of
Judgments*.—A further characteristic that has been ascribed
to all judgments, when considered in relation to the judging mind, is
*necessity*. This too is connected with the claim to objective
validity. When we judge, we are not free to judge as we will. No doubt
by controlling the intellectual influences to which we subject
ourselves we may indirectly and in the long run modify within certain
limits our beliefs. This is a question belonging to psychology into
which we need not now enter. But at any given moment the judgments we
form are determined by our mental history and the circumstances in
which we are placed. We are bound to judge as we do judge; so far as
we feel a question to be an 78 open one our judgment regarding it is
suspended. It must be granted that we not unfrequently make statements
which do not betray the doubts which as a matter of fact we feel with
regard to the point at issue; but such statements do not represent our
real judgments. The propositions we utter are the expressions of
possible judgments, but not of *our* judgments.

In any discussion of the modality of judgments, other senses in
which the term “necessary” may be applied to judgments
have to be considered. In here affirming necessity as a characteristic
of *all* judgments, we are merely declaring over again in another
aspect their objective character. The merely subjective sequence of
ideas in our minds is more or less under our own control. At any rate
we can at will bring given ideas together in our mind. But a judgment
is more than a relation between ideas. It claims to be true of some
system of reality; and hence it is not so much determined *by*
us, as *for* us by the knowledge which we have come to possess or
think we have come to possess about that system of reality.

52. “What is once true
is always true.”

“What is true to-day may be false to-morrow.”

Examine these statements. [L.]

53. *The Classification of
Judgments*.—It is customary for logicians to offer a
classification of judgments or propositions. There is, however, so
much variation in the objects they have in view in drawing up their
classifications, that very often their results are not really
comparable.

(1) Our object in classifying propositions may, in the first place,
be to produce a working scheme for the formulation of judgments. An
illustration of this is afforded by the traditional scheme of
propositions (*All S is P*, *No S is P*, etc.), or by the
Hamiltonian scheme based upon the quantification of the predicate. A
classification of this kind is essentially formal. The different
propositional forms that are recognised must receive clearly defined
interpretations; and the resulting scheme, if it is worth anything at
all, will be orderly and compact. On the other hand, it is not likely
to be comprehensive or exhaustive; for many natural modes of judgment
will not find a place in it, at any rate until they have been
expressed in a modified, though as nearly as possible equivalent,
form.

There are many ways of formulating judgments, each of which has its special merits and is from some particular point of view specially appropriate. We must, however, give up the idea that any one of these ways can hold the field as a fundamental and essentially suitable classification of judgments looked at from the psychological point of view.

(2) From the psychological standpoint our endeavour must be to give rather what may be called a natural history 80 classification of judgments. Primitive types of judgment, which in a logical scheme of formulation are not likely to find a place at all, will now be regarded as of equal importance with more developed and scientific types. Our object may indeed be (as with Mr Bosanquet) to sketch the development of judgments from the most primitive types to those which give expression to the ideal of knowledge.

In a classification of this kind the dividing lines are not so clear and sharply defined as in a scheme framed for the logical formulation of judgments. The different types, moreover, do not stand out in marked distinction from one another, and it is difficult to arrange the different classes in due subordination, and with complete avoidance of cross divisions. The underlying plan is indeed apt to be obscured by details, so that the whole discussion tends to become somewhat cumbrous.

(3) A classification of propositions of still another kind is given by Mill in the later part of his chapter on the Import of Propositions. The conclusion at which he arrives is that every proposition affirms, or denies, either simple existence, or else some sequence, coexistence, causation, or resemblance. This classification is certainly not a formal one; it is not a scheme for the logical formulation of judgments. Nor, on the other hand, can it be regarded as a psychological classification of types of judgment, designed to illustrate the nature and growth of thought. Mill’s point of view is objective and material. In one place he describes his scheme as a classification of matters of fact, of all things that can be believed; and the main use that he subsequently makes of it is in connexion with the enquiry as to the methods of proof that are appropriate according to the nature of the matter of fact that is asserted.

In the pages that follow various schemes for formulating judgments will be considered. For reasons already stated, however, no scheme of this kind can be regarded as constituting an exhaustive classification of judgments. The traditional scheme, for example, is ludicrously unsatisfactory and incomplete if put forward as affording such a classification.

We shall not attempt to give what has been spoken of above as a natural history classification of judgments. The really 81 important distinctions involved in such a classification can be raised independently, and the general plan of this work is to dwell principally on the more formal aspects of logic. It may be added that even from a broader point of view the problem of the evolution of thought is hardly to be regarded as primarily a logical problem.

Again, such a classification as Mill’s involves material considerations that are outside the scope of this treatise.

Without, however, professing to give any complete scheme of classification, we shall endeavour to touch upon the most fundamental differences that may exist between judgments.

54. *Kant’s
Classification of Judgments*.—Kant classified judgments
according to four different principles (*Quantity*,
*Quality*, *Relation*, and *Modality*) each yielding
three subdivisions, as follows:

(1) | Quantity. |
(i) | Singular |
This S is P. |

(ii) | Particular | Some S is P. | ||

(iii) | Universal | All S is P. | ||

(2) | Quality. | (i) | Affirmative | All S is P. |

(ii) | Negative | No S is P. | ||

(iii) | Infinite |
All S is not-P. | ||

(3) | Relation. | (i) | Categorical | S is P. |

(ii) |
Hypothetical | If S is P then Q is R. | ||

(iii) |
Disjunctive | Either S is P or Q is R. | ||

(4) | Modality. | (i) | Problematic | S may be P. |

(ii) | Assertoric | S is P. | ||

(iii) | Apodeictic | S must be P. |

This arrangement is open to criticism from several points of view; and its symmetry, although attractive, is not really defensible. At the same time it has the great merit of making prominent what really are the fundamental distinctions between judgments.

The first distinction that we shall consider is that between simple and compound judgments (replacing Kant’s distinction according to relation). We shall then consider in turn distinctions of modality, of quantity, and of quality. 82

55. *Simple Judgments and
Compound Judgments*.—Under the head of *relation*, Kant
gave the well-known threefold division of judgments into
*categorical*, where the affirmation or denial is absolute (*S
is P*); *hypothetical* (or *conditional*), where the
affirmation or denial is made under a condition (*If A is B then S
is P*); and *disjunctive*, where the affirmation or denial is
made with an alternative (*Either S is P or Q is R*).

These three kinds of judgment cannot, however, properly be
co-ordinated as on an equality with one another in a threefold
division. For the categorical judgment appears as an element in both
the others, and hence the distinction between the categorical, on the
one hand, and the hypothetical and the disjunctive, on the other,
appears to be on a different level from that between the two latter.
Moreover, the hypothetical and the disjunctive do not exhaust the
modes in which categorical judgments may be combined so as to form
further judgments. It is, therefore, better not to start from the
above threefold division, but from a twofold, namely, into
*simple* and *compound*.

A *compound judgment* may be defined as a judgment into the
composition of which other judgments enter as elements.78
There are three principal ways in which judgments may be combined, and
in each case the denial of the validity of the combination yields a
further form of judgment, so that there are six kinds of compound
judgments to be considered.

^{78} The distinction here implied
has been criticised on the ground that (*a*) if the so-called
elements are really judgments, the combination of them yields no fresh
judgment; while (*b*) if the combination is really an independent
judgment, the elements into which it can be analysed are not
themselves judgments. It will be seen that (*a*) is intended to
apply to conjunctive syntheses, and (*b*) to hypotheticals and
disjunctives. We shall consider the argument under these heads
severally.

(1) We may affirm two or more simple judgments together. Thus,
given that *P* and *Q* stand separately for judgments, we
may affirm “*P and Q*.”

It has been held that a synthesis of two independent judgments in
this way does not really yield any fresh judgment distinct from the
two judgments themselves.79 In a sense this is true. Anyone may, however,
be challenged for holding two 83 judgments together on grounds which would
have no application to either taken separately. Hence it is convenient
to regard the combination as constituting a distinct logical whole,
which demands some kind of separate treatment; and on this ground the
description of “*P and Q*” as a compound judgment may
be justified.

^{79} Compare Sigwart, *Logic*, i. p. 214.

The synthesis involved is conjunctive. Hence *P and Q* may be
spoken of more distinctively as a *conjunctive judgment*. Its
denial yields “*Not both P and Q*” and this form is
more truly disjunctive than the form (*P or Q*) to which that
designation is more commonly applied.

(2) Without committing ourselves to the affirmation of either
*P* or *Q* we may hold them to be so related that the truth
of the former involves that of the latter. This yields the
hypothetical judgment, “*If P then Q*.”

It has been held that to regard this as a combination of judgments,
and to speak of it as in this sense a compound judgment, is
misleading, since P and Q are here not judgments at all, that is to
say, they are not at the moment intended as statements. Neither
*P* nor *Q* is affirmed to be true. What is affirmed to be
true is a certain relation between them.80

^{80} Compare Sigwart, *Logic*, i. p. 219.

It is certainly the case that when I judge “*If P then
Q*,” *P* need not be *my* judgment, nor need *Q* ; my
object may even be to establish the falsity of *P* on the ground
of the known falsity of *Q*. A more impersonal view, however,
being taken, *P* and *Q* are suppositions, that is, possible
judgments, so that they have meaning as judgments; and *If P then
Q* may fairly be said to express a relation between judgments in
the sense of its force being that the acceptance of *P* as a true
judgment involves the acceptance of *Q* as a true judgment also.
The description of the hypothetical judgment as compound appears
therefore to be in this sense justified. Such a judgment as *If P
then Q* cannot be interpreted except on the supposition that
*P* and *Q* taken separately have meaning as judgments.

As we get a compound judgment when we declare two judgments to be
so related that if one is accepted the other must be accepted also, so
we get a compound judgment when 84 we deny that this relation subsists between
them. Thus in addition to the judgment “*If P then
Q*,” we have its denial, namely, “*If P then not
necessarily Q*.”81 The best mode of describing this form of
proposition will be considered in a subsequent chapter.

^{81} In giving this as the
contradictory of *If P then Q*, we are assuming a particular
doctrine of the import of the hypothetical judgment. The question will
be discussed more fully later on.

(3) We have another form of compound judgment when we affirm that
*one or other* of two given judgments is true. This form of
judgment, “*P or Q*,” is usually called
*disjunctive*, though *alternative* would be a better name.
It has been already pointed out that *Not both P and Q* is the
more distinctively disjunctive form.

It may be denied that *P or Q* is a compound judgment on the
same grounds as those on which this is denied of *If P then Q*.
Since, however, the points at issue are practically the same as
before, the discussion need not be repeated.

The denial of “*P or Q*” yields “*Neither
P nor Q*.” This may be called a *remotive* judgment if a
distinctive name is wanted for it.

It should be added that not all forms of proposition which would
ordinarily be described as hypothetical or disjunctive are really the
expressions of compound judgments as above described. Thus the forms
*If any S is P it is Q* (*If a triangle is isosceles the angles
at its base are equal*). *Every S is either P or Q* (*Every
blood vessel is either a vein or an artery*), do not—like the
forms *If P is true Q is true* (*If there is a righteous God
the wicked will not escape their just punishment*), *Either P or
Q is true* (*Either free will is a fact or the sense of
obligation is an illusion*)—express any relation between two
independent judgments or propositions. This point will be developed
subsequently in a distinction that will be drawn between the true
hypothetical (*If P is true Q is true*) and the conditional
(*If any S is P it is Q*).

56. *The Modality of
Judgments*.—Very different accounts of the modality of
judgments or propositions are given by different writers, and the
problems to which distinctions of modality give 85 rise are as a rule not
easy of solution. At the same time such distinctions are of a
fundamental character, and they are apt to present themselves in a
disguised form, thus obscuring many questions that at first sight
appear to have no connexion with modality at all. It is a drawback to
have to deal with so difficult a problem nearly at the commencement of
our treatment of judgments, and the space at our disposal will not
admit of our dealing with it in great detail. Moreover, it can hardly
be hoped that the solution offered will be accepted by all readers.
Still a brief consideration of modal distinctions at this stage will
help to make some subsequent discussions easier.

The main point at issue is whether distinctions of modality are subjective or objective. In attempting to decide this question it will be convenient to deal separately with simple judgments and compound judgments.

57. *Modality in relation
to Simple Judgments*.—The Aristotelian doctrine of modals,
which was also the scholastic doctrine, gave a fourfold division into
(*a*) *necessary*, (*b*) *contingent*, (*c*)
*possible*, and (*d*) *impossible*, according as a
proposition expresses (*a*) that which is necessary and
unchangeable, and which cannot therefore be otherwise; or (*b*)
that which happens to be at any given time, but might have been
otherwise; or (*c*) that which is not at any given time, but may
be at some other time; or (*d*) that which cannot be. The point
of view here taken is objective, not subjective; that is to say, the
distinctions indicated depend upon material considerations, and do not
relate to the varying degrees of belief with which different
propositions are accepted.82

^{82} The consideration of modality
as above conceived has sometimes been regarded as extra-logical on the
ground that necessity, contingency, possibility, and impossibility
depend upon matters of fact with which the logician as such has no
concern. But it also depends upon matters of fact whether any given
predicate can rightly be predicated affirmatively or negatively,
universally or particularly, of any given subject. Distinctions of
quality and quantity can nevertheless be formally expressed, and if
distinctions of modality can also be formally expressed, there is no
initial reason why they should not be recognised by the logician, even
though he is not competent to determine the validity of any given
modal. In so far, however, as the modality of a proposition is
something that does not admit of formal expression, so that
propositions of the same form may have a different modality, then the
argument that the doctrine of modals is extra-logical is more worthy
of consideration.

86 Kant’s doctrine of modality is distinguished from the
scholastic doctrine in that the point of view taken is subjective, not
objective, according to one of the senses in which Kant uses these
terms. Kant divides judgments according to modality into (*a*)
*apodeictic* judgments—*S must be P*, (*b*)
*assertoric* judgments—*S is P*, and (*c*)
*problematic* judgments—*S may be P* ; and the
distinctions between these three classes have come to be interpreted
as depending upon the character of the belief with which the judgments
are accepted.

The distinction between these two doctrines is fundamental; for, as Sigwart puts it,83 the statement that a judgment is possible or necessary is not the same as the statement that it is possible or necessary for a predicate to belong to a subject. The former (which is the Kantian doctrine) refers to the subjective possibility or necessity of judgment; the latter (which is the Aristotelian doctrine) refers to the objective possibility or necessity of what is stated in the judgment.

^{83} *Logic*, i. p. 176.

58. *Subjective
Distinctions of Modality*.—We must reject the view that
subjective distinctions of modality can be drawn in relation to simple
judgments.84 For all judgments, as we have seen, possess
the characteristic of necessity, and hence this characteristic cannot
be made the distinguishing mark of a particular class of judgments,
the apodeictic.

^{84} What follows in this section
is based mainly on Sigwart’s treatment of the subject
(*Logic*, § 31).

We may touch on two ways in which it has been attempted to draw a distinction, from the subjective point of view, between assertoric and apodeictic judgments.

The assertoric judgment has been regarded as expressing what has only subjective validity, that is, what can be affirmed to be true only for the person forming the judgment, while the apodeictic judgment expresses what has universal validity and can be affirmed to be true for everyone.

This again conflicts with the general doctrine of judgment already laid down. We hold that every judgment claims to be true, and that truth cannot be relative to the individual. The assertoric judgment, therefore, as thus defined is no true 87 judgment at all, and we find that all judgments are really apodeictic.

Another suggested ground of distinction is that between immediate knowledge and knowledge that is based on inference, the former being expressed by the assertoric judgment, and the latter by the apodeictic.

There is no doubt that we often say a thing *is* so and so when this
is a matter of direct perception, while we say it *must be* so and so
when we cannot otherwise account for certain perceived facts. Thus, if
I have been out in the rain, I say *it has rained* ; if, without
having observed any rain fall, I notice that the roads and roofs are
wet, I say *it must have rained*.

It is obvious, however, that this distinction is quite inconsistent
with the ascription of any superior certainty to the apodeictic
judgment. For that which we know mediately must always be based on
that which we know immediately; and, since in the process of inference
error may be committed, it follows that that which we know mediately
must have inferior certainty to that of which we have immediate
knowledge. Accordingly in ordinary discourse the statement that
anything *must be* so and so would generally be understood as
expressing a certain degree of doubt.

We cannot then justify the recognition of the apodeictic judgment as expressing a higher degree of certainty than the merely assertoric.

On the other hand, the so-called problematic judgment, interpreted as expressing mere uncertainty,85 cannot be regarded as in itself expressing a judgment at all. It may imply a judgment in regard to the validity of arguments brought forward in support or in disproof of a given thesis; and it implies also a judgment as to the state of mind of the person who is in a state of uncertainty; but it is in itself a mere suspension of judgment.

^{85} The problematic judgment as
interpreted in the following section does more than express mere
uncertainty. The form of proposition *S may be P* is no doubt
ambiguous.

59. *Objective Distinctions
of Modality*.—We have next to consider whether, having regard
not to the judgment as a 88 subjective product, but to the objective
fact expressed in a judgment, any valid distinction can be drawn
between the *necessary*, the actual (or *contingent*), and
the *possible* ; and our answer must be in the affirmative,
provided that we are prepared to admit the conception of the operation
of law.

Thus the judgment *Planets move in elliptic orbits* is in this
sense a *judgment of necessity*. It expresses something which we
regard as the manifestation of a law, and it has an indefinitely wide
application. For we believe it to hold good not only of the planets
with which we are acquainted, but also of other planets (if such there
be) which have not yet been discovered.

Now take the judgment, *All the kings who ruled in France in the
eighteenth century were named Louis*. This is a statement of fact,
but clearly is not the expression of any law. The proposition relates
to a limited number of individuals who happened to have the same name
given to them; but we recognise that their names might have been
different, and that their being kings of France was not dependent on
their possessing the name in question. This then we may call a
*judgment of actuality*.

We have a *judgment of possibility* when we make such a
statement as that *a seedling rose may be produced different in
colour from any roses with which we are at present acquainted*,
meaning that there is nothing in the inherent nature of roses (or in
the laws regulating the production of roses) to render this
impossible.

We have then a *judgment of necessity* (an *apodeictic*
judgment) when the intention is to give expression to some law
relating to the class of objects denoted by the subject-term; we have
a *judgment of actuality* (an *assertoric* judgment) when
the intention is to state a fact, as distinguished from the
affirmation or denial of a law; we have a *judgment of
possibility* (a *problematic* judgment) when the intention is
to deny the operation of any law rendering some complex of properties
impossible.86

^{86} The case of a proposition
which may be regarded as expressing a particular instance of the
operation of a law needs to be specially considered. Granting, for
instance, that the proposition *Every triangle has its angles equal
to two right angles* is apodeictic, are we to describe the
proposition *This triangle has its angles equal to two right
angles* as apodeictic or as assertoric? The right answer seems to
be that, as thus barely stated, the proposition may be merely
assertoric; for it may do no more than express a fact that has been
ascertained by measurement. If, however, the proposition is
interpreted as meaning *This figure, being a triangle, has its
angles equal to two right angles*, then it is apodeictic.

89 I shall not attempt to give here any adequate philosophic analysis of the conception of objective necessity. It must suffice to say that we all have the conception of the operation of law, and that for our present purpose the validity of this conception is assumed.

With regard to this treatment of modality the objection may perhaps be raised that, whatever their value in themselves, the distinctions involved are not of a kind with which formal logic has any concern. It is true that, in a sense, judgments of necessity are the peculiar concern of inductive, as distinguished from formal, logic. The main function of inductive logic is indeed to determine how apodeictic judgments (as above defined) are to be established on the basis of individual observations; for what we mean by induction is the process of passing from particulars to the laws by which they are governed. Granting this, however, there are also many problems, with which logic in its more formal aspects has to deal, in the solution of which some recognition of the distinctions under discussion is desirable, if not essential.

But it will be said that the distinctions cannot be applied
formally: that, for example, given a proposition in the bare form *S
is P*, or given an ordinary universal affirmative proposition
*All S is P*, it cannot be determined, apart from the matter of
the proposition, whether it is apodeictic (in the sense in which that
term is used in this section) or merely assertoric. This is true if we
are limited to the traditional schedule of propositions. But it is to
be remembered that the formulation and the interpretation of
propositions are within certain limits under our own control, and that
it is within our power so to interpret propositional forms for logical
purposes as to bring out distinctions that are not made clear in
ordinary discourse or in the traditional logic. Thus, the form *S as
such is P* might be used for giving formal expression to the
apodeictic judgment, *S is P* being interpreted as merely
assertoric.

90 Another solution, however, and one that may be made to yield a symmetrical scheme, is to utilise the conditional (as distinguished from the true hypothetical,87) proposition, and to differentiate it from the categorical, by interpreting it as modal,88 while the categorical remains merely assertoric.

^{88} Here and elsewhere in
speaking of a proposition as modal (in contradistinction to
assertoric) we mean a proposition that is either apodeictic or
problematic.

Thus, we should have,—

*If anything is S it is P*,—apodeictic;

*All S is P*,—assertoric;

*If anything is S it may be P*,—problematic.89

^{89} It will be observed that in
this scheme (leaving on one side the question of existential import)
the categorical proposition *All S is P* is inferable from the
conditional *If anything is S it is P*, but not *vice
versâ*.

It is of course not pretended that the differentiation here proposed is adopted in the ordinary use of the propositional forms in question; we shall, for example, have presently to point out that in the customary usage of categoricals the universal affirmative has frequently an apodeictic force. We shall return to a discussion of the suggested scheme later on.

60. *Modality in relation
to Compound Judgments*.—We may now consider the application
of distinctions of modality to compound judgments, that is, to
judgments which express a relation in which simple judgments stand one
to another. It is one thing to say that as a matter of fact two
judgments are not both true; it is another thing to say that two
judgments are so related to one another that they cannot both be true.
We may describe the one statement as assertoric, the other as
apodeictic. An apodeictic judgment thus conceived expresses a relation
of ground and consequence; an obligation, therefore, to affirm the
truth of a certain proposition when the truth of a certain other
proposition or combination of propositions is admitted. The obligation
may sometimes depend upon the assistance of certain other propositions
which are left unexpressed.90

^{90} In an apodeictic compound
judgment, the necessity may (at any rate in certain cases) be
described as *subjective*. This is so in the case of a formal
hypothetical; as, for example, in the proposition *If all S is P
then all not-P is not-S*, or in the proposition *If all S is M
and all M is P then all S is P*.

91 In section 55 a threefold classification of compound judgments was
given; the distinction now under consideration points, however, to a
more fundamental twofold classification. From this point of view a
scheme may be suggested in which conjunctives (*P and Q*) and
so-called disjunctives (*P or Q*) would be regarded as
assertoric, while hypotheticals (*If P then Q*) would be regarded
as modal. The enquiry as to how far this is in accordance with the
ordinary usage of the propositional forms in question must be
deferred. It may, however, be desirable to point out at once that, if
this scheme is adopted, certain ordinarily recognised logical
relations are not valid. For the hypothetical *If P then Q* is
ordinarily regarded as equivalent to the disjunctive *Either not-P
or Q*, and this as equivalent to the denial of the conjunctive
*Both P and not-Q*. If, however, the conjunctive (and, therefore,
its denial) and also the disjunctive are merely assertoric, while the
hypothetical is apodeictic, it is clear that this equivalence no
longer holds good. The disjunctive can indeed still be inferred from
the hypothetical, but not the hypothetical from the disjunctive. This
result will be considered further at a later stage.

So far we have spoken only of the *apodeictic* form, *If P then
Q*. The corresponding *problematic* form is, *If P then
possibly Q* ; for example, *If all S is P it is still possible
that some P is not S*. This denies the obligation to admit that
*all P is S* when it has been admitted that *all S is P*. It
is to be observed that in any treatment of modality, the apodeictic
and the problematic involve one another, since the one form is always
required to express the contradictory of the other.

61. *The Quantity and the
Quality of Propositions*.—Propositions are commonly divided
into *universal* and *particular*, according as the
predication is made of the whole or of a part of the subject. This
division of propositions is said to be according to their
*quantity*.

Kant added a third subdivision, namely, *singular* ; and other
logicians have added a fourth, namely, *indefinite*. Under the
head of quantity there have also to be considered what are called
*plurative* and *numerically definite* propositions; and the
possibility of *multiple quantification* has to be recognised.
The 92 question may
also be raised whether there are not some propositions, *e.g.*,
hypothetical propositions, which do not admit of division according to
quantity at all. The discussion of the various points here indicated
may, however, conveniently be deferred until the traditional scheme of
categorical propositions, which is based on the definitive division
into universal and particular, has been briefly touched upon.

Another primary division of propositions is into *affirmative*
and *negative*, according as the predicate is affirmed or denied
of the subject. This division of propositions is said to be according
to their *quality*.

Here, again, Kant added a third subdivision, namely,
*infinite*. This threefold division and the more fundamental
question as to the true significance of logical denial, will also be
deferred until some account has been given of the traditional scheme
of propositions.

62. *The traditional Scheme
of Propositions*.—The traditional scheme of formulating
propositions is intended primarily for categoricals, and it is based
on distinctions of quantity and quality only, distinctions of modality
not being taken into account. For the purposes of the traditional
scheme the following analysis of the categorical proposition may be
given.

A categorical proposition consists of two terms (which are
respectively the *subject* and the *predicate*), united by a
*copula*, and usually preceded by a *sign of quantity*. It
thus contains four elements, two of which—the subject and the
predicate—constitute its *matter*, while the remaining
two—the copula and the sign of quantity—constitute its
*form*.91

^{91} The *logical* analysis of a
proposition must be distinguished from its *grammatical* analysis.
Grammatically only two elements are recognised, namely, the subject
and the predicate. Logically we further analyse the grammatical
subject into quantity and logical subject, and the grammatical
predicate into copula and logical predicate.

The *subject* is that term about which affirmation or denial
is made. The *predicate* is that term which is affirmed or denied
of the subject.

When propositions are brought into one of the forms recognised in
the traditional scheme the subject precedes the predicate. In ordinary
discourse, however, this order is sometimes 93 inverted for the sake
of literary effect, for example, in the proposition—*Sweet are
the uses of adversity*.

The *sign of quantity* attached to the subject indicates the
extent to which the individuals denoted by the subject-term are
referred to. Thus, in the proposition *All S is P* the sign of
quantity is *all*, and the affirmation is understood to be made
of each and every individual denoted by the term *S*.

The *copula* is the link of connexion between the subject and
the predicate, and indicates whether the latter is *affirmed* or
*denied* of the former.

The different elements of the proposition as here distinguished are by no means always separately expressed in the propositions of ordinary discourse; but by analysis and expansion they may be made to appear without any change of meaning. Some grammatical change of form is, therefore, often necessary before propositions can be dealt with in the traditional scheme. Thus in such a proposition as “All that love virtue love angling,” the copula is not separately expressed. The proposition may, however, be written—

sign of quantity | subject | copula | predicate |

All | lovers of virtue | are | lovers of angling ; |

and in this form the four different elements are made distinct. The
older logicians distinguished between propositions *secundi
adjacentis* and propositions *tertii adjacentis*. In the
former, the copula and the predicate are not separated, *e.g.*,
The man runs, All that love virtue love angling; in the latter, they
are made distinct, *e.g.*, The man is running, All lovers of
virtue are lovers of angling.

The traditional scheme of propositions is obtained by a combination
of the division (according to quantity) into universal and particular,
and the division (according to quality) into affirmative and negative.
This combination yields four fundamental forms of proposition as
follows:—

(1) the *universal affirmative*—*All S is P* (or
*Every S is P*, or *Any S is P*, or *All S’s are
P’s*)—usually denoted by the symbol **A**; 94

(2) the *particular affirmative*—*Some S is P* (or
*Some S’s are P’s*)—usually denoted by the
symbol **I**;

(3) the *universal negative*—*No S is P* (or *No
S’s are P’s*)—usually denoted by the symbol
**E**;

(4) the *particular negative*—*Some S is not P* (or
*Not all S is P*, or *Some S’s are not P’s*, or
*Not all S’s are P’s*)—usually denoted by the
symbol **O**.

These symbols **A**, **I**, **E**, **O**, are taken
from the Latin words *affirmo* and *nego*, the affirmative
symbols being the first two vowels of the former, and the negative
symbols the two vowels of the latter.

Besides these symbols, it will sometimes be found convenient to use the following,—

*SaP* = *All S is P* ;

*SiP* = *Some S is P* ;

*SeP* = *No S is P* ;

*SoP* = *Some S is not P*.

These forms are useful when it is desired that the symbol which is used to denote the proposition as a whole should also indicate what symbols have been chosen for the subject and the predicate respectively. Thus,

*MaP* = *All M is P* ;

*PoQ* = *Some P is not Q*.

It will further be found convenient sometimes to denote
*not-S* by *Sʹ*, *not-P* by *Pʹ*, and so on. Thus we shall have

*SʹaPʹ* = *All not-S is not-P* ;

*PiQʹ* = *Some P is not-Q*.

It is better not to write the universal negative in the form *All
S is not P* ;92 for this form is ambiguous and would usually
be interpreted as being merely particular, the *not* being taken
to qualify the *all*, so that we have *All S is not P* =
*Not-all S is P*. Thus, “All that glitters is not
gold” is intended for an **O** proposition, and is equivalent
to “Some things that glitter are not gold.”

^{92} Similar remarks apply to the form *Every S is not P*.

95 The traditional scheme of formulation is somewhat limited in its scope, and from more points of view than one it is open to criticism. It has, however, the merit of simplicity, and it has met with wide acceptation. For these reasons it is as a rule convenient to adopt it as a basis of discussion, though it is also not infrequently necessary to look beyond it.

63. *The Distribution of
Terms in a Proposition*.—A term is said to be distributed
when reference is made to *all* the individuals denoted by it; it
is said to be undistributed when they are only referred to
*partially*, that is, when information is given with regard to a
portion of the class denoted by the term, but we are left in ignorance
with regard to the remainder of the class. It follows immediately from
this definition that the subject is distributed in a universal, and
undistributed in a particular,93 proposition. It can
further be shewn that the predicate is distributed in a negative, and
undistributed in an affirmative proposition. Thus, if I say *All S
is P*, I identify every member of the class *S* with some
member of the class *P*, and I therefore imply that at any rate
*some P* is *S*, but I make no implication with regard to
the whole of *P*. It is left an open question whether there is or
is not any *P* outside the class *S*. Similarly if I say
*Some S is P*. But if I say *No S is P*, in excluding the
whole of *S* from *P*, I am also excluding the whole of
*P* from *S*, and therefore *P* as well as *S* is
distributed. Again, if I say *Some S is not P*, although I make
an assertion with regard to a part only of *S*, I exclude this
part from the whole of *P*, and therefore the whole of *P*
from it. In this case, then, the predicate is distributed, although
the subject is not.94

^{93} *Some* being used in the
sense of *some, it may be all*. If by *some* we understand
*some, but not all*, then we are not really left in ignorance
with regard to the remainder of the class which forms the subject of
our proposition.

^{94} Hence we may say that the
quantity of a proposition, so far as its predicate is concerned, is
determined by its quality. The above results, however, no longer hold
good if we explicitly quantify the predicate as in Hamilton’s
doctrine of the quantification of the predicate. According to this
doctrine, the predicate of an affirmative proposition is sometimes
expressly distributed, while the predicate of a negative proposition
is sometimes given undistributed. For example, such forms are
introduced as *Some S is all P*, *No S is some P*. This
doctrine will be discussed in chapter 7.

96 Summing up our
results, we find that

**A** distributes its subject only,

**I** distributes neither
its subject nor its predicate,

**E** distributes both its subject
and its predicate,

**O** distributes its predicate only.

64. *The Distinction
between Subject and Predicate in the traditional Scheme of
Propositions*.—The nature of the distinction ordinarily drawn
between the subject and the predicate of a proposition may be
expressed by saying that the subject is that of which something is
affirmed or denied, the predicate that which is affirmed or denied of
the subject; or we may say that the subject is that which we regard as
the determined or qualified notion, while the predicate is that which
we regard as the determining or qualifying notion.

It follows that the subject must be given first in idea, since we
cannot assert a predicate until we have something about which to
assert it. Can it, however, be said that because the subject logically
comes first in order of thought, it must necessarily do so in order of
statement, the subject always preceding the copula, and the predicate
always following it? In other words, can we consider the order of the
terms in a proposition to suffice as a criterion? If the subject and
the predicate are pure synonyms95 or if the
proposition is practically reduced to an equation, as in the doctrine
of the quantification of the predicate, it is difficult to see what
other criterion can be taken; or it may rather be said that in these
cases the distinction between subject and predicate loses all
importance. The two are placed on an equality, and nothing is left by
which to distinguish them except the order in which they are stated.
This view is indicated by Professor Baynes in his *Essay on the New
Analytic of Logical Forms*. In such a proposition, for example, as
“Great is Diana of the Ephesians,” he would call
“great” the subject, reading the proposition,
“(Some) great is (all) Diana of the Ephesians.”

^{95} For illustrations of this
point, and on the general question raised in this section, compare
Venn, *Empirical Logic*, pp. 208 to 214.

With reference to the traditional scheme of propositions, however,
it cannot be said that the order of terms is always a 97 sufficient criterion. In
the proposition just quoted, “Diana of the Ephesians”
would generally be accepted as the subject. What further criterion
then can be given? In the case of **E** and **I** propositions
(propositions, as will be shewn, which can be simply converted) we
must appeal to the context or to the question to which the proposition
is an answer. If one term clearly conveys information regarding the
other term, it is the predicate. It will be shewn also that it is more
usual for the subject to be read in extension and the predicate in
intension.96 If these considerations are not decisive,
then the order of the terms must suffice. In the case of **A** and
**O** propositions (propositions, as will be shewn, which cannot be
simply converted) a further criterion may be added. From the rules
relating to the distribution of terms in a proposition it follows that
in affirmative propositions the distributed term (if either term is
distributed) is the subject; whilst in negative propositions, if only
one term is distributed, it is the predicate. It is doubtful if the
inversion of terms ever occurs in the case of an **O** proposition;
but in **A** propositions it is not infrequent. Applying the above
considerations to such a proposition as “Workers of miracles
were the Apostles,” it is clear that the latter term is
distributed while the former is not; the latter term is, therefore,
the subject. Since a singular term is equivalent to a distributed
term, it follows further as a corollary that in an affirmative
proposition if one and only one term is singular it is the subject.
This decides such a case as “Great is Diana of the
Ephesians.”

65. *Universal
Propositions*.—In discussing the import of the universal
proposition *All S is P*, attention must first be called to a
certain ambiguity resulting from the fact that the word *all* may
be used either distributively or collectively. In the proposition,
*All the angles of a triangle are less than two right angles*, it
is used distributively, the predicate applying to each and every angle
of a triangle taken separately. In the proposition. *All the angles
of a triangle are equal to two right angles*, it is used
collectively, the predicate applying to all the 98 angles taken together,
and not to each separately. This ambiguity attaches to the symbolic
form *All S is P*, but not to the form *All S’s are
P’s*. Ambiguity may also be avoided by using *every*
instead of *all*, as the sign of quantity. In any case the
ambiguity is not of a dangerous character, and it may be assumed that
*all* is to be interpreted distributively, unless by the context
or in some other way an indication is given to the contrary.

A more important distinction between propositions expressed in the
form *All S is P* remains to be considered. For such propositions
may be merely assertoric or they may be apodeictic, in the sense given
to these terms in section 59.

It will be convenient here to commence with a threefold distinction.

(1) The proposition *All S is P* may, in the first place, make
a predication of a limited number of particular objects which admit of
being enumerated: *e.g.*, *All the books on that shelf are
novels*, *All my sons are in the army*, *All the men in this
year’s eleven were at public schools*. A proposition of this
kind may be called distinctively an *enumerative universal*. It
is clear that such a proposition cannot claim to be apodeictic.

(2) The proposition *All S is P* may, in the second place,
express what is usually described as an empirical law or uniformity:
*e.g.*, *All lions are tawny*, *All scarlet flowers are
without sweet scent*, *All violets are white or yellow or have a
tinge of blue in them*. Many propositions relating to the use of
drugs, to the succession of certain kinds of weather to certain
appearances of sky, and so on, fall into this class. A proposition of
this kind expresses a uniformity which has been found to hold good
within the range of our experience, but which we should hesitate to
extend much beyond that range either in space or in time. The
predication which it makes is not limited to a definite number of
objects which can be enumerated, but at the same time it cannot be
regarded as expressing a necessary relation between subject and
predicate. Such a proposition is, therefore, assertoric, not
apodeictic.

(3) The proposition *All S is P* may, in the third place,
express a law in the strict sense, that is to say, a uniformity 99 that we believe to hold
good universally and unconditionally: *e.g.*, *All equilateral
triangles are equiangular*, *All bodies have weight*, *All
arsenic is poisonous*. A proposition of this kind is to be regarded
as expressing a necessary relation between subject and predicate, and
it is, therefore, apodeictic.

Propositions falling under the first two of the above categories
may be described as *empirically universal*, and those falling
under the third as *unconditionally universal*.97

^{97} I have borrowed these terms
from Sigwart, *Logic*, § 27; but I cannot be sure that my usage
of them corresponds exactly with his. In section 27 he appears to
include under empirically universal judgments only such judgments as
belong to the first of the three classes distinguished from one
another above. At the same time, his description of the
unconditionally universal judgment applies to the third class only:
such a judgment, he says, expresses a necessary connexion between the
predicate *P* and the subject *S* ; it means, *If anything
is S it must also be P*. And it seems clear from his subsequent
treatment (in § 96) of judgments belonging to the second class that he
does not regard them as unconditionally universal.

Lotze (*Logic*, § 68) indicates the distinction we are
discussing by the terms *universal* and *general*. But again
there seems some uncertainty as to which term he would apply to
judgments belonging to our second class. In the universal judgment, he
says, we have merely a summation of what is found to be true in every
individual instance of the subject; in the general judgment the
predication is of the whole of an indefinite class, including both
examined and unexamined cases. From this it would appear that the
universal judgment corresponds to (1) only, while the general judgment
includes both (2) and (3). Lotze, however, continues, “The
universal judgment is only a collection of many singular judgments,
the sum of whose subjects does as a matter of fact fill up the whole
extent of the universal concept; … the universal proposition, *All
men are mortal*, leaves it still an open question whether, strictly
speaking, they might not all live for ever, and whether it is not
merely a remarkable concatenation of circumstances, different in every
different case, which finally results in the fact that no one remains
alive. The general judgment, on the other hand, *Man is mortal*,
asserts by its form that it lies in the character of mankind that
mortality is inseparable from everyone who partakes in it.” The
illustration here given seems to imply that a judgment may be regarded
as universal, though it relates to a class of objects, not all of
which can be enumerated.

If this distinction is regarded merely as a distinction between
different ways in which judgments may be obtained (for example, by
enumeration or empirical generalisation on the one hand, or by
abstract reasoning or the aid of the principle of causality on the
other hand), without any real difference of content, it becomes merely
genetic and can hardly be retained as a 100 distinction between judgments considered
in and by themselves. If we are so to retain it, it must be as a
distinction between the merely assertoric and the apodeictic in the
sense already explained. In order to be able to deal with it as a
*formal* distinction, we must further be prepared to assign
distinctive forms of expression to the two kinds of universal
judgments respectively. Lotze appears to regard the forms *All S is
P* and *S is P* as sufficiently serving this purpose. But this
is hardly borne out by the current usage of these forms. *All the
S’s are P* might serve for the enumerative universal and *S
as such is P* for the unconditionally universal. These forms do
not, however, fit into any generally recognised schedules; and our
second class of universal would be left out. Another solution, which
has been already indicated in section 59, would be to use the
categorical form for the empirically universal judgment only, adopting
the conditional form for the unconditionally universal judgment.

The most important outcome of the above discussion is that a
proposition ordinarily expressed in the form *All S is P* may be
either assertoric or apodeictic. It will be found that this
distinction has an important bearing on several questions subsequently
to be raised.

66. *Particular
Propositions*.—In dealing with particular propositions it is
necessary to assign a precise signification to the sign of quantity
*some*.

In its ordinary use, the word *some* is always understood to
be exclusive of *none*, but in its relation to *all* there
is ambiguity. For it is sometimes interpreted as excluding *all*
as well as *none*, while sometimes it is not regarded as carrying
this further implication. The word may, therefore, be defined in two
conflicting senses: *first*, as equivalent simply to *one at
least*, that is, as the pure contradictory of *none*, and
hence as covering every case (including *all*) which is
inconsistent with *none* ; *secondly*, as any quantity
intermediate between *none* and *all* and hence carrying
with it the implication *not all* as well as *not none*. In
ordinary speech the latter of these two meanings is probably the more
usual.98 It has, however, been customary with 101 logicians in
interpreting the traditional scheme to adopt the other meaning, so
that *Some S is P* is not inconsistent with *All S is P*.
Using the word in this sense, if we want to express *Some, but not
all, S is P*, we must make use of two propositions—*Some S
is P*, *Some S is not P*. The particular proposition as thus
interpreted is *indefinite*, though with a certain limit; that
is, it is indefinite in so far that it may apply to any number from a
single one up to all, but on the other hand it is definite in so far
as it excludes *none*. We shall henceforth interpret *some*
in this indefinite sense unless an explicit indication is given to the
contrary.

^{98} We might indeed go further
and say that in ordinary speech *some* usually means
*considerably less than all*, so that it becomes still more
limited in its signification. In common language, as is remarked by De
Morgan, “*some* usually means a rather small fraction of
the whole; a larger fraction would be expressed by *a good many* ;
and somewhat more than half by *most* ; while a still larger
proportion would be *a great majority* or *nearly
all*” (*Formal Logic*, p. 58).

Mr Bosanquet regards the particular proposition as unscientific, on
the ground that it always depends either upon imperfect description or
upon incomplete enumeration.99 I may, for instance,
know that all *S*’s of some particular description are
*P*, but not caring or not troubling to define them I content
myself with saying *Some S is P*, for example, *Some truth is
better kept to oneself*.100 Contrasted with
this, we have the particular proposition of incomplete enumeration
where our ground for asserting it is simply the observation of
individual instances in which the proposition is found to hold good.

^{99} *Essentials of Logic*, pp. 116, 117.

^{100} It is implied that a
proposition of this kind might be expanded into the proposition *All
S that is A is P*, that is, *All AS is P*. Mr Bosanquet gives,
as an example, *Some engines can drag a train at a mile a minute for
a long distance*. “This does not mean a certain number of
engines, though of course there are a certain number. It means certain
engines of a particular make, not specified in the
judgment.”

It is true that the particular proposition is not in itself of much
scientific importance; and its indefinite character naturally limits
its practical utility. It seems, however, hardly correct to describe
it as unscientific, since—as will subsequently be shewn in more
detail—it may be regarded as possessing distinctive functions.
Two such functions may be distinguished, though they are often
implicated the one in the other. In the first place, the utility of
the particular proposition often depends 102 rather on what it denies than on what it
affirms, and the proposition that it denies is not indefinite. One of
the principal functions of the particular affirmative is to deny the
universal negative, and of the particular negative to deny the
universal affirmative. In the second place, the distinctive purpose of
the particular proposition may be to affirm existence; and this is
probably as a rule the case with propositions which are described as
resulting from incomplete description. If, for example, we say that
“some engines can drag a train at a mile a minute for a long
distance,” our object is primarily to affirm that there
*are* such engines; and this would not be so clearly expressed in
the universal proposition of which the particular is said to be the
incomplete and imperfect expression.

The relation of the particular proposition, *Some S is P*, to
the problematic proposition, *S may be P*, will be considered
subsequently.

67. *Singular
Propositions*.—By a *singular* or *individual*
proposition is meant a proposition in which the affirmation or denial
is made of a single individual only: for example, *Brutus is an
honourable man* ; *Much Ado about Nothing is a play of
Shakespeare’s* ; *My boat is on the shore*.

Singular propositions may be regarded as forming a sub-class of
universals, since in every singular proposition the affirmation or
denial is of the *whole* of the subject.101 More definitely,
the singular proposition may be said to fall into line, as a rule,
with the enumerative universal proposition.

^{101} It is argued by Father
Clarke that singulars ought to be included under particulars, on the
ground that when a predicate is asserted of one member only of a
class, it is asserted of a portion only of the class. “Now if I
say, *This Hottentot is a great rascal*, my assertion has
reference to a smaller portion of the Hottentot nation than the
proposition *Some Hottentots are great rascals*. The same is the
case even if the subject be a proper name. *London is a large
city* must necessarily be a more restricted proposition than
*Some cities are large cities* ; and if the latter should be
reckoned under particulars, much more the former” (*Logic*,
p. 274). This view fails to recognise that what is really
characteristic of the particular proposition is not its *restricted*
character—since the particular is not inconsistent with the
universal—but its *indefinite* character.

Hamilton distinguishes between universal and singular propositions,
the predication being in the former case of a *whole undivided*,
and in the latter case of a *unit indivisible*. The 103 distinction here
indicated is sometimes useful; but it can with advantage be expressed
somewhat differently. A singular proposition may generally without
risk of confusion be denoted by one of the symbols **A** or
**E**; and in syllogistic inferences a singular may ordinarily be
treated as equivalent to a universal proposition. The use of
independent symbols for singular propositions (affirmative and
negative) would introduce considerable additional complexity into the
treatment of the syllogism; and for this reason it seems desirable as
a rule to include singulars under universals. Universal propositions
may, however, be divided into *general* and *singular*, and
there will then be terms whereby to call attention to the distinction
whenever it may be necessary or useful to do so.

There is also a certain class of propositions which, while
*singular*, inasmuch as they relate but to a single individual,
possess also the indefinite character which belongs to the
*particular* proposition: for example, *A certain man had two
sons* ; *A great statesman was present* ; *An English officer
was killed*. Having two such propositions in the same discourse we
cannot, apart from the context, be sure that the same individual is
referred to in both cases. Carrying the distinction indicated in the
preceding paragraph a little further, we have a fourfold division of
propositions:—*general definite*, “All *S* is
*P*”; *general indefinite*, “Some *S* is
*P*”; *singular definite*, “This *S* is
*P*”; *singular indefinite*, “A certain *S*
is *P*.” This classification admits of our working with the
ordinary twofold distinction into universal and particular—or,
as it is here expressed, definite and indefinite—wherever this
is adequate, as in the traditional doctrine of the syllogism; while at
the same time it introduces a further distinction which may in certain
connexions be of importance.

68. *Plurative Propositions
and Numerically Definite Propositions*.—Other signs of
quantity besides *all* and *some* are sometimes recognised
by logicians. Thus, propositions of the forms *Most S’s are
P’s*, *Few S’s are P’s*, are called
*plurative* propositions. *Most* may be interpreted as equivalent
to *at least one more than half*. *Few* has a negative
force; and *Few S’s are P’s* may be regarded as
equivalent to *Most S’s are not* 104 *P’s*.102 Formal logicians
(excepting De Morgan and Hamilton) have not as a rule recognised these
additional signs of quantity; and it is true that in many logical
combinations they cannot be regarded as yielding more than particular
propositions, *Most S’s are P’s* being reduced to
*Some S’s are P’s*, and *Few S’s are
P’s* to *Some S’s are not P’s*. Sometimes,
however, we are able to make use of the extra knowledge given us;
*e.g.*, from *Most M’s are P’s*, *Most
M’s are S’s*, we can infer *Some S’s are
P’s*, although from *Some M’s are P’s*,
*Some M’s are S’s*, we can infer nothing.

^{102} With perhaps the further
implication “although *some S’s* are
*P’s*”; thus, *Few S’s are P’s* is
given by Kant as an example of the *exponible* proposition (that
is, a proposition which, though not compound in form, can nevertheless
be resolved into a conjunction of two or more simpler propositions,
which are independent of one another), on the ground that it contains
both an affirmation and a negation, though one of them in a concealed
way. It should be added that *a few* has not the same
signification as *few*, but must be regarded as affirmative, and
generally, as simply equivalent to *some* ; *e.g.*, *A few
S’s are P’s* = *Some S’s are P’s*.
Sometimes, however, it means *a small number*, and in this case
the proposition is perhaps best regarded as singular, the subject
being collective. Thus, “a few peasants successfully defended
the citadel” may be rendered “a small band of peasants
successfully defended the citadel,” rather than “some
peasants successfully defended the citadel,” since the stress is
intended to be laid at least as much on the paucity of their numbers
as on the fact that they were peasants. Whilst the proposition
interpreted in this way is singular, not general, it is *singular
indefinite*, not singular definite; for what small band is alluded
to is left indeterminate.

*Numerically definite* propositions are those in which a
predication is made of some definite proportion of a class;
*e.g.*, *Two-thirds of S are P*. A certain ambiguity may
lurk in numerically definite propositions; *e.g.*, in the above
proposition is it meant that *exactly two-thirds of S neither more
nor less are P*, so that we are also given implicitly *one-third
of S are not P*, or is it merely meant that *at least two-thirds
of S but perhaps more are P*? In ordinary discourse we should no
doubt mean sometimes the one and sometimes the other. If we are to fix
our interpretation, it will probably be best to adopt the first
alternative, on the ground that if figures are introduced at all we
should aim at being quite determinate.103 Some such words
105 as *at
least* can then be used when it is not professed to state more than
the minimum proportion of *S’s* that are *P’s*.

^{103} De Morgan remarks that
“a perfectly *definite particular*, as to quantity, would
express how many X’s are in existence, how many Y’s, and
how many of the X’s are or are not Y’s; as in 70 *of
the* 100 *X’s are among the* 200 *Y’s*”
(*Formal Logic*, p. 58). He contrasts the *definite
particular* with the *indefinite particular* which is of the
form *Some X’s are Y’s*. It will be noticed that De
Morgan’s *definite particular*, as here defined, is still
more explicit than the *numerically definite* proposition, as
defined in the text.

69. *Indefinite
Propositions*.—According to quantity, propositions have by
some logicians been divided into (1) Universal, (2) Particular, (3)
Singular, (4) Indefinite. Singular propositions have already been
discussed.

By an *indefinite* proposition is meant one “in which
the quantity is not explicitly declared by one of the designatory
terms *all*, *every*, *some*, *many*,
&c.”; *e.g.*, *S is P*, *Cretans are liars*.
We may perhaps say with Hamilton, that *indesignate* would be a
better term to employ. At any rate the so-called *indefinite
proposition* is not the expression of a distinct form of
*judgment*. It is a form of proposition which is the imperfect
expression of a judgment. For reasons already stated, the particular
has more claim to be regarded as an *indefinite judgment*.

When a proposition is given in the indesignate form, we can
generally tell from our knowledge of the subject-matter or from the
context whether it is meant to be universal or particular. Probably in
the majority of cases indesignate propositions are intended to be
understood as universals, *e.g.*, “Comets are subject to
the law of gravitation”; but if we are really in doubt with
regard to the quantity of the proposition, it must logically be
regarded as particular.104

^{104} In the *Port Royal
Logic* a distinction is drawn between *metaphysical
universality* and *moral universality*. “We call
metaphysical universality that which is perfect and without exception;
and moral universality that which admits of some exception, since in
moral things it is sufficient that things are generally such”
(*Port Royal Logic*, Professor Baynes’s translation, p.
150). The following are given as examples of moral universals: *All
women love to talk* ; *All young people are inconstant* ; *All
old people praise past times*. Indesignate propositions may almost
without exception be regarded as universals either metaphysical or
moral. But it seems clear that moral universals have in reality no
valid claim to be called universals at all. Logically they ought not
to be treated as more than particulars, or at any rate
pluratives.

70. *Multiple
Quantification*.—The application of a predicate to a subject
is sometimes limited with reference to times or conditions, and this
may be treated as yielding a *secondary* quantification of the
proposition; for example, *All men are* 106 *sometimes unhappy*, *In some
countries all foreigners are unpopular*. This differentiation may
be carried further so as to yield triple or any higher order of
quantification. Thus, we have triple quantification in the
proposition, *In all countries all foreigners are sometimes
unpopular*.105

^{105} For a further development
of the notion of multiple quantification see Mr Johnson’s
articles on *The Logical Calculus* in *Mind*, 1892.

In this way a proposition with a singular term for subject may,
with reference to some secondary quantification, be classified as
universal or particular as the case may be; for example, *Gladstone
is always eloquent*, *Browning is sometimes obscure*.

71. *Infinite or Limitative
Propositions*.—In place of the ordinary twofold division of
propositions in respect of quality, Kant gave a *threefold*
division, recognising a class of *infinite* (or
*limitative*) judgments, which are neither affirmative nor
negative. Thus, *S is P* being affirmative, and *S is not P*
negative, *S is not-P* is spoken of as infinite or limitative.106
It is, however, difficult to justify the separate recognition of this
third class, whether we take the purely formal stand-point, or have
regard to the real content of the propositions. From the formal
stand-point we might substitute some other symbol, say *Q*, for
*not-P*, and from this point of view *Some S is not-P* must
be regarded as simply affirmative. On the other hand, *Some S is
not-P* is equivalent in meaning to *Some S is not P*, and
(assuming *P* to be a positive term) these two propositions must,
having regard to their real content, be equally negative in force.

^{106} An *infinite*
judgment, in the sense in which the term is here used, may be
described as the affirmative predication of a negative. Some writers,
however, include under *propositiones infinitae* those whose
subject, as well as those whose predicate, is negative. Thus Father
Clarke defines *propositiones infinitae* as propositions in which
“the subject or predicate is indefinite in extent, being limited
only in its exclusion from some definite class or idea: as, *Not to
advance is to recede*” (*Logic*, p. 268).

Some writers go further and appear to deny that the so-called
infinite judgment has any meaning at all. This point is closely
connected with a question that we have already discussed, namely,
whether the negative term *not-P* has any meaning. If we
recognise the negative term—and we have endeavoured to 107 shew that we ought to
do so—then the proposition *S is not-P* is equivalent to
the proposition *S is not P*, and the former proposition must,
therefore, have just as much meaning as the latter.

The question of the utility of so called infinite propositions has been further mixed up with the question as to the nature of significant denial. But it is better to keep the two questions distinct. Whatever the true character of denial may be, it is not dependent on the use of negative terms.

72. Determine the quality of
each of the following propositions, and the distribution of its terms:
(*a*) A few distinguished men have had undistinguished sons;
(*b*) Few very distinguished men have had very distinguished
sons; (*c*) Not a few distinguished men have had distinguished
sons. [J.]

73. Examine the significance
of *few*, *a few*, *most*, *any*, in the following
propositions; *Few* artists are exempt from vanity; *A few*
facts are better than a great deal of rhetoric; *Most* men are
selfish; If *any* philosophers have been wise, Socrates and Plato
must be numbered among them. [M.]

74. *Everything is either X
or Y* ; *X and Y are coextensive* ; *Only X is Y* ; *The
class X comprises the class Y and something more*. Express each of
these statements by means of ordinary *A*, *I*, *E*,
*O* categorical propositions. [C.]

75. Express each of the following statements in one or more of the forms recognised in the traditional scheme of categorical propositions: (i) No one can be rich and happy unless he is also temperate and prudent, and not always then; (ii) No child ever fails to be troublesome if ill taught and spoilt; (iii) It would be equally false to assert that the rich alone are happy, or that they alone are not. [V.]

76. Express, as nearly as you
can, each of the following statements in the form of an ordinary
categorical proposition, and determine its quality and the
distribution of its terms:

(*a*) It cannot be maintained that pleasure is the sole good;
108

(*b*) The trade of a country does not always suffer, if its
exports are hampered by foreign duties;

(*c*) The man who shews fear cannot be presumed to be guilty;

(*d*) One or other of the members of the committee must have
divulged the secret. [C.]

77. Find the categorical
propositions, expressed in terms of cases of *Q* or *non-Q*
and of *R* or *non-R*, which are directly or indirectly
implied by each of the following statements:

(*a*) The presence of *Q* is a necessary, but not a
sufficient, condition for the presence of *R* ;

(*b*) The absence of *Q* is a necessary, but not a
sufficient, condition for the presence of *R* ;

(*c*) The presence of *Q* is a necessary, but not a
sufficient, condition for the absence of *R*.

In what respects, if any, does the categorical form fail to express
the full significance of such propositions as the above? [J.]

78. “Honesty of purpose
is perfectly compatible with blundering ignorance.”

“The
affair might have turned out otherwise than it did.”

“It
may be that *Hamlet* was not written by the actor known by his
contemporaries as Shakespeare.”

Employ the above propositions to illustrate your views in regard to
the modality of propositions; and examine the relations between each
of the propositions and any assertoric proposition which may be taken
to be its ground or to be partially equivalent to it. [C.]

^{107} This chapter will be mainly
concerned with the opposition of *categorical* propositions; and,
as regards categoricals, complications arising in connexion with their
existential interpretation will for the present be postponed.

79. *The Square of
Opposition*.—In dealing with the subject of this chapter it
will be convenient to begin with the ancient square of opposition
which relates exclusively to the traditional schedule of propositions.
It will, however, ultimately be found desirable to give more general
accounts of what is to be understood by the terms
*contradictory*, *contrary*, &c., so that they may be
adapted to other schedules of propositions.

Two propositions are technically said to be *opposed* to each
other when they have the same subject and predicate respectively, but
differ in quantity or quality or both.108

^{108} This definition, according
to which opposed propositions are not necessarily incompatible with
one another, is given by Aldrich (p. 53 in Mansel’s edition).
Ueberweg (*Logic*, § 97) defines opposition in such a way as to
include only contradiction and contrariety; and Mansel remarks that
“subalterns are improperly classed as *opposed*
propositions” (*Aldrich*, p. 59). Modern logicians,
however, usually adopt Aldrich’s definition, and this seems on
the whole the best course. Some term is wanted to signify the above
general relation between propositions; and though it might be possible
to find a more convenient term, no confusion is likely to result from
the use of the term *opposition* if the student is careful to
notice that it is here employed in a technical sense.

Taking the propositions *SaP*, *SiP*, *SeP*,
*SoP*, in pairs, we find that there are four possible kinds of
relation between them.

(1) The pair of propositions may be such that they can neither both
be true nor both false. This is called *contradictory*
opposition, and subsists between *SaP* and *SoP*, and
between *SeP* and *SiP*. 110

(2) They may be such that whilst both cannot be true, both may be
false. This is called *contrary* opposition. *SaP* and
*SeP*.

(3) They may be such that they cannot both be false, but may both
be true. *Subcontrary* opposition. *SiP* and *SoP*.

(4) From a given universal proposition, the truth of the particular
having the same quality follows, but not *vice versâ*.109
This is *subaltern opposition*, the universal being called the
*subalternant*, and the particular the *subalternate* or
*subaltern*. *SaP* and *SiP*. *SeP* and
*SoP*.

^{109} This result and some of our
other results may need to be modified when, later on, account is taken
of the existential interpretation of propositions. But, as stated in
the note at the beginning of the chapter, all complications resulting
from considerations of this kind are for the present put on one
side.

All the above relations are indicated in the ancient square of opposition.

The doctrine of opposition may be regarded from two different
points of view, namely, as a relation between two given propositions;
and, secondly, as a process of inference by which one proposition
being given either as true or as false, the truth or falsity of
certain other propositions may be determined. Taking the second of
these points of view, we have the following table:— 111

**A** being given *true*, **E** is *false*,
**I** *true*, **O** *false* ;

**E** being given
*true*, **A** is *false*, **I** *false*, **O**
*true* ;

**I** being given *true*, **A** is unknown,
**E** *false*, **O** unknown;

**O** being given
*true*, **A** is *false*, **E** unknown, **I**
unknown;

**A** being given *false*, **E** is unknown,
**I** unknown, **O** *true* ;

**E** being given
*false*, **A** is unknown, **I** *true*, **O**
unknown;

**I** being given *false*, **A** is *false*,
**E** *true*, **O** *true* ;

**O** being given
*false*, **A** is *true*, **E** *false*, **I**
*true*.

80. *Contradictory
Opposition*.—The doctrine of opposition in the preceding
section is primarily applicable only to the fourfold schedule of
propositions ordinarily recognised. We must, however, look at the
question from a wider point of view. It is, in particular, important
that we should understand clearly the nature of contradictory
opposition whatever may be the schedule of propositions with which we
are dealing.

The nature of significant denial will be considered in some detail
in the concluding section of this chapter. At this point it will
suffice to say that to deny the truth of a proposition is equivalent
to affirming the truth of its *contradictory* ; and *vice
versâ*. The criterion of contradictory opposition is that *of the
two propositions, one must be true and the other must be false* ;
they cannot be true together, but on the other hand no mean is
possible between them. The relation between two contradictories is
mutual; it does not matter which is given true or false, we know that
the other is false or true accordingly. Every proposition has its
contradictory, which may however be more or less complicated in form.

It will be found that attention is almost inevitably called to any ambiguity in a proposition when an attempt is made to determine its contradictory. It has been truly said that we can never fully understand the meaning of a proposition until we know precisely what it denies; and indeed the problem of the import of propositions sometimes resolves itself at least partly into the question how propositions of a given form are to be contradicted.

The nature of contradictory opposition may be illustrated by
reference to a discussion entered into by Jevons (*Studies in*
112 *Deductive
Logic*, p. 116) as to the precise meaning of the assertion that a
proposition—say, *All grasses are edible*—is false.
After raising this question, Jevons begins by giving an answer, which
may be called the orthodox one, and which, in spite of what he goes on
to say, must also be considered the correct one. When I assert that a
proposition is false, I mean that its contradictory is true. The given
proposition is of the form **A**, and its contradictory is the
corresponding **O** proposition—*Some grasses are not
edible*. When, therefore, I say that it is false that all grasses
are edible, I mean that some grasses are not edible. Jevons, however,
continues, “But it does not seem to have occurred to logicians
in general to enquire how far similar relations could be detected in
the case of disjunctive and other more complicated kinds of
propositions. Take, for instance, the assertion that ‘all
endogens are *all* parallel-leaved plants.’ If this be
false, what is true? Apparently that one or more endogens are not
parallel-leaved plants, or else that one or more parallel-leaved
plants are not endogens. But it may also happen that no endogen is a
parallel-leaved plant at all. There are three alternatives, and the
simple falsity of the original does not shew which of the possible
contradictories is true.”

This statement is open to criticism in two respects. In the first
place, in saying that one or more endogens are not parallel-leaved
plants, we do not mean to exclude the possibility that no endogen is a
parallel-leaved plant at all. Symbolically, *Some S is not P*
does not exclude *No S is P*. The three alternatives are,
therefore, at any rate reduced to the two first given. But in the
second place, it is incorrect to speak of either of these alternatives
as being by itself a contradictory of the original proposition. The
true contradictory is the affirmation of the truth of *one or other* of
these alternatives. If the original proposition is false, we certainly
know that the new proposition limiting us to such alternatives is
true, and *vice versâ*.

The point at issue may be made clearer by taking the proposition in
question in a symbolic form. *All S is all P* is a condensed
expression, resolvable into the form, *All S is P and* 113 *all P is S*. It
has but one contradictory, namely, *Either some S is not P or some P
is not S*.110 If either of these alternatives holds good,
the original statement must in its entirety be false; and, on the
other hand, if the latter is false, one at least of these alternatives
must be true. *Some S is not P* is not by itself a contradictory
of *All S is all P*. These two propositions are indeed
inconsistent with one another; but they may both be false.

^{110} The contradictory of *All
S is all P* may indeed be expressed in a different form, namely,
*S and P are not coextensive*, but this has precisely the same
force as the contradictory given in the text. We go on to shew that
two different forms of the contradictory of the same proposition must
necessarily be equivalent to one another.

It follows that we must reject Jevons’s further statement
that “a proposition of moderate complexity has an almost
unlimited number of contradictory propositions, which are more or less
in conflict with the original. The truth of any one or more of these
contradictories establishes the falsity of the original, but the
falsity of the original does not establish the truth of any one or
more of its contradictories.”111 No doubt a
proposition which is complicated in form may yield an indefinite
number of other non-equivalent propositions the truth of any one of
which is *inconsistent with* its own. It will also be true that
its contradictory can be expressed in more than one form. But these
forms will necessarily be equivalent to one another, since it is
impossible for a proposition to have two or more non-equivalent
contradictories. This position may be formally established as follows.
Let *Q* and *R* be both contradictories of *P*. They
will be equivalent if it can 114 be shewn that *if Q then R*, and
*if R then Q*. Since *P* and *Q* are contradictories,
we have *If Q then not P*, and since *P* and *R* are
contradictories we have *If not P then R*. Combining these two
propositions we have the conclusion *If Q then R*. *If R then
Q* follows similarly. Hence we have established the desired result.

^{111} It must be admitted that it
has not been uncommon for logicians to use the word *contradict*
somewhat loosely. For example, in the *Port Royal Logic*, we find
the following: “*Except the wise man* (said the
Stoics) *all men are truly fools*. This may be contradicted (1)
by maintaining that the wise man of the Stoics was a fool as well as
other men; (2) by maintaining that there were others, besides their
wise man, who were not fools; (3) by affirming that the wise man of
the Stoics was a fool, and that other men were not” (p. 140).
The affirmation of any one of these three propositions certainly
renders it necessary to deny the truth of the given proposition, but
no one of them is by itself the *contradictory* of the given
proposition. The true contradictory is the alternative proposition:
*Either the wise man of the Stoics is a fool or some other
men are not fools*.

In connexion with the same point, Jevons raises another question,
in regard to which his view is also open to criticism. He says,
“But the question arises whether there is not confusion of ideas
in the usual treatment of this ancient doctrine of opposition, and
whether a contradictory of a proposition is not any proposition which
involves the falsity of the original, but is not the sole condition of
it. I apprehend that any assertion is false which is made without
sufficient grounds. It is false to assert that the hidden side of the
moon is covered with mountains, not because we can prove the
contradictory, but because we know that the assertor must have made
the assertion without evidence. If a person ignorant of mathematics
were to assert that ‘all involutes are transcendental
curves,’ he would be making a false assertion, because, whether
they are so or not, he cannot know it.” We should, however,
involve ourselves in hopeless confusion were we to consider the truth
or falsity of a proposition to depend upon the knowledge of the person
affirming it, so that the same proposition would be now true, now
false. It will be observed further that on Jevons’s view both
the propositions *S is P* and *S is not P* would be false to
a person quite ignorant of the nature of *S*. This would mean
that we could not pass from the falsity of a proposition to the truth
of its contradictory; and such a result as this would render any
progress in thought impossible.

81. *Contrary
Opposition*.—Seeking to generalise the relation between
**A** and **E**, we might naturally be led to characterize the
contrary of a given proposition by saying that it goes beyond mere
denial, and sets up a further assertion as far as possible removed
from the original assertion; so that, whilst the contradictory of a
proposition denies its entire truth, its contrary may be said to
assert its entire falsehood. A pair of contraries as thus defined may
be regarded as standing at the opposite 115 ends of a scale on which there are a
number of intermediate positions.

On this definition, however, the notion of contrariety cannot very satisfactorily be extended much beyond the particular case contemplated in the ordinary square of opposition. For if we have a proposition which cannot itself be regarded as standing at one end of a scale, but only as occupying an intermediate position, such proposition cannot be regarded as forming one of a pair of contraries. Plurative and numerically definite propositions may be taken as illustrations.

Hence if it is desired to define contrariety so that the conception
may be generally applicable, the idea of two propositions standing, as
it were, furthest apart from each other must be given up, and any two
propositions may be described as contraries if they are inconsistent
with one another without at the same time exhausting all
possibilities. Contraries must on this definition always admit of a
mean, but they may not always be what we should speak of as
diametrical opposites, and any given proposition is not limited to a
single contrary, but may have an indefinite number of non-equivalent
contraries. At the same time, it will be observed that this definition
still suffices to identify **A** and **E** as a pair of
contraries, and as the only pair in the traditional scheme of
opposition.

82. *The Opposition of
Singular Propositions*.—Taking the proposition *Socrates is
wise*, its contradictory is *Socrates is not wise* ;112
and so long as we keep to the same terms, we cannot go beyond this
simple denial. The proposition has, therefore, no formal contrary.113
This opposition of singulars has been called *secondary
opposition* (Mansel’s *Aldrich*, p. 56).

^{112} This must be regarded as
the correct contradictory from the point of view reached in the
present chapter. The question becomes a little more difficult when the
existential interpretation of propositions is taken into
account.

^{113} We can obtain what may be
called a *material* contrary of the given proposition by making
use of the contrary of the predicate instead of its mere
contradictory; thus, *Socrates has not a grain of sense*. This is
spoken of as *material* contrariety because it necessitates the
introduction of a fresh term that could not be formally obtained out
of the given proposition. It should be added that the distinction
between formal and material contrariety might also be applied in the
case of general propositions.

116 If, however,
there is secondary quantification in a proposition having a singular
subject, then we may obtain the ordinary square of opposition. Thus,
if our original proposition is *Socrates is always* (or *in all
respects*) *wise*, it is contradicted by the statement that
*Socrates is sometimes* (or in *some respects*) *not
wise*, while it has for its contrary, *Socrates is never* (or
*in no respects*) *wise*, and for its subaltern, *Socrates
is sometimes* (or *in some respects*) *wise*. It may be
said that when we thus regard Socrates as having different
characteristics at different times or under different conditions, our
subject is not strictly singular, since it is no longer a whole
indivisible. This is in a sense true, and we might no doubt replace
our proposition by one having for its subject “the judgments or
the acts of Socrates.” But it does not appear that this
resolution of the proposition is necessary for its logical treatment.

The possibility of implicit secondary quantification, although no
such quantification is explicitly indicated, is a not unfruitful
source of fallacy in the employment of propositions having singular
subjects. If we take such propositions as *Browning is obscure*,
*Epimenides is a liar*, *This flower is blue*, and give as
their contradictories *Browning is not obscure*, *Epimenides is
not a liar*, *This flower is not blue*, shall we say that the
original proposition or its contradictory is true in case Browning is
sometimes (but not always) obscure, or in case Epimenides sometimes
(but not often) speaks the truth, or in case the flower is partly (but
not wholly) blue? There is certainly a considerable risk in such
instances as these of confusing contradictory and contrary opposition,
and this will be avoided if we make the secondary quantification of
the propositions explicit at the outset by writing them in the form
*Browning is always* (or *sometimes*) *obscure*,
&c.114 The contradictory will then be particular or
universal accordingly.

^{114} Or we might reduce them to
the forms,—All (or some) of the poems of Browning are obscure,
All (or some) of the statements of Epimenides are false, All (or some)
of the surface of this flower is blue.

83. *The Opposition of
Modal Propositions*.—So far in this chapter our attention has
been confined to assertoric propositions. For the present, a very
brief reference to the opposition 117 of modals will suffice. The main points
involved will come up for further consideration later on.

We have seen that the unconditionally universal proposition,
whether expressed in the ordinary categorical form *All S is P*,
or as a conditional *If anything is S it is P*, affirms a
necessary connexion, by which is meant not merely that all the
*S*’s are as a matter of fact *P*’s, but that it
is inherent in their nature that they should be so. The statement that
some *S*’s *are* not *P*’s is
*inconsistent* with this proposition, but is not its
contradictory, since both the propositions might be false: the
*S*’s might all happen to be *P*’s, and yet
there might be no law of connexion between *S* and *P*. The
proposition in question being *apodeictic* will have for its
contradictory a modal of another description, namely, a
*problematic* proposition; and this may be written in the form
*S need not be P*, or *If anything is S still it need not be
P*, according as our original proposition is expressed as a
categorical or as a conditional

Similarly, the contradictory of the hypothetical *If P is true
then Q is true*, this proposition being interpreted modally, is
*If P is true still Q need not be true*.

84. *Extension of the
Doctrine of Opposition*.115—If we do not
confine ourselves to the ordinary square of opposition, but consider
any pair of propositions (whatever may be the schedule to which they
belong), it becomes necessary to amplify the list of formal relations
recognised in the square of opposition, and also to extend the meaning
of certain terms. We may give the following classification:

^{115} The illustrations given in
this section presuppose a knowledge of immediate inferences. The
section may accordingly on a first reading be postponed until part of
the following chapter has been read.

(1) Two propositions may be *equivalent* or
*equipollent*, each proposition being formally inferable from the
other. Hence if either one of the propositions is true, the other is
also true; and if either is false, the other is also false. For
example, as will presently be shewn, *All S is P* and *All
not-P is not-S* stand to each other in this relation.

(2) and (3) One of the two propositions may be formally inferable
from the other, but not *vice versâ*. If we are 118 considering two given
propositions *Q* and *R*, this yields two cases: for
*Q* may carry with it the truth of *R*, but not conversely;
or *R* may carry with it the truth of *Q*, but not
conversely. Ordinary subaltern propositions with their subalternants
fall into this class; and it will be convenient to extend the meaning
of the term *subaltern*, so as to apply it to any pair of
propositions thus related, whether they belong to the ordinary square
of opposition or not. It will indeed be found that any pair of simple
propositions of the forms **A**, **E**, **I**, **O**, that
are subaltern in the extended sense, are equivalent to some pair that
are subaltern in the more limited sense.116 Thus *All S is
P* and *Some P is S*, which are subaltern in the extended
sense, are equivalent to *All S is P* and *Some S is P*.
*All S is P* and *Some not-S is not P* are another pair of
subalterns. Here it is not so immediately obvious in what direction we
are to look for a pair of equivalent propositions belonging to the
ordinary square of opposition. *No not-P is S* and *Some not-P
is not S* will, however, be found to satisfy the required
conditions.

^{116} This will of course not
hold good when we apply the term subaltern to compound propositions,
*e.g.*, to the pair *Some S is not P and some P is not S*,
*Some S is not P or some P is not S*.

(4) The propositions may be such that they can both be true
together, or both false, or either one true and the other false. For
example, *All S is P* and *All P is S*. Such propositions
may be called *independent* in their relation to one another.

(5) The propositions may be such that *one or other* of them *must be*
true while *both may be* true. A pair of propositions which are thus
related—for example, *Some S is P* and *Some not-S is
P*—may, by an extension of meaning as in the case of the term
*subaltern*, be said to be *subcontrary*. It can be shewn
that any pair of subcontraries of the forms **A**, **E**,
**I**, **O** are equivalent to some pair of subcontraries
belonging to the ordinary square of opposition; thus, the above pair
are equivalent to *Some P is S* and *Some P is not S*.

(6) The two propositions may be *contrary* to one another, in the
sense that they cannot both be true, but can both be false. It can as
before be shewn that any pair of contraries of 119 the forms **A**,
**E**, **I**, **O** are equivalent to some pair of contraries
in the more ordinary sense. For example, the contraries *All S is
P* and *All not-S is P* are equivalent to *No not-P is S*
and *All not-P is S*.

(7) The two propositions may be *contradictory* to one another
according to the definition given in section 80, that is, they can
neither both be true nor both false. *All S is P* and *Some
not-P is S* afford an example outside the ordinary square of
opposition. It will be observed that these two propositions are
equivalent to the pair *All S is P* and *Some S is not P*.

Two propositions, then, may, in respect of inferability,
consistency, or inconsistency, be formally (1) equivalent, (2) and (3)
subaltern, (4) independent, (5) subcontrary, (6) contrary, (7)
contradictory, the terms *subaltern*, &c., being used in the
most extended sense. What pairs of categorical propositions (into
which only the same terms or their contradictories enter) actually
fall into these categories respectively will be shewn in sections 106 and 107.

These seven possible relations between propositions (taken in pairs) will be found to be precisely analogous to the seven possible relations between classes (taken in pairs) as brought out in a subsequent chapter (section 130).

85. *The Nature of
Significant Denial*.—It is desirable that, before concluding
this chapter, we should briefly discuss a more fundamental question
than any that has yet been raised, namely, the meaning and nature of
negation and denial.

We observe, in the first place, that negation always finds expression in a judgment, and that it always involves the denial of some other judgment. The question therefore arises whether negation always presupposes an antecedent affirmation. This question must be answered in the negative if it is understood to mean that in order to be able to deny a proposition we must begin by regarding it as true. The proposition which we deny may be asserted or suggested by someone else; or it may occur to us as one of several possible alternatives; or it may be put in the form of a question.

It is, however, to be added that if a denial is to have any value
as a statement of matter of fact, the corresponding 120 affirmation must be
consistent with the meaning of the terms employed. Thus if *A*
connotes *m*, *n*, *p*, and *B* connotes
*not-p*, *q*, *r*, then the denial that *A* is
*B* gives no real information respecting *A*. For the
affirmation that *A* is *B* cannot be made by anyone who
knows what is meant by *A* and *B* respectively. The same
point may be otherwise expressed by saying that just as the
affirmation of a verbal proposition is insignificant regarded as a
real affirmation concerning the subject (and not merely as an
affirmation concerning the meaning to be attached to the
subject-term), so the denial of a contradiction in terms is
insignificant from the same point of view. Such a denial yields merely
what is tautologous and practically useless.

For example, the denial that *the soul is a ship in full sail*
is insignificant regarded as a statement of matter of fact; for such
denial gives no information to anyone who is already acquainted with
the meaning of the terms involved.

The nature of logical negation is of so fundamental and ultimate a
character that any attempt to explain it is apt to obscure rather than
to illumine. It cannot be expressed more simply and clearly than by
the laws of contradiction and excluded middle: *a judgment and its
contradictory cannot both be true; nor can they both be false*.

Because every negative judgment involves the denial of some other
judgment, it has been argued that a negative judgment such as *S is
not P* is primarily a judgment concerning the positive judgment
*S is P*, not concerning the subject *S* ; and hence that a
negative judgment is not co-ordinate with a positive judgment, but
dependent upon it.117

^{117} Compare Sigwart, *Logic*, i. pp. 121, 2.

Passing by the point that a positive judgment also involves the
denial of some other judgment, we may observe that a distinction must
be drawn between “*S is P*” *is not true* (which
is a judgment about *S is P*), and *S is not P* (which is a
judgment about *S*). Denial no doubt presents itself to the mind
most simply in the first of these two forms. But in contradicting a
given judgment our method usually is to establish another judgment
involving the same terms which stands to the given judgment in the
relation expressed by the laws of contradiction 121 and excluded middle;
and when we oppose the judgment *S is not P* to the judgment *S
is P* we have reached the less direct mode of denial in which we
have again a judgment concerning our original subject.

The example here taken tends perhaps to obscure the point at issue
because the distinction between “*S is P*” *is not
true* and *S is not P* may appear to be so slight as to be
immaterial. That there is a real distinction will, however, appear
clear if we take such pairs of propositions as “*All S is
P*” *is not true*, *Some S is not P* ; “*All
S is all P*” *is not true*, *Either some S is not P or
some P is not S* ; “*If any P is Q it is R*” *is
not true*, *P might be Q without being R*.

It will be convenient if in general we understand by the
*contradictory* of a proposition *P* not its simple denial
“*P is not true*,” but the proposition *Q*
involving the same terms, which is formally so related to *P*,
that *P* and *Q* cannot both be true or both false.

Sigwart observes that the ground of a denial may be either
(*a*) a deficiency, or (*b*) an opposition.118
I may, for example, pronounce that a certain thing does not possess a
given attribute either (*a*) because I fail to discover the
presence of the attribute, or (*b*) because I recognise the
presence of some other attribute which I know to be incompatible with
the one suggested.

^{118} *Logic*, i. p. 127.

This distinction may be illustrated by one or two further examples.
Thus, I may deny that a man travelled by a certain train either
(*a*) because I searched the train through just before it started
and found he was not there, or (*b*) because I know he was
elsewhere when the train started,—I may, for instance, have seen
him leave the station at the same moment in another train in the
opposite direction. Similarly, I may deny a universal proposition
either (*a*) because I have discovered certain instances of its
not holding good, or (*b*) because I accept another universal
proposition which is inconsistent with it. Again, I may deny that a
given metal, or the metal contained in a certain salt, is copper
(*a*) on the ground of deficiency, namely, that it does not
answer to a certain test, or (*b*) on the ground 122 of opposition, namely,
that I recognise it to be another metal, say, zinc.

The ground of denial always involves something positive, for example, the search through the train, or the discovery of individual exceptions. But it is clear that when we establish an opposition we get a result that is itself positive in a way that is not the case when we merely establish a deficiency. This may lead up to a brief examination of a doctrine of the nature of significant denial that is laid down by Mr Bosanquet.

Mr Bosanquet holds that *bare* denial has in itself no significance,
and he apparently denies that the *contradictory* of a judgment,
apart from the grounds on which it is based, conveys any
information.119 For the meaning of significant negation we
must, he says, look to the grounds of the negation; or else for
*contradictory* denial we must substitute *contrary* denial.
As a consequence, a judgment can, strictly and properly, “only
be denied by another judgment of the same nature; a singular by a
singular judgment, a generic by a generic, a hypothetical by a
hypothetical”;120 and, presumably, a particular by a
particular, an apodeictic by an apodeictic.

^{119} *Logic*, i. p. 305.

^{120} *Ibid*, p. 383.

It is of course true that every denial must have some kind of positive basis, but it is also necessary that a judgment should be distinguished from the grounds on which it is based. We cannot say that a judgment of given content is different for two people because they accept it on different grounds; and if it is said that this is to beg the question, since a difference in ground constitutes in itself a difference in content, the reply is that such a doctrine must render the content of every judgment so elusive and uncertain as to make it impossible of analysis.

The view that identifies the denial of a judgment with its contrary not only mixes up a judgment with its grounds, but also overlooks one of the two principal grounds of denial. When the ground of negation is an opposition, we may no doubt be said to reach denial through the contrary, though we should still hold that the denial is in itself something less than the contrary; but when the ground of denial is a deficiency, even this cannot be allowed. If, for example, I have arrived 123 at the conclusion that a man did not start by a given train because I searched the train through before its departure and did not find him there; or if I conclude that a given metal is not copper because it does not satisfy a given test; I have obtained no contrary judgment, and yet my denial is justified.

These would be cases of *bare* denial. I have gained no
positive knowledge of the whereabouts of the man in question, nor can
I identify the given metal. But surely it cannot be seriously
maintained that the denial is meaningless or useless, say, to a
detective in the first instance, or to an analytical chemist in the
second.

Of course we seldom or never rest content with bare denial. The contrary rather than the contradictory represents our ultimate aim. But it is often the case that, temporarily at any rate, we cannot get beyond bare denial; and we ought not to consider that we have altogether failed to make progress when all that we have achieved is the exclusion of a possible alternative or the overthrow of a false theory. Recent researches, for example, into the origin of cancer have led to no positive results; but it is claimed for them that by destroying preconceived ideas on the subject they have cleared the way for future advance. Will anyone affirm that this was not worth doing or that the time spent on the researches was wasted?

Looking at the question from another point of view, it is surely absurd to say that we cannot deny a universal unless we are able to substitute another universal in its place. Various algebraical formulae have from time to time been suggested as necessarily yielding a prime number. They have all been overthrown, and no valid formula has been established in their place. But knowledge that these formulae are false is not quite appropriately described as ignorance.

Elsewhere Mr Bosanquet says that mere enumerative exceptions are
futile and cannot touch the essence of the unconditionally universal
judgments they apparently oppose.121 He appears to have
in view cases where nothing more than some modification of the
original judgment is shewn to be 124 necessary. But even so the enumerative
exceptions *have* overthrown the original judgment. No doubt a
scientific law which has had a great amount of evidence in its favour
is likely to contain elements of truth even if it is not altogether
true; and the object of a man of science who overthrows a law will be
to set up some other law in its place. But, says Mr Bosanquet, even if
the first generic judgment were a sheer blunder and confusion, as has
been the case from time to time with judgments propounded in science,
it is scarcely possible to rectify the confusion except by
substituting for it the true positive conceptions that arise out of
the cases which overthrew it.” Here it is admitted that the
exceptions do overthrow the law, and the rest of the argument is
surely an instance of *ignoratio elenchi*. It is moreover a pure,
and in many cases an unjustifiable, assumption that the cases which
suffice to overthrow a false law will also suffice as the basis for
the establishment of a true law in its place.

^{121} *Logic*, i. p. 313.

86. Examine the nature of the opposition between each pair of the following propositions:—None but Liberals voted against the motion; Amongst those who voted against the motion were some Liberals; It is untrue that those who voted against the motion were all Liberals. [K.]

87. If *some* were used
in its ordinary colloquial sense, how would the scheme of opposition
between propositions have to be modified? [J.]

88. Explain the technical
terms “contradictory” and “contrary” applying
them to the following propositions: *Few S are P* ; *He was not
the only one who cheated* ; *Two-thirds of the army are
abroad*. [V.]

89. Give the contradictory of
each of the following propositions:—*Some but not all S is
P* ; *All S is P and some P is not R* ; *Either all S is P or
some P is not R* ; *Wherever the property A is found, either the
property B or the property C will be found with it, but not both of
them together*. [K.]

125 90. Give the contradictory, and also a contrary, of each of the following propositions:

Half the candidates failed;

Wellington was always successful both in beating
the enemy and in utilising his victory;

All men are either not knaves or not fools;

All but he had fled;

Few of them are honest;

Sometimes all our efforts fail;

Some of our efforts always fail. [L.]

91. Give the contradictory,
and also a contrary, of each of the following propositions:

I am certain you are wrong;

Sometimes when it rains I find myself without an umbrella;

Whatever you say, I shall not believe you. [C.]

92. Define the terms
*subaltern*, *subcontrary*, *contrary*,
*contradictory*, in such a way that they may be applicable to
pairs of propositions generally, and not merely to those included in
the ordinary square of opposition. Do the above exhaust the formal
relations (in respect of inferability, consistency, or inconsistency)
that are possible between pairs of propositions?

Illustrate your answer by considering the relation (in respect of
inferability, consistency, or inconsistency) between each of the
following propositions and each of the remainder: *S and P are
coincident* ; *Some S is P* ; *Not all S is P* ; *Either
some S is not P or some P is not S* ; *Anything that is not P is
S*. [K.]

93. Given that the
propositions *X* and *Z* are contradictory, *Y* and
*V* contradictory, and *X* and *Y* contrary, shew
(without assuming that *X*, *Y*, *V*, *Z* belong
to the ordinary schedule of propositions) that the relations of
*V* to *X*, *Z* to *Y*, *V* to *Z* are
thereby deducible. [J.]

94. Prove formally that if two propositions are equivalent, their contradictories will also be equivalent. [K.]

95. Examine the doctrine that
a judgment can properly be denied only by another judgment of the same
type. Illustrate by reference to (*a*) universal judgments,
(*b*) particular judgments (*c*) disjunctive judgments,
(*d*) apodeictic judgments. [K.]

^{122} In this chapter we concern
ourselves mainly with the traditional scheme of propositions, and
except where an explicit statement is made to the contrary we proceed
on the assumption that each class represented by a simple term exists
in the universe of discourse, while at the same time it does not
exhaust that universe. This assumption appears to have been made
implicitly in the traditional treatment of logic.

96. *The Conversion of
Categorical Propositions*.—By *conversion*, in a broad
sense, is meant a change in the position of the terms of a
proposition.123 Logic, however, is concerned with conversion
only in so far as the truth of the new proposition obtained by the
process is a legitimate inference from the truth of the original
proposition. For example, the change from *All S is P* to *All
P is S* is not a legitimate logical conversion, since the truth of
the latter proposition does not follow from the truth of the former.
In other words, logical conversion is a case of *immediate
inference*, which may be defined as the inference of a proposition
from a single other proposition.124

^{123} Ueberweg (*Logic*, §
84) defines conversion thus. Compare also De Morgan, *Formal
Logic*, p. 58. In geometry, *all equiangular triangles are
equilateral* would be regarded as the converse of *all
equilateral triangles are equiangular*. In this sense of the term
conversion, which is its ordinary non-technical sense, we may
say—as we frequently do say—“Yes, such and such a
proposition is true; but its converse is not true.”

^{124} In discussing immediate
inferences we “pursue the content of an enunciated judgment into
its relations to judgments not yet uttered” (Lotze). Instead of
“immediate inferences” Professor Bain prefers to speak of
“equivalent propositional forms.” It will be found,
however, that the new propositions obtained by immediate inference are
not always equivalent to the original proposition, *e.g.*, in
conversion *per accidens*. Miss Jones suggests the term
*eduction* as a synonym for *immediate inference*
(*General Logic*, p. 79); and she then distinguishes between
*eversions* and *transversions*, an *eversion* being an
eduction from categorical form to categorical, or from hypothetical to
hypothetical, &c., and *transversion* an eduction from
categorical form to conditional, or from conditional to categorical,
&c. For the present we shall be concerned with eversions
only.

127 The simplest
form of logical conversion, and that which is understood in logic when
we speak of conversion without further qualification, may be defined
as *a process of immediate inference in which from a given
proposition we infer another, having the predicate of the original
proposition for subject, and its subject for predicate*. Thus,
given a proposition having *S* for its subject and *P* for
its predicate, our object in the process of conversion is to obtain by
immediate inference a new proposition having *P* for its subject
and *S* for its predicate. The original proposition may be called
the *convertend*, and the inferred proposition the
*converse*.

The process will be valid if the two following rules are observed:

(1) The converse must be the same in quality as the convertend
(*Rule of Quality*);

(2) No term must be distributed in the converse unless it was
distributed in the convertend (*Rule of Distribution*).

Applying these rules to the four fundamental forms of proposition, we have the following table:—

Convertend. | Converse. |

All S is P. A. | Some P is S.
I. |

Some S is P. I. | Some P is
S. I. |

No S is P. E. | No P
is S. E. |

Some S is not P. O. | (None) |

It is desirable at this stage briefly to call attention to a point
which will receive fuller consideration later on in connexion with the
reading of propositions in extension and intension, namely, that,
generally speaking, in any judgment we have naturally before the mind
the objects denoted by the 128 subject, but the qualities connoted by the
predicate. In the process of converting a proposition, however, the
extensive force of the predicate is made prominent, and an import is
given to the predicate similar to that of the subject. At the same
time the distribution of the predicate has to be made explicit in
thought. It is in passing from the *predicative* to the
*class* reading (*e.g.* from *all men are mortal* to
*all men are mortals*), that the difficulty sometimes found in
correctly converting propositions probably consists. We shall at any
rate do well to recognise that conversion and other immediate
inferences usually involve a distinct mental act of the above nature.

It follows from what has been said above that some propositions
lend themselves to the process of conversion much more readily than
others. When the predicate of a proposition is a substantive little or
no effort is required in order to convert the proposition; more effort
is necessary when the predicate is an adjective; and still more when
in the original proposition the logical predicate is not expressed
separately at all, as in propositions *secundi adjacentis*.
Compare for purposes of conversion the propositions, *Whales are
mammals*, *Lions are carnivorous*, *A stitch in time saves
nine*. In some cases, in consequence of the awkwardness of changing
adjectives and verbal predicates into substantives, the conversion of
a proposition appears to be a very artificial production.125

^{125} Compare Sigwart, *Logic*, i. p. 340.

97. *Simple Conversion and
Conversion per accidens*.—It will be observed that in the
case of **I** and **E**, the converse is of the same form as the
original proposition; moreover we do not lose any part of the
information given us by the convertend, and we can pass back to it by
re-conversion of the converse. The convertend and its converse are
accordingly *equivalent* propositions. The conversion under these
conditions is said to be *simple*.

In the case of **A**, it is different; we cannot pass by
immediate inference from *All S is P* to *All P is S*,
inasmuch as *P* is distributed in the latter of these
propositions but undistributed in the former. Hence, although we start
with a universal proposition, we obtain by conversion a particular
129 proposition
only,126 and by no means of operating upon the
converse can we regain the original proposition. The convertend and
its converse are accordingly non-equivalent propositions. The
conversion in this case is called conversion *per accidens*,127
or conversion *by limitation*.128

^{126} The failure to recognise or
to remember that universal affirmative propositions are not simply
convertible is a fertile source of fallacy.

^{127} The conversion of **A**
is said by Mansel to be called conversion *per accidens*
‘because it is not a conversion of the universal *per se*,
but by reason of its containing the particular. For the proposition
‘Some *B* is *A*’ is *primarily* the
converse of ‘Some *A* is *B*,’
*secondarily* of ‘All *A* is *B*’”
(Mansel’s *Aldrich*, p. 61). Professor Baynes seems to deny
that this is the correct explanation of the use of the term (*New
Analytic of Logical Forms*, p. 29); but however this may be, we
certainly need not regard the converse of **A** as necessarily
obtained through its subaltern. It is possible to proceed directly
from *All A is B* to *Some B is A* without the intervention
of *Some A is B*.

^{128} Simple conversion and
conversion *per accidens* are also called respectively
*conversio pura* and *conversio impura*. Compare Lotze,
*Logic*, § 79.

For concrete illustrations of the process of conversion we may take
the propositions,—A stitch in time saves nine; None but the
brave deserve the fair. The first of these may be written in the
form,—All stitches in time are things that save nine stitches.
This, being an **A** proposition, is only convertible *per
accidens*, and we have for our converse,—Some things that
save nine stitches are stitches in time. The second of the given
propositions may be written,—No one who is not brave is
deserving of the fair. This, being an **E** proposition, may be
converted simply, giving, No one deserving of the fair is not brave.
Our results may be expressed in a more natural form as follows: One
way of saving nine stitches is by a stitch in time; No one deserving
of the fair can fail to be brave.

No difficulty ought ever to be found in converting or performing other immediate inferences upon any given proposition when once it has been brought into the traditional logical form, its quantity and quality being determined, its subject, copula, and predicate being definitely distinguished from one another, and its predicate as well as its subject being read in extension. If, however, this rule is neglected, mistakes are pretty sure to follow.

130 98. *Inconvertibility of
Particular Negative Propositions*.—It follows immediately
from the rules of conversion given in section 96 that *Some S is not P* does not admit of ordinary conversion; for *S* which is
undistributed in the convertend would become the predicate of a
negative proposition in the converse, and would therefore be
distributed.129 It will be shewn presently, however, that
although we are unable to infer anything about *P* in this case,
we are able to draw an inference concerning *not-P*.

Jevons considers that the fact that the particular negative
proposition is incapable of ordinary conversion “constitutes a
blot in the ancient logic” (*Studies in Deductive Logic*,
p. 37). There is, however, no sufficient justification for this
criticism. We shall find subsequently that just as much can be
inferred from the particular negative as from the particular
affirmative (since the latter unlike the former does not admit of
contraposition). No logic, symbolic or other, can actually obtain more
from the given information than the ancient logic does. It has been
suggested that what Jevons means is that the inconvertibility of
**O** results in a want of symmetry and that logicians ought
specially to aim at symmetry. With this last contention we may
heartily agree. The want of symmetry, however, in the case before us
is apparent only and results from taking an incomplete view. It will
be found that symmetry reappears later on.130

99. *Legitimacy of
Conversion*.—Aristotle proves the conversion of **E**
*indirectly*, as follows;131 *No S is P*,
therefore, *No P is S* ; for if not, *Some individual P, say Q,
is S* ; and hence *Q is both S and P* ; but this is inconsistent
with the original proposition.

^{131} “By the method called
ἔκθεσις, *i.e.*, by the *exhibition* of an individual instance.” See Mansel’s *Aldrich*, pp. 61, 2.

Having shewn that the simple conversion of **E** is legitimate,
we can prove that the conversion *per accidens* of **A** is
also legitimate. *All S is P*, therefore, *Some P is S* ;
for, if not, *No P is S*, and therefore (by conversion) *No S
is P* ; but this 131 is inconsistent with the original
supposition. The legitimacy of the simple conversion of **I**
follows similarly.

The above proof appears to involve nothing beyond the principles of
contradiction and excluded middle. The proof itself, however, is not
satisfactory; for it practically assumes the validity of the very
process that it seeks to justify, that is to say, it assumes the
equivalence of the propositions *S is Q* and *Q is S*.

A better justification of the process of conversion may be obtained
by considering the class relations involved in the propositions
concerned. Thus, taking an **E** proposition, it is self-evident
that if one class is entirely excluded from another class, this second
class is entirely excluded from the first.132 In the case of an
**A** proposition it is clear on reflection that the statement
*All S is P* is consistent with either of two relations of the
classes *S* and *P*, namely, *S* and *P*
coincident, or *P* containing *S* and more besides, and
further that these are the only two possible relations with which it
is consistent. It is self-evident that in each of these cases *Some
P is S* ; and hence the inference by conversion from an **A**
proposition is shewn to be justified.133 In the case of an
**O** proposition, if we consider all the relationships of classes
in which it holds good, we find that nothing is true of *P* in
terms of *S* in *all* of them. Hence **O** is
inconvertible.134 The inconvertibility of **O** can also be
established 132 by
shewing that *Some S is not P* is compatible with every one of
the following propositions—*All P is S*, *Some P is
S*, *No P is S*, *Some P is not S*.

^{132} It is impossible to agree
with Professor Bain, who would establish the rules of conversion by a
kind of inductive proof. He writes as follows:—“When we
examine carefully the various processes in Logic, we find them to be
material to the very core. Take *Conversion*. How do we know
that, if No *X* is *Y*, No *Y* is *X*? By
examining cases in detail, and finding the equivalence to be true.
Obvious as the inference seems on the mere formal ground, we do not
content ourselves with the formal aspect. If we did, we should be as
likely to say, All *X* is *Y* gives All *Y* is
*X* ; we are prevented from this leap merely by the examination of
cases” (*Logic*, *Deduction*, p. 251). But no one
would on reflection maintain it to be self-evident that the simple
conversion of **A** is legitimate; for when the case is put to us
we recognise immediately that the contradictory of *All P is S*
is compatible with *All S is P*. On the other hand, no one can
deny that in the case of **E** the legitimacy of the process of
conversion is self-evident.

^{133} Compare section 126, where this and other similar inferences are illustrated by the aid of the
Eulerian diagrams.

100. *Table of
Propositions connecting any two terms*.—There
are—connecting any two terms *S* and *P*—eight
propositions of the forms **A**, **E**, **I**, **O**,
namely, four with *S* as subject, and four with *P* as
subject. The results at which we have arrived concerning the
conversion of propositions shew that of these eight, the two **E**
propositions are equivalent to one another, and that the same is true
of the two **I** propositions, **E** and **I** being simply
convertible; also that these are the only equivalences obtainable. We
have, therefore, the following table of propositions connecting any
two terms *S* and *P*:—

SaP, |

PaS, |

SeP = PeS, |

SiP = PiS, |

SoP, |

PoS. |

The pair of propositions *SaP* and *PaS* are independent
(see section 84); and the same is true of the pairs *SoP* and
*PoS*, *SaP* and *PoS*, *PaS* and *SoP*. The
first pair taken together indicate that the classes *S* and
*P* are coextensive, and they may be called *complementary*
propositions. The second pair taken together indicate that the classes
*S* and *P* are neither coextensive nor either included
within the other; they may be called *sub-complementary*
propositions. The third pair taken together indicate that the class
*S* is included within the class *P* but that it does not
exhaust that class; they may be called *contra-complementary*
propositions. The fourth pair taken together indicate that the class
*P* is included within the class *S* but that it does not
exhaust that class; they are, therefore, also
*contra-complementary*.135

^{135} The new technical terms
here introduced have been suggested by Mr Johnson.

The above table will be supplemented in section 106 by a table of propositions connecting any two terms and their 133 contradictories,
*S*, *P*, *not-S*, *not-P*. It will then be found that we have a symmetry that is at present wanting.

101. *The Obversion of
Categorical Propositions*.136—Obversion is
*a process of immediate inference in which the inferred
proposition* (or *obverse*), *whilst retaining the original
subject, has for its predicate the contradictory of the predicate of
the original proposition* (or *obvertend*). This process is
legitimate for a proposition of any form if at the same time the
quality of the proposition is changed. The inferred proposition is,
moreover, in all cases equivalent to the original proposition, so that
we can always pass back from the obverse to the obvertend.

^{136} The process of immediate
inference discussed in this section has been called by a good many
different names. The term *obversion*, which is used by Professor
Bain, is the most convenient. Other names which have been used are
*permutation* (Fowler), *aequipollence* (Ueberweg),
*infinitation* (Bowen), *immediate inference by private
conception* (Jevons), *contraversion* (De Morgan),
*contraposition* (Spalding). Professor Bain distinguishes between
*formal obversion* and *material obversion*. By *formal
obversion* is meant the kind of obversion discussed in the above
section, and this is the only kind of obversion that can properly be
recognised by the formal logician. *Material obversion* is
described as the process of making “obverse inferences which are
justified only on an examination of the matter of the
proposition” (*Logic*, vol. i., p. 111); and the following
are given as examples—“Warmth is agreeable; therefore,
cold is disagreeable. War is productive of evil; therefore, peace is
productive of good. Knowledge is good; therefore, ignorance is
bad.” It is very doubtful if these are legitimate inferences,
formal or otherwise. The conclusions appear to require quite
independent investigations to establish them. Apart from this,
however, it is a mistake to regard the process as analogous to formal
obversion. In the latter, the inferred proposition has the same
subject as the original proposition, whilst its quality is different;
but neither of these conditions is fulfilled in the above examples.
The process is really more akin to the immediate inference presently
to be discussed under the name of *inversion*.

We have the following table:—

Obvertend. | Obverse. |

All S is P. A. | No S is
not-P. E |

Some S is P. I. | Some S is not not-P. O. |

No S is P. E. | All S is not-P. A. |

Some S is not P. O. | Some S is not-P. I. |

134 It will be
observed that the obversion of *All S is P* depends upon the
principle of contradiction, which tells us that if anything is
*P* then it is not *not-P*; but that we pass back from *No S is not-P* to *All S is P* by the principle of excluded middle, which tells us that if anything is not *not-P* then it is
*P*. The remaining inferences by obversion also depend upon one
or other of these two principles.

102. *The Contraposition
of Categorical Propositions*.137—Contraposition may be defined as *a
process of immediate inference in which from a given proposition
another proposition is inferred having for its subject the
contradictory of the original predicate*. Thus, given a proposition
having *S* for its subject and *P* for its predicate, we
seek to obtain by immediate inference a new proposition having
*not-P* for its subject.

^{137} This form of immediate
inference is called by some logicians *conversion by negation* ;
Miss Jones suggests the name *contraversion*. More strictly we
might speak of *conversion by contraposition*. The word
*contrapositive* was used by Boethius for the opposite of a term
(*e.g.*, *not-A*), the word contradictory being confined to
propositional forms; and the passage from *All S is P* to *All
not-P is not-S* was called *Conversio per contrapositionem
terminorum*. In this usage Boethius was followed by the medieval
logicians. Compare Minto, *Logic*, pp. 151, 153.

It will be observed that in the above definition it is left an open
question whether the contrapositive of a proposition has the original
subject or the contradictory of the original subject for its
predicate; and every proposition which admits of contraposition will
accordingly have two contrapositives, each of which is the obverse of
the other. For example, in the case of *All S is P* there are the
two forms *No not-P is S* and *All not-P is not-S*. For many
purposes the distinction may be practically neglected without risk of
confusion. It will be observed, however, that when *not-S* is
taken as the predicate of the contrapositive, the quality of the
original proposition is preserved and there is greater symmetry.138
On the other hand, 135 if we regard contraposition as compounded
out of obversion and conversion in the manner indicated in the
following paragraph, the form with *S* as predicate is the more
readily obtained. Perhaps the best solution (in cases in which it is
necessary to mark the distinction) is to speak of the form with
*not-S* as predicate as the full contrapositive, and the form
with *S* as predicate as the partial contrapositive.139

^{138} The following is from
Mansel’s *Aldrich*, p. 61,—“Conversion by
contraposition, which is not employed by Aristotle, is given by
Boethius in his first book, *De Syllogismo Categorico*. He is
followed by Petrus Hispanus. It should be observed, that the old
logicians, following Boethius, maintain that in conversion by
contraposition, as well as in the others, the *quality* should
remain unchanged. Consequently the converse of ‘All *A* is
*B*’ is ‘All not-*B* is not-*A*,’ and
of ‘Some *A* is not *B*,’ ‘Some
not-*B* is not not-*A*.’ It is simpler, however, to
convert **A** into **E**, and **O** into **I**, (‘No
not-*B* is *A*,’ ‘Some not-*B* is
*A*’), as is done by Wallis and Archbishop Whately; and
before Boethius by Apuleius and Capella, who notice the conversion,
but do not give it a name. The principle of this conversion may be
found in Aristotle, *Top.* II. 8. 1,
though he does not employ it for logical purposes.”

^{139} In previous editions the
form with *S* as predicate was called the contrapositive, and the
form with *not-S* as predicate was called the obverted
contrapositive.

The following rule may be adopted for obtaining the full
contrapositive of a given proposition:—Obvert the original
proposition, then convert the proposition thus obtained, and then once
more obvert. For given a proposition with *S* as subject and
*P* as predicate, obversion will yield an equivalent proposition
with *S* as subject and *not-P* as predicate; the conversion
of this will make *not-P* the subject and *S* the predicate;
and a repetition of the process of obversion will yield a proposition
with *not-P* as subject and *not-S* as predicate.

Applying this rule, we have the following table:—

Original Proposition | Obverse | Partial Contrapositive | Full Contrapositive |

All S is P. A. |
No S is not-P. E. |
No not-P is S. E. |
All not-P is not-S A. |

Some S is P. I. |
Some S is not not-P. O. |
(None.) | (None.) |

No S is P. E. |
All S is not-P. A. |
Some not-P is S. I. |
Some not-P is not not-S. O. |

Some S is not P. O. |
Some S is not-P. I. |
Some not-P is S. I. |
Some not-P is not not-S. O. |

It will be observed that in the case of **A** and **O**, the
contrapositive is equivalent to the original proposition, the quantity
136 being unchanged,
whereas in the case of **E** we pass from a universal to a
particular.140 In order to emphasize this difference, and
following the analogy of ordinary conversion, the contraposition of
**A** and **O** has been called *simple contraposition*,
and that of **E** *contraposition per accidens*.141

^{140} In most text-books, no
*definition* of contraposition is given at all, and it may be pointed
out that, in the attempt to generalise from special examples, Jevons
in his *Elementary Lessons in Logic* involves himself in
difficulties. For the contrapositive of **A** he gives *All not-P
is not-S* ; **O** he says has no contrapositive (but only a
converse by negation, *Some not-P is S*); and for the
contrapositive of **E** he gives *No P is S*. It is impossible
to discover any definition of contraposition that can yield these
results. Assuming that in contraposition the quality of the
proposition is to remain unchanged as in Jevons’s contrapositive
of **A**, then the contrapositive of both **E** and **O** is
*Some not-P is not not-S*.

^{141} Compare Ueberweg, *Logic*, § 90.

That **I** has no contrapositive follows from the
inconvertibility of **O**. For when *Some S is P* is obverted
it becomes a particular negative, and the conversion of this
proposition would be necessary in order to render the contraposition
of the original proposition possible.

As regards the utility of the investigation as to the inferences
that can be drawn from given propositions by the aid of
contraposition, De Morgan142 points out that
the recognition that *Every not-P is not-S* follows from *Every
S is P*, whatever *S* and *P* may stand for, renders
unnecessary the special proofs that Euclid gives of certain of his
theorems.143

^{142} *Syllabus of Logic*, p. 32.

^{143} It will be found that,
taking Euclid’s first book, proposition 6 is obtainable by
contraposition from proposition 18, and 19 from 5 and 18 combined; or
that 5 can be obtained by contraposition from 19, and 18 from 6 and
19. Similar relations subsist between propositions 4, 8, 24, and 25;
and, again, between axiom 12 and propositions 16, 28, and 29. Other
examples might be taken from Euclid’s later books. In some of
the cases the logical relations in which the propositions stand to one
another are obvious; in other cases some supplementary steps are
necessary.

In consequence of his dislike of negative terms Sigwart regards the
passage from *All S is P* to *No not-P is S* as an artificial perversion. But he recognises the value of the inference
from *If anything is S it is P* to *If anything is not P it is
not S*. This distinction seems to be little more than verbal. It is
to 137 be observed
that we can avoid the use of negative terms without having recourse to
the conditional form of proposition: for example, *Whatever is S is
P*, therefore, *Whatever is not P is not S* ; *Anything that
is S is P*, therefore, *Anything that is not P is not S*.

103. *The Inversion of
Categorical Propositions*.—In discussing conversion and
contraposition we have enquired in what cases it is possible, having
given a proposition with *S* as subject and *P* as
predicate, to infer (*a*) a proposition with *P* as subject,
(*b*) a proposition with *not-P* as subject. We may now enquire further in what cases it is possible to infer (*c*) a
proposition with *not-S* as subject.

If such a proposition can be inferred at all, it will be obtainable by a certain combination of the more elementary processes of ordinary conversion and obversion.144 We will, therefore, take each of the fundamental forms of proposition and see what can be inferred (1) by first converting it, and then performing alternately the operations of obversion and conversion; (2) by first obverting it, and then performing alternately the operations of conversion and obversion. It will be found that in each case the process can be continued until a particular negative proposition is reached whose turn it is to be converted.

^{144} It might also be obtained
directly; by the aid, for example, of Euler’s circles. See the
following chapter.

(1) The results of performing alternately the processes of
conversion and obversion, commencing with the *former*, are as
follows:—

(i) All *S* is *P*,

therefore (by conversion), Some *P* is *S*,

therefore (by obversion), Some *P* is not
not-*S*.

Here comes the turn for conversion; but as we have to deal with an
**O** proposition, we can proceed no further.

(ii) Some *S* is *P*,

therefore (by conversion), Some *P* is *S*,

therefore (by obversion), Some *P* is not
not-*S* ;

and again we can go no further. 138

(iii) No *S* is *P*,

therefore (by conversion), No *P* is *S*,

therefore (by obversion), All *P* is not-*S*,

therefore (by conversion), *Some not-S is P*,

therefore (by obversion), *Some not-S is not not-P*.

In this case either of the propositions in italics is the immediate
inference that was sought.

(iv) Some *S* is not *P*.

In this case we are not able even to commence our series of
operations.

(2) The results of performing alternately the processes of
conversion and obversion, commencing with the *latter*, are as
follows:—

(i) All *S* is *P*,

therefore (by obversion), No *S* is not-*P*,

therefore (by conversion), No not-*P* is *S*,

therefore (by obversion), All not-*P* is not-*S*,

therefore (by conversion), *Some not-S is* *not-P*,

therefore (by obversion), *Some not-S is not P*.

Here again we have obtained the desired form.

(ii) Some *S* is *P*,

therefore (by obversion), Some *S* is not not-*P*.

(iii) No *S* is *P*,

therefore (by obversion), All *S* is not-*P*,

therefore (by conversion), Some not-*P* is *S*,

therefore (by obversion), Some not-*P* is not not-*S*.

(iv) Some *S* is not *P*,

therefore (by obversion), Some *S* is not-*P*,

therefore (by conversion), Some not-*P* is *S*,

therefore (by obversion), Some not-*P* is not not-*S*.

We can now answer the question with which we commenced this
enquiry. The required proposition can be obtained only if the given
proposition is universal; we then have, according as it is affirmative
or negative,—

*All S is P*, therefore, *Some not-S is not P* (= *Some
not-S is* *not-P*); 139

*No S is P*, therefore, *Some not-S is P* (= *Some
not-S is not* *not-P*).

This form of immediate inference has been more or less casually
recognised by various logicians, without receiving any distinctive
name. Sometimes it has been vaguely classed under contraposition
(compare Jevons, *Elementary Lessons in Logic*, pp. 185, 6), but
it is really as far removed from the process to which that designation
has been given as the latter is from ordinary conversion. The term
*inversion* was suggested in an earlier edition of this work, and
has since been adopted by some other writers. Inversion may be defined
as *a process of immediate inference in which from a given
proposition another proposition is inferred having for its subject the
contradictory of the original subject*. Thus, given a proposition
with *S* as subject and *P* as predicate, we obtain by
inversion a new proposition with *not-S* as subject. The original
proposition may be called the *invertend*, and the inferred
proposition the *inverse*.

In the above definition it is not specified whether the inverse is
to have for its predicate *P* or *not-P*. Hence two forms
(each being the obverse of the other) have been obtained as in the
case of contraposition. So far as it is necessary to mark the
distinction, we may speak of the form in which *P* is the
predicate as the partial inverse, and of that in which *not-P* is the predicate as the full inverse.

104. *The Validity of
Inversion*.—It will be remembered that we are at present
working on the assumption that each class represented by a simple term
exists in the universe of discourse, while at the same time it does
not exhaust that universe; in other words, we assume that *S*,
*not-S*, *P*, *not-P*, all represent existing classes.
This assumption is perhaps specially important in the case of
inversion, and it is connected with certain difficulties that may have
already occurred to the reader. In passing from *All S is P* to
its inverse *Some not-S is not P * there is an apparent illicit process, which it is not quite easy either to account for or explain
away. For the term *P*, which is undistributed in the premiss, is
distributed in the conclusion, and yet if the universal validity of
obversion and 140
conversion is granted, it is impossible to detect any flaw in the
argument by which the conclusion is reached. It is in the assumption
of the existence of the contradictory of the original predicate that
an explanation of the apparent anomaly may be found. That assumption
may be expressed in the form *Some things are not P*. The
conclusion *Some not-S is not P* may accordingly be regarded as
based on this premiss combined with the explicit premiss *All S is
P* ; and it will be observed that, in the additional premiss,
*P* is distributed.145

^{145} The question of the
validity of inversion under other assumptions will be considered in
chapter 8.

105. *Summary of
Results*.—The results obtained in the preceding sections are
summed up in the following table:—

A. | E. | I. | O. | ||

i | Original proposition | SaP | SiP | SeP | SoP |

ii | Obverse | SePʹ | SoPʹ
| SaPʹ | SiPʹ |

iii | Converse | PiS | PiS | PeS
| |

iv | Obverted Converse | PoSʹ | PoSʹ
| PaSʹ | |

v | Partial Contrapositive146 | PʹeS
| PʹiS | PʹiS | |

vi | Full Contrapositive146 | PʹaSʹ | PʹoSʹ | PʹoSʹ | |

vii | Partial Inverse146 | SʹoP | SʹiP | ||

viii | Full Inverse146 | SʹiPʹ |
SʹoPʹ |

^{146} In previous editions what
are here called the partial contrapositive and the full contrapositive
respectively were called the contrapositive and the obverted
contrapositive; and what are here called the partial inverse and the
full inverse were called the inverse and the obverted
inverse.

It may be pointed out that the following rules apply to all the
above immediate inferences:— 141

*Rule of Quality*.—The total number of negatives
admitted or omitted in subject, predicate, or copula must be even.

*Rules of Quantity*.—If the new subject is *S*, the
quantity may remain unchanged; if *Sʹ*, the quantity must be
depressed;147 if *P*, the quantity must be depressed
in **A** and **O**; if *Pʹ*, the quantity must be depressed
in **E** and **I**.

^{147} In speaking of the quantity
as depressed, it is meant that a universal yields a particular, and a
particular yields nothing.

106. *Table of
Propositions connecting any two terms and their
contradictories*.—Taking any two terms and their
contradictories, *S*, *P*, *not-S*, *not-P*, and
combining them in pairs, we obtain thirty-two propositions of the
forms **A**, **E**, **I**, **O**. The following table,
however, shews that only eight of these thirty-two propositions are
non-equivalent.

(i) | (ii) | (iii) | (iv) | ||||

Universals | |||||||

A | SaP | = | SePʹ | = | PʹeS | = |
PʹaSʹ |

Aʹ | SʹaPʹ | = | SʹeP | = | PeSʹ | = | PaS |

E |
SaPʹ | = | SeP | = | PeS | = | PaSʹ |

Eʹ | SʹaP | = | SʹePʹ | = |
PʹeSʹ | = | PʹaS |

Particulars | |||||||

O | SoP | = | SiPʹ | = | PʹiS | = |
PʹoSʹ |

Oʹ | SʹoPʹ | = |
SʹiP | = | PiSʹ | = | PoS |

I |
SoPʹ | = | SiP | = | PiS | = | PoSʹ |

Iʹ | SʹoP | = | SʹiPʹ | = |
PʹiSʹ | = | PʹoS |

In this table, columns (i) and (ii) contain the propositions in
which *S* or *Sʹ* is subject, and columns (iii) and (iv)
the propositions in which *P* or *Pʹ* is subject. In
columns (i) and (iv) we have the forms which admit of simple
contraposition (*i.e.*, **A** and **O**), and in columns
(ii) and (iii) those which admit of simple conversion (*i.e.*,
**E** and **I**). Contradictories are shewn by identical places
in the universal and particular rows. We pass from column (i) to
column (ii) by obversion; from column (ii) to column (iii) by simple
conversion; and from column (iii) to column (iv) by obversion.

The forms in black type shew that we may take for our 142 eight non-equivalent
propositions the four propositions connecting *S* and *P*,
and a similar set connecting *not-S* and *not-P*.148
To establish their non-equivalence we may proceed as follows:
*SaP* and *SeP* are already known to be non-equivalent, and
the same is true of *SʹaPʹ* and *SʹePʹ* ; but
no universal proposition can yield a universal inverse; therefore, no
one of these four propositions is equivalent to any other. Again,
*SiP* and *SoP* are already known to be non-equivalent, and
the same is true of *SʹiPʹ* and *SʹoPʹ* ; but
no particular proposition has any inverse; therefore, no one of these
propositions is equivalent to any other. Finally, no universal
proposition can be equivalent to a particular proposition.149

^{148} The former set being
denoted by **A**, **E**, **I**, **O**, the latter set may
be denoted by **Aʹ**, **Eʹ**, **Iʹ**,
**Oʹ**.

^{149} Mrs Ladd Franklin, in an
article on *The Proposition* in *Baldwin’s Dictionary of
Philosophy and Psychology*, reaches the result arrived at in this
section from a different point of view. Mrs Franklin shews that, if we
express everything that can be said in the form of existential
propositions (that is, propositions affirming or denying existence),
it is at once evident that the actual number of different statements
possible in terms of *X* and *Y* and their contradictories
*x* and *y* is eight. For the combinations of *X* and
*Y* and their contradictories are *XY*, *Xy*,
*xY*, *xy*, and we can affirm each of these combinations to
exist or to be non-existent. Hence it is clear that eight different
statements of fact are possible, and that these eight must remain
different, no matter what the form in which they may be expressed.

It may be worth adding that the conditional and disjunctive forms
as well as the categorical may here be included on the understanding
that all the propositions are interpreted assertorically. Thus, the
four following propositions are, on the above understanding,
equivalent to one another: *All X is Y* (categorical); *If
anything is X, it is Y* (conditional); *Nothing is Xy*
(existential); *Everything is x or Y* (disjunctive).

107. *Mutual Relations of
the non-equivalent Propositions connecting any two terms and their
contradictories*.150—We may now investigate the mutual
relations of our eight non-equivalent propositions. *SaP*,
*SeP*, *SiP*, *SoP* form an ordinary square of
opposition; and so do *SʹaPʹ*, *SʹePʹ*,
*SʹiPʹ*, *SʹoPʹ*. Reference to columns (iii)
and (iv) in the table will shew further that *SaP*,
*SʹePʹ*, *SʹiPʹ*, *SoP* are equivalent
to another square of opposition; and that the same is true of
*SʹaPʹ*, *SeP*, *SiP*, *SʹoPʹ*.
This leaves only the following pairs unaccounted for: 143 *SaP*,
*SʹaPʹ* ; *SeP*, *SʹePʹ* ; *SoP*,
*SʹoPʹ* ; *SiP*, *SʹiPʹ* ; *SaP*,
*SʹoPʹ* ; *SʹaPʹ*, *SoP* ; *SeP*,
*SʹiPʹ* ; *SʹePʹ*, *SiP* ; and it will be
found that in each of these cases we have an independent pair.

^{150} This section may be omitted on a first reading.

*SaP* and *SʹaPʹ* (which are equivalent to
*SaP*, *PaS*, and also to *PʹaSʹ*,
*SʹaPʹ*) taken together serve to identify the classes
*S* and *P*, and also the classes *Sʹ* and
*Pʹ*. They are therefore *complementary* propositions,
in accordance with the definition given in section 100. Similarly,
*SeP* and *SʹePʹ* (which are equivalent to
*SaPʹ*, *PʹaS*, and also to *PaSʹ*,
*SʹaP*) are complementary; they serve to identify the classes
*S* and *Pʹ*, and also the classes *Sʹ* and
*P*. It will be observed that the complementary of any universal
proposition may be obtained by replacing the subject and predicate
respectively by their contradictories. A not uncommon fallacy is the
tacit substitution of the complementary of a proposition for the
proposition itself.

The complementary relation holds only between universals.
Particulars between which there is an analogous relation (the subject
and predicate of the one being respectively the contradictories of the
subject and predicate of the other) will be found to be
*sub-complementary* in accordance with the definition in section
100; this relation holds between *SoP* and *SʹoPʹ*,
and between *SiP* and *SʹiPʹ*. *SoP* and
*SʹoPʹ* (which are equivalent to *SoP*, *PoS*,
and also to *PʹoSʹ*, *SʹoPʹ*) indicate that
the classes *S* and *P* are neither coextensive nor either
included within the other, and also that the same is true of
*Sʹ* and *Pʹ* ; *SiP* and *SʹiPʹ*
(which are equivalent to *SoPʹ*, *PʹoS*, and also to
*PoSʹ*, *SʹoP*) indicate the same thing as regards
*S* and *Pʹ*, *Sʹ* and *P*.

The four remaining pairs are *contra-complementary*, each pair
serving conjointly to *subordinate* a certain class to a certain
other class; or, rather, since each such subordination implies a
supplementary subordination, we may say that each pair subordinates
two classes to two other classes. Thus, *SaP* and
*SʹoPʹ* (which are equivalent to *SaP*, *PoS*,
and also to *PʹaSʹ*, *SʹoPʹ*) taken together
shew that the class *S* is contained in but does not exhaust the
class *P*, and also that the class *Pʹ* is contained in
but does not exhaust the class *Sʹ* ; *SʹaPʹ* and
*SoP* (which are equivalent to *SʹaPʹ*,
*PʹoSʹ*, and also to *PaS*, *SoP*) yield the
same results as regards the classes *Sʹ* and *Pʹ*,
and the classes *P* and *S* ; *SeP* and
*SʹiPʹ* (which are equivalent 144 to *SaPʹ*, *PʹoS*, and
also to *PaSʹ*, *SʹoP*) as regards *S* and
*Pʹ*, and *P* and *Sʹ* ; and
*SʹePʹ* and *SiP* (which are equivalent to
*SʹaP*, *PoSʹ*, and also to *PʹaS*,
*SoPʹ*) as regards *Sʹ* and *P*, *Pʹ*
and *S*.

Denoting the complementaries of **A** and **E** by
**Aʹ** and **Eʹ**, and the sub-complementaries of
**I** and **O** by **Iʹ** and **Oʹ**, the various
relations between the non-equivalent propositions connecting any two
terms and their contradictories may be exhibited in the following
*octagon* of opposition:

Each of the *dotted* lines in the above takes the place of
*four* connecting lines which are not filled in; for example, the
dotted line marked as connecting contraries indicates the relation
between **A** and **E**, **A** and **Eʹ**,
**Aʹ** and **E**, **Aʹ** and **Eʹ**.151

^{151} For the octagon of opposition in the form in which it is here given I am indebted to Mr Johnson.

108. *The Elimination of
Negative Terms*.152—The process of obversion enables us by
the aid of negative terms to reduce all propositions to the
affirmative form; and the question may be 145 raised whether the various processes of
immediate inference and the use, where necessary, of negative
propositions will not equally enable us to eliminate negative terms.

^{152} This section may be omitted on a first reading.

It is of course clear that by means of obversion we can get rid of
a negative term occurring as the predicate of a proposition. The
problem is more difficult when the negative term occurs as subject,
but in this case elimination may still be possible; for example,
*SʹiP* = *PoS*. We may even be able to get rid of two
negative terms; for example, *SʹaPʹ* = *PaS*. So
long, however, as we are limited to categorical propositions of the
ordinary type we cannot eliminate a negative term (without introducing
another in its place) where such a term occurs as subject either
(*a*) in a universal affirmative or a particular negative with a
positive term as predicate, or (*b*) in a universal negative or a
particular affirmative with a negative term as predicate.

The validity of the above results is at once shewn by reference to the table of equivalences given in section 106. At least one proposition in which there is no negative term will be found in each line of equivalences except the fourth and the eighth, which are as follows:

SʹaP | = | SʹePʹ | = |
PʹeSʹ | = | PʹaS ; |

SʹoP | = | SʹiPʹ | = |
PʹiSʹ | = | PʹoS. |

In these cases we may indeed get rid of *Sʹ* (as, for
example, from *SʹaP*), but it is only by introducing
*Pʹ* (thus, *SʹaP* = *PʹaS*); there is no
getting rid of negative terms altogether. We may here refer back to
the results obtained in sections 100 and 106; with two terms
*six* non-equivalent propositions were obtained, with two terms
and their contradictories *eight* non-equivalent propositions.
The ground of this difference is now made clear.

If, however, we are allowed to enlarge our scheme of propositions
by recognising certain additional types, and if we work on the
assumption that universal propositions are existentially negative
while particular propositions are existentially affirmative,153
then negative terms may always be eliminated.146 Thus, *No not-S is not-P* is
equivalent to the statement *Nothing is both not-S and not-P*,
and this becomes by obversion *Everything is either S or P*.
Again, *Some not-S is not-P* is equivalent to the statement
*Something is both not-S and not-P*, and this becomes by
obversion *Something is not either S or P*, or, as this
proposition may also be written, *There is something besides S and
P*. The elimination of negative terms has now been accomplished in
all cases. It will be observed further that we now have *eight*
non-equivalent propositions containing only *S* and
*P*—namely, *All S is P*, *No S is P*, *Some S
is P*, *Some S is not P*, *All P is S*, *Some P is not
S*, *Everything is either S or P*, *There is something
besides S and P*.

^{153} It is necessary here to
anticipate the results of a discussion that will come at a later
stage. See chapter 8.

Following out this line of treatment, the table of equivalences given in section 106 may be rewritten as follows [columns (ii) and (iii) being omitted, and columns (v) and (vi) taking their places]:

(i) | (iv) | (v) | (vi) | |||

SaP | = | PʹaSʹ | = | Nothing is SPʹ | = | Everything is Sʹ or P. |

SʹaPʹ | = | PaS | = | Nothing is SʹP | = | Everything is S or Pʹ. |

SaPʹ | = | PaSʹ | = | Nothing is SP | = | Everything is Sʹ or Pʹ. |

SʹaP | = | PʹaS | = |
Nothing is SʹPʹ | = | Everything is S or P. |

SoP | = | PʹoSʹ | = | Something is SPʹ | = | There is something besides Sʹ
and P. |

SʹoPʹ | = | PoS | = | Something is SʹP | = | There is something besides S and Pʹ. |

SoPʹ | = | PoSʹ | = | Something is SP | = | There is something besides Sʹ and Pʹ. |

SʹoP | = | PʹoS | = |
Something is SʹPʹ | = | There is
something besides S and P. |

Taking the propositions in two divisions of four sets each, the two
diagonals from left to right give propositions containing *S* and
*P* only.154

^{154} The first four propositions
in column (v) may be expressed symbolically *SPʹ* = 0,
&c.; the second four *SPʹ* > 0, &c.; the first four
in column (vi) *Sʹ* + *P* = 1, &c.; and the second
four *Sʹ* + *P* < 1, &c.; where 1 = the universe of
discourse, and 0 = nonentity, *i.e.*, the contradictory of
the universe of discourse. Compare section 138.

147 The scheme of propositions given in this section may be brought into interesting relation with the three fundamental laws of thought.155 The scheme is based upon the recognition of the following propositional forms and their contradictories:

*Every S is P* ;

*Every not-P is not-S* ;

*Nothing is both S and not-P* ;

*Everything is either P or not-S* ;

and these four propositions
have been shewn to be equivalent to one another.

^{155} Compare Mrs Ladd Franklin
in *Mind*, January, 1890, p. 87.

If in the above propositions we now write *S* for *P*, we
have the following:

*Every S is S* ;

*Every not-S is not-S* ;

*Nothing is both S and not-S* ;

*Everything is either S or not-S*.

But the first two of these propositions express the law of identity, with positive and negative terms respectively, the third is an expression of the law of contradiction, and the fourth of the law of excluded middle. The scheme of propositions with which we have been dealing may, therefore, be said to be based upon the recognition of just those propositional forms which are required in order to express the fundamental laws of thought.

Since the propositional forms in question have been shewn to be
mutually equivalent to one another, the further argument may suggest
itself that if the validity of the immediate inferences involved be
granted, then it follows that the fundamental laws of thought have
been shewn to be mutually inferable from one another. But it may, on
the other hand, be held that this argument is open to the charge of
involving a *circulus in probando* on the ground that the
validity of the immediate inferences themselves requires that the laws
of thought be first postulated as an antecedent condition.

109. *Other Immediate
Inferences*.—Some other commonly recognised forms of
immediate inference may be briefly touched upon. 148

(1) *Immediate inferences based on the square of opposition*
have been discussed in the preceding chapter.

(2) *Immediate inference by change of relation* is the process
whereby we pass from a categorical proposition to a conditional or a
disjunctive, or from a conditional to a disjunctive or a categorical,
or from a disjunctive to a categorical or a conditional.156
For example, *All S is P*, therefore, *If anything is S it is
P* ; *Every S is P or Q*, therefore, *Any S that is not P is
Q*. References have been already made to inferences such as these,
and they will be further discussed later on.

(3) *Immediate inference by added determinants* is a process
of immediate inference which consists in limiting both the subject and
the predicate of the original proposition by means of the same
determinant. For example,—*All P is Q*, therefore, *All
AP is AQ* ; A negro is a fellow creature, therefore, A suffering
negro is a suffering fellow creature. The formal validity of the
reasoning may be shewn as follows: *AP* is a subdivision of the
class *P*, namely, that part of it which also belongs to the
class *A* ; and, therefore, whatever is true of the whole of
*P* must be true of *AP* ; hence, given that *All P is
Q*, we can infer that *All AP is Q* ; moreover, by the law of
identity, *All AP is A* ; therefore, *All AP is AQ*.157

^{157} It must be observed,
however, that the validity of this argument requires an assumption in
regard to the existential import of propositions, which differs from
that which we have for the most part adopted up to this point. It has
to be assumed that universals do not imply the existence of their
subjects. Otherwise this inference would not be valid in the case of
no *P* being *A*. *P* might exist, and all *P*
might be *Q*, but we could not pass to *AP is AQ*, since
this would imply the existence of *AP*, which would be incorrect.
It is necessary briefly to call attention to the above at this point,
but our aim through all these earlier chapters has been to avoid as
far as possible the various complications that arise in connexion with
the difficult problem of existential import.

The formal validity of immediate inference by added determinants
has been denied on the ground of the obvious fallacy of arguing from
such a premiss as *an elephant is an animal* to the conclusion
*a small elephant is a small animal*, or from such a premiss as
*cricketers are men* to the conclusion *poor cricketers are
poor men*. In these cases, however, the fallacy really results from
the ambiguity of language, the added determinant 149 receiving a different
interpretation when it qualifies the subject from that which it has
when it qualifies the predicate. A term of comparison like
*small* can indeed hardly be said to have an independent
interpretation, its force always being relative to some other term
with which it is conjoined. While then the inference in its symbolic
form (*P is Q*, therefore, *AP is AQ*) is perfectly valid,
it is specially necessary to guard against fallacy in its use when
significant terms are employed. All that we have to insist upon is
that the added determinant shall receive the same interpretation in
both subject and predicate. There is, for example, no fallacy in the
following: An elephant is an animal, therefore, A small elephant is an
animal which is small compared with elephants generally; Cricketers
are men, therefore, Poor cricketers are men who in their capacity as
cricketers are poor.

(4) *Immediate inference by complex conception* is a process
of immediate inference which consists in employing the subject and the
predicate of the original proposition as parts of a more complex
conception. Symbolically we can only express it somewhat as follows:
*P is Q*, therefore, *Whatever stands in a certain relation to
P stands in the same relation to Q*. The following is a concrete
example: An elephant is an animal, therefore, the ear of an elephant
is the ear of an animal. A systematic treatment of this kind of
inference belongs to the special branch of formal logic known as the
*logic of relatives*, any detailed consideration of which is
beyond the scope of the present work. Attention may, however, be
called to the danger of our committing a fallacy, if we perform the
process carelessly. For example, Protestants are Christians,
therefore, A majority of Protestants are a majority of Christians; A
negro is a man, therefore, the best of negroes is the best of men. The
former of these fallacies is akin to the fallacy of composition (see
section 11), since we pass from the distributive to the collective use of a term.

(5) *Immediate inference by converse relation* is a process of
immediate inference analogous to ordinary conversion but belonging to
the logic of relatives. It consists in passing from a statement of the
relation in which *P* stands to *Q* to a 150 statement of the
relation in which *Q* consequently stands to *P*. The two
terms are transposed and the word by which their relation is expressed
is replaced by its correlative. For example, *A* is greater than
*B*, therefore, *B* is less than *A* ; Alexander was the
son of Philip, therefore, Philip was the father of Alexander; Freedom
is synonymous with liberty, therefore, Liberty is synonymous with
freedom.

Mansel gives the first two of the above as examples of *material
consequence* as distinguished from *formal consequence*.
“A Material Consequence is defined by Aldrich to be one in which
the conclusion follows from the premisses solely by the force of the
terms. This in fact means from some understood Proposition or
Propositions, connecting the terms, by the addition of which the mind
is enabled to reduce the Consequence to logical form…… The failure of
a Material Consequence takes place when no such connexion exists
between the terms as will warrant us in supplying the premisses
required; *i.e.*, when one or more of the premisses so supplied
would be *false*. But to determine this point is obviously beyond
the province of the Logician. For this reason, Material Consequence is
rightly excluded from Logic…… Among these material, and therefore
extralogical, Consequences, are to be classed those which Reid adduces
as cases for which Logic does not provide; *e.g.*,
‘Alexander was the son of Philip, therefore, Philip was the
father of Alexander’; ‘*A* is greater than *B*,
therefore, *B* is less than *A*.’ In both these it is
our material knowledge of the relations ‘father and son,’
‘greater and less,’ that enables us to make the
inference” (*Aldrich*, p. 199).

The distinction between what is formal and what is material is not
in reality so simple or so absolute as is here implied.158
It is usual to recognise as formal only those relations which can be
expressed by the ordinary copula *is* or *is not* ; and there
is very good reason for proceeding upon this basis in the greater part
of our logical discussions. No other relation is of the same
fundamental importance or admits of an equally developed logical
superstructure. But it is important to recognise that there are other
relations which may remain the 151 same while the things related vary; and
wherever this is the case we may regard the relation as constituting
the form and the things related the matter. Accordingly with each such
relation we may have a different formal system. The logic of relatives
deals with such systems as are outside the one ordinarily recognised.
Each immediate inference by converse relation will, therefore, be
formal in its own particular system. This point is admirably put by
Miss Jones: “A proposition containing a relative term
furnishes—besides the ordinary immediate inferences—other
immediate inferences to any one acquainted with the system to which it
refers. These inferences cannot be educed except by a person knowing
the ‘system’; on the other hand, no knowledge is needed of
the objects referred to, except a knowledge of their place in the
system, and this knowledge is in many cases coextensive with ordinary
intelligence; consider, *e.g.*, the relations of magnitude of
objects in space, of the successive parts of time, of family
connexions, of number” (*General Logic*, p. 34).

(6) *Immediate inference by modal consequence* or, as it is
also called, inference by change of modality, is somewhat analogous to
subaltern inference. It consists in nothing more than weakening a
statement in respect of its modality; and hence it is never possible
to pass back from the inferred to the original proposition. Thus, from
the *validity* of the apodeictic judgment we can pass to the
validity of the assertoric, and from that to the validity of the
problematic; but not *vice versâ*. On the other hand, from the
*invalidity* of the problematic judgment we can pass to the
invalidity of the assertoric, and from that to the invalidity of the
apodeictic; but again not *vice versâ*.159

^{159} Compare Ueberweg, *Logic*, § 98.

110. *Reduction of
immediate inferences to the mediate form*160—Immediate
inference has been defined as the inference of a proposition from a
single other proposition; mediate inference, on the other hand, is the
inference of a proposition from at least two other propositions.

^{160} Students who have not
already a technical knowledge of the syllogism may omit this section
until they have read the earlier chapters of Part III.

We may briefly consider various ways of establishing the validity of immediate inferences by means of mediate inferences.

152 (1) One of the old Greek logicians, Alexander of Aphrodisias,
establishes the conversion of **E** by means of a syllogism in *Ferio*.

No S is P, | |

therefore, | No P is S ; |

for, if
not, then by the law of contradiction, *Some P is S* ; and we have
this syllogism,—

No S is P, | |

Some P is S, | |

therefore, | Some P is not P, |

a *reductio ad absurdum*.161

^{161} Compare Mansel’s
*Aldrich*, p. 62. The conversion of **A** and the conversion
of **I** may be established similarly.

(2) It may be plausibly maintained that in Aristotle’s proof
of the conversion of **E** (given in section 99), there is an
implicit syllogism: namely,—*Q is P*, *Q is S*,
therefore, *Some S is P*.

(3) The contraposition of **A** may be established by means of a
syllogism in *Camestres* as follows:—

Given | All S is P, | |

we have also | No not-P is P, | by the law of contradiction, |

therefore, | No not-P is
S.162 |

^{162} Similarly, granting the
validity of obversion, the contraposition of **O** may be
established by a syllogism in *Datisi* as follows:—

Given *Some S is not P*, then we have

All S is S, | by the law of identity, | |

and | Some S is not-P, | by obversion of the given proposition, |

therefore, | Some not-P is S. |

It will be found that, adopting the same method, the contraposition
of **E** is yielded by a syllogism in *Darapti*.

(4) We might also obtain the contrapositive of *All S is P* as
follows:—

By the law of excluded middle, *All not-P is S or **not-S*, and, by hypothesis, *All S is P*,

therefore, | All not-P is P or
not-S ; |

but, by the law of contradiction, | No not-P is P, |

therefore, | All not-P is not-S.163 |

^{163} Compare Jevons,
*Principles of Science*, chapter 6, § 2; and *Studies in
Deductive Logic*, p. 44.

153 (5) The
contraposition of **A** may also be established indirectly by means
of a syllogism in *Darii*:—

All S is P, | |

therefore, | No not-P is S ; |

for, if not,
*Some not-P is S* ; and we have the following syllogism,—

All S is P, | |

Some not-P is S, | |

therefore, | Some not-P is P, |

which is absurd.164

^{164} Compare De Morgan,
*Formal Logic*, p. 25. Granting the validity of obversion, the
contraposition of **E** and the contraposition of **O** may be
established similarly.

All the above are interesting, as illustrating the processes of immediate inference; but regarded as proofs they labour under the disadvantage of deducing the less complex by means of the more complex.

111. Give all the logical
opposites of the proposition,—Some rich men are virtuous; and
also the converse of the contrary of its contradictory. How is the
latter directly related to the given proposition?

Does it follow that a proposition admits of simple conversion
because its predicate is distributed? [K.]

112. Point out any ambiguities in the following propositions, and give the contradictory and (where possible) the converse of each of them:—(i) Some of the candidates have been successful; (ii) All are not happy that seem so; (iii) All the fish weighed five pounds. [K.]

113. State in logical form
and convert the following propositions:—(*a*) He jests at
scars who never felt a wound; (*b*) Axioms are self-evident;
(*c*) Natives alone can stand the climate of Africa; (*d*)
Not one of the Greeks at Thermopylae escaped; (*e*) All that
glitters is not gold. [O.]

114. “The angles at the base of an isosceles triangle are equal.” What can be inferred from this proposition by obversion, conversion, and contraposition respectively? [L.]

154 115. Give the obverse, the contrapositive, and the inverse of each of the following propositions:—The virtuous alone are truly noble; No Athenians are Helots. [M.]

116. Give the contrapositive and (where possible) the inverse of the following propositions:—(i) A stitch in time saves nine; (ii) None but the brave deserve the fair; (iii) Blessed are the peacemakers; (iv) Things equal to the same thing are equal to one another; (v) Not every tale we hear is to be believed. [K.]

117. If it is false that
“Not only the virtuous are happy,” what can we infer
(*a*) with regard to the non-virtuous, (*b*) with regard to
the non-happy? [J.]

118. Write down the
contradictory, and also—where possible—the converse, the
contrapositive, and the inverse of each of the following propositions:

A bird in the hand is worth two in the bush;

No unjust acts are expedient;

All are not saints that go to church. [K.]

119. Give the contrapositive
and the inverse of each of the following propositions,—They
never fail who die in a great cause; Whom the Gods love die young.

If *A* is either *B* or else both *C* and *D*,
what do we know about that which is not *D*? [K.]

120. Take the following
propositions in pairs, and in regard to each pair state whether the
two propositions are consistent or inconsistent with each other; in
the former case, state further whether either proposition can be
inferred from the other, and, if it can be, point out the nature of
the inference; in the latter case, state whether it is possible for
both the propositions to be false:—(*a*) *All S is
P* ; (*b*) *All not-S is P* ; (*c*) *No P is S* ;
(*d*) *Some not-P is S*. [K.]

121. Transform the following
propositions in such a way that, without losing any of their force,
they may all have the same subject and the same predicate:—*No
not-P is S* ; *All P is not-S* ; *Some P is S* ; *Some
not-P is not not-S*. [K.]

122. Describe the logical
relations, if any, between each of the following propositions, and
each of the others:—

(i) There are no inorganic substances which do not contain carbon;
155

(ii) All organic substances contain carbon;

(iii) Some substances not containing carbon are organic;

(iv) Some inorganic substances do not contain carbon.
[C.]

123. “All that love
virtue love angling.”

Arrange the following propositions in the three following
groups:—(*α*) those which can be inferred from the above
proposition; (*β*) those which are consistent with it, but
which cannot be inferred from it; (*γ*) those which are
inconsistent with it.

(i) None that love not virtue love angling.

(ii) All that love angling love virtue.

(iii) All that love not angling love virtue.

(iv) None that love not angling love virtue.

(v) Some that love not virtue love angling.

(vi) Some that love not virtue love not angling

(vii) Some that love not angling love virtue.

(viii) Some that love not angling love not virtue. [K.]

124. Determine the logical
relation between each pair of the following propositions:—

(1) All crystals are solids.

(2) Some solids are not crystals.

(3) Some not crystals are not solids.

(4) No crystals are not solids.

(5) Some solids are crystals.

(6) Some not solids are not crystals.

(7) All solids are crystals. [L.]

125. *The use of Diagrams
in Logic*.—In representing propositions by geometrical
diagrams, our object is not that we may have a new set of symbols, but
that the relation between the subject and predicate of a proposition
may be exhibited by means of a sensible representation, the
signification of which is clear at a glance. Hence the first
requirement that ought to be satisfied by any diagrammatic scheme is
that the interpretation of the diagrams should be intuitively obvious,
as soon as the principle upon which they are based has been
explained.165

^{165} Hamilton’s
“geometric scheme,” which he himself describes as
“easy, simple, compendious, all-sufficient, consistent,
manifest, precise, complete” (*Logic*, II. p. 475), fails to satisfy this condition. He represents an affirmative copula by a
horizontal tapering line (), the broad end of which is
towards the subject. Negation is marked by a perpendicular line
crossing the horizontal one (). A colon (:) placed at
either end of the copula indicates that the corresponding term is
distributed; a comma (,) that it is undistributed. Thus, for *All S
is P* we have,—

*S* : , *P* ;

and similarly for the other propositions.

Dr Venn rightly observes that this scheme is purely symbolical, and
does not deserve to rank as a diagrammatic scheme at all. There is
clearly nothing in the two ends of a wedge to suggest subjects and
predicates, or in a colon and comma to suggest distribution and
non-distribution” (*Symbolic Logic*, p. 432).
Hamilton’s scheme may certainly be rejected as valueless. The
schemes of Euler and Lambert belong to an altogether different
category.

A second essential requirement is that the diagrams should be adequate; that is to say, they should give a complete, and 157 not a partial, representation of the relations which they are intended to indicate. Hamilton’s use of Euler’s diagrams, as described in the following section, will serve to illustrate the failure to satisfy this requirement.

In the third place, the diagrams should be capable of representing
all the propositional forms recognised in the schedule of propositions
which are to be illustrated, *e.g.*, particulars as well as
universal. One scheme of diagrams may, however, be better suited for
one purpose, and another scheme for another purpose. It will be found
that Dr Venn’s diagrams, to be described presently, are not
quite so well adapted to the representation of particulars as of
universals.

Lastly, it is advantageous that a diagrammatic scheme should be as little cumbrous as possible when it is desired to represent two or more propositions in combination with one another. This is the weak point of Euler’s method. A scheme of diagrams may, however, serve a very useful function in making clear the full force of individual propositions, even when it is not well adapted for the representation of combined propositions.

A further requirement is sometimes added, namely, that each propositional form should be represented by a single diagram, not by a set of alternative diagrams. This is, however, by no means essential. For if we adopt a schedule of propositions some of which yield only an indeterminate relation in respect of extension between the terms involved, it is important that this should be clearly brought out, and a set of alternative diagrams may be specially helpful for the purpose. This point will be illustrated, with reference to Euler’s diagrams, in the following section.166

^{166} It must be borne in mind
that in all the schemes described in this chapter the terms of the
propositions which are represented diagrammatically are taken in
extension, not in intension.

126. *Euler’s
Diagrams*.—We may begin with the well-known scheme of
diagrams, which was first expounded by the Swiss mathematician and
logician, Leonhard Euler, and which is usually called after his
name.167

^{167} Euler lived from 1707 to
1783. His diagrammatic scheme is given in his *Lettres à une
Princesse d’Allemagne* (Letters 102 to 105).

158 Representing the individuals included in any class, or denoted by any name, by a circle, it will be obvious that the five following diagrams represent all possible relations between any two classes:—

The force of the different propositional forms is to exclude one or
more of these possibilities.

*All S is P* limits us to one of the two *α*, *β* ;

*Some S is P* to one of the four *α*, *β*, *γ*, *δ* ;

*No S is P* to *ε* ;

*Some S is not P* to one of the three *γ*, *δ*, *ε*.

It will be observed that there is great want of symmetry in the
number of circles corresponding to the different propositional forms;
also that there is an apparent inequality in the amount of information
given by **A** and by **E**, and again by **I** and by
**O**. We shall find that these anomalies disappear when account is
taken of negative terms.

It is most misleading to attempt to represent *All S is P* by
a single pair of circles, thus

or *Some S is P* by a single pair, thus

159 for in each
case the proposition really leaves us with other alternatives. This
method of employing the diagrams has, however, been adopted by a good
many logicians who have used them, including Sir William Hamilton
(*Logic*, I. p. 255), and Professor Jevons (*Elementary Lessons in Logic*, pp. 72 to 75); and the attempt at such simplification
has brought their use into undeserved disrepute. Thus, Dr Venn
remarks, “The common practice, adopted in so many manuals, of
appealing to these diagrams—Eulerian diagrams as they are often
called—seems to me very questionable. The old four propositions
**A**, **E**, **I**, **O**, do not exactly correspond to
the five diagrams, and consequently none of the moods in the syllogism
can in strict propriety be represented by these diagrams”
(*Symbolic Logic*, pp. 15, 16; compare also pp. 424, 425). This
criticism, while perfectly sound as regards the use of Euler’s
circles by Hamilton and Jevons, loses most of its force if the
diagrams are employed with due precautions. It is true that the
diagrams become somewhat cumbrous in relation to the syllogism; but
the logical force of propositions and the logical relations between
propositions can in many respects be well illustrated by their aid.
Thus, they may be employed:—

(1) To illustrate the distribution of the predicate in a proposition. In the case of each of the four fundamental propositions we may shade the part of the predicate concerning which information is given us.

We then have,—

We see that with **A** and **I**, only part of *P* is in
some of the cases shaded; whereas with **E** and **O**, the
whole of *P* is in every case shaded; and it is thus made clear
that negative propositions distribute, while affirmative propositions
do not distribute, their predicates.

(2) To illustrate the opposition of propositions. Comparing two
contradictory propositions, *e.g.*, **A** and **O**, we see
that they have no case in common, but that between them they exhaust
all possible cases. Hence the truth, that two contradictory
propositions cannot be true together but that one of them must be
true, is brought home to us under a new aspect. Again, comparing two
subaltern propositions, *e.g.*, **A** and **I**, we notice
that the former gives us all the information given by the latter and
something more, since it still further limits the possibilities. The
other relations involved in the doctrine of opposition may be
illustrated similarly.

(3) To illustrate the conversion of propositions. Thus it is made
clear by the diagrams how it is that **A** admits only of
conversion *per accidens*. *All S is P* limits us to one or
other of the following,—

What then do we know of *P*? In the first case we have *All
P is S*, in the second *Some P is S* ; and since we are
ignorant as to which of the two cases holds good, we can only state
what is common to them both, namely, *Some P is S*.

Again, it is made clear how it is that **O** is inconvertible.
*Some S is not P* limits us to one or other of the
following,—

161 What then do
we know concerning *P*? The three cases give us
respectively,—(i) *All P is S* ; (ii) *Some P is S* and
*Some P is not S* ; (iii) *No P is S*. But (i) and (iii) are
inconsistent with one another. Hence nothing can be affirmed of
*P* that is true in all three cases indifferently.

(4) To illustrate the more complicated forms of immediate
inference. Taking, for example, the proposition *All S is P*, we
may ask, What does this enable us to assert about *not-P* and
*not-S* respectively? We have one or other of these cases,—

As regards *not-P*, these yield respectively (i) *No not-P
is S* ; (ii) *No not-P is S*. And thus we obtain the
contrapositive of the given proposition.

As regards *not-S*, we have (i) *No not-S is P*, (ii)
*Some not-S is P* and *some not-S is not P*.168
Hence in either case we may infer *Some not-S is not P*.

^{168} It is assumed in the use of
Euler’s diagrams that *S* and *P* both exist in the
universe of discourse, while neither of them exhausts that universe.
This assumption is the same as that upon which our treatment of
immediate inferences in the preceding chapter has been
based.

**E**, **I**, **O** may be dealt with similarly.

(5) To illustrate the joint force of a pair of complementary or
contra-complementary or sub-complementary propositions (compare
section 100). Thus, the pair of complementary propositions, *SaP*
and *PaS*, taken together, limit us to

Similarly the pair of contra-complementary propositions, *SaP*
and *PoS*, limit us to the relation marked *β* on page 158; and the
pair of contra-complementary propositions, *SoP* and 162 *PaS*, to *γ* ; while the pair of sub-complementary propositions, *SoP* and
*PoS*, give us a choice between *δ* and *ε*.

The application of the diagrams to syllogistic reasonings will be considered in a subsequent chapter.

With regard to all the above, it may be said that the use of the circles gives us nothing that could not easily have been obtained independently. This is of course true; but no one, who has had experience of the difficulty that is sometimes found by students in properly understanding the elementary principles of formal logic, and especially in dealing with immediate inferences, will despise any means of illustrating afresh the old truths, and presenting them under a new aspect.

The fact that we have not a single pair of circles corresponding to each fundamental form of proposition is fatal if we wish to illustrate any complicated train of reasoning in this way; but in indicating the real nature of the information given by the propositions themselves, it is rather an advantage than otherwise, inasmuch as it shews how limited in some cases this information actually is.169

^{169} Dr Venn writes in criticism
of Euler’s scheme, “A fourfold scheme of propositions will
not very conveniently fit in with a fivefold scheme of diagrams… What
the five diagrams are competent to illustrate is the actual relation
of the classes, not our possibly imperfect knowledge of that relation” (*Empirical Logic*, p. 229). The reply to this criticism
is that inasmuch as the fourfold scheme of propositions gives but an
imperfect knowledge of the actual relation of the classes denoted by
the terms, the Eulerian diagrams are specially valuable in making this
clear and unmistakeable. By the aid of dotted lines it is indeed
possible to represent each proposition by a single Eulerian figure;
but the diagrams then become so much more difficult to interpret that
the loss is considerably greater than the gain. The first and second
of the following diagrams are borrowed from Ueberweg (*Logic*, §
71). In the case of **O**, Ueberweg’s diagram is rather
complicated; and I have substituted a simpler one.

In the last of these diagrams we get the three cases yielded by an
**O** proposition by (1) filling in the dotted line to the left and
striking out the other, (2) filling in both dotted lines, (3) filling
in the dotted line to the right and striking out the other. These
three cases are respectively those marked *γ*, *δ*,
*ε*, on page 158.

163 127.
*Lambert’s Diagrams*.—A scheme of diagrams was
employed by Lambert170 in which horizontal straight lines take the
place of Euler’s circles. The extension of a term is represented
by a horizontal straight line, and so far as two such lines overlap it
is indicated that the corresponding classes are coincident, while so
far as they do not overlap these classes are shewn to be mutually
exclusive. Both the absolute and the relative length of the lines is
of course arbitrary and immaterial.

^{170} Johann Heinrich Lambert was
a German philosopher and mathematician who lived from 1728 to 1777.
His *Neues Organon* was published at Leipzig in 1768.
Lambert’s own diagrammatic scheme differs somewhat from both of
those given in the text; but it very closely resembles the one in
which portions of the lines are dotted. The modifications in the text
have been introduced in order to obviate certain difficulties involved
in Lambert’s own diagrams. See note 2 on page 165.

We may first shew how Lambert’s lines may be used in such a manner as to be precisely analogous to Euler’s circles. 164 Thus, the four fundamental propositions may be represented as follows:—

These diagrams occupy less space than Euler’s circles. But they seem also to be less intuitively clear and less suggestive. The different cases too are less markedly distinct from one another. It is probable that one would in consequence be more liable to error in employing them.

The different cases may, however, be combined by the use of dotted
lines so as to yield but a single diagram for each proposition much
more satisfactorily than in Euler’s scheme. Thus, *All S is
P* may be represented by the diagram

where the dotted line indicates that we are uncertain as to whether
there is or is not any *P* which is not *S*. We obviously
get two cases according as we strike out the dots or fill them in, and
these are the two cases previously shewn to be compatible with an
**A** proposition.

Again, *Some S is P* may be represented by the diagram

and here we get the four cases previously given for an **I**
165 proposition by
(*a*) filling in the dots to the left and striking out those to
the right, (*b*) filling in all the dots, (*c*) striking
them all out, (*d*) filling in those to the right and striking
out those to the left.

Two complete schemes of diagrams may be constructed on this plan,
in one of which no part of any *S* line, and in the other no part
of any *P* line, is dotted.171 These two schemes
are given below to the left and right respectively of the
propositional forms themselves.

^{171} It is important to give
both these schemes as it will be found that neither one of them will
by itself suffice when this method is used for illustrating the
syllogism. For obvious reasons the **E** diagram is the same in
both schemes.

It must be understood that the two diagrams given above in the
cases of **A**, **I**, and **O** are alternative in the sense
that we may select which we please to represent our proposition; but
either represents it completely.

We shall find later on that for the purpose of illustrating the
syllogistic moods, Lambert’s method is a good deal less cumbrous
than Euler’s.172 An adaptation of Lambert’s diagrams in
which the contradictories of *S* and *P* are introduced as
well 166 as *S*
and *P* themselves will be given in section 131. This more
elaborated scheme will be found useful for illustrating the various
processes of immediate inference.

^{172} Dr Venn (*Symbolic
Logic*, p. 432) remarks, “As a whole Lambert’s scheme
seems to me distinctly inferior to the scheme of Euler, and has in
consequence been very little employed by other logicians.” The
criticism offered in support of this statement is directed chiefly
against Lambert’s own representation of the particular
affirmative proposition, namely,

This diagram certainly seems as appropriate to **O** as it does
to **I**; but the modification introduced in the text, and indeed
suggested by Dr Venn himself, is not open to a similar
objection.

128. *Dr Venn’s
Diagrams*.—In the diagrammatic scheme employed by Dr Venn
(*Symbolic Logic*, chapter 5) the diagram

does not itself represent any proposition, but the framework into
which propositions may be fitted. Denoting *not-S* by
*Sʹ* and what is both *S* and *P* by *SP*,
&c., it is clear that everything must be contained in one or other
of the four classes *SP*, *SPʹ*, *SʹP*,
*SʹPʹ* ; and the above diagram shews four compartments
(one being that which lies outside both the circles) corresponding to
these four classes. Every universal proposition denies the existence
of one or more of such classes, and it may therefore be
diagrammatically represented by shading out the corresponding
compartment or compartments. Thus, *All S is P*, which denies the
existence of *SPʹ*, is represented by

*No S is P* by

167 With three terms we have three circles and eight compartments, thus,—

*All S is P or Q* is represented by

*All S is P and Q* by

It is in cases involving three or more terms that the advantage of
this scheme over the Eulerian scheme is most manifest. The diagrams
are not, however, quite so well adapted to the case of particular
propositions. Dr Venn (in *Mind*, 1883, pp. 599, 600) suggests
that we might draw a bar across the compartment declared to be saved
by a particular proposition;173 thus, *Some S is
P* would be represented by drawing a bar across the *SP*
compartment. This plan can be worked out satisfactorily; but in
representing a combination of propositions in this way special care is
needed in the interpretation of the diagrams. For example, if we have
the diagram for three terms *S*, *P*, *Q*, and are
given *Some S is P*, 168 we do not know that *both* the compartments
*SPQ*, *SPQʹ*, are to be saved, and in a case like this
a bar drawn across the *SP* compartment is in some danger of
misinterpretation.

^{173} Dr Venn’s scheme
differs from the schemes of Euler and Lambert, in that it is not based
upon the assumption that our terms and their contradictories all
represent existing classes. It involves, however, the doctrine that
particulars are existentially affirmative, while universals are
existentially negative.

129. *Expression of the
possible relations between any two classes by means of the
propositional forms* **A**, **E**, **I**,
**O**.—Any information given with respect to two classes
limits the possible relations between them to something less than the
five *à priori* possibilities indicated diagrammatically by
Euler’s circles as given at the beginning of section 126. It
will be useful to enquire how such information may in all cases be
expressed by means of the propositional forms **A**, **E**,
**I**, **O**.

The five relations may, as before, be designated respectively
*α*, *β*, *γ*, *δ*, *ε*.174
Information is given when the possibility of one or more of these is
denied; in other words, when we are limited to one, two, three, or
four of them. Let limitation to *α*, or *β*, the
exclusion of *γ*, *δ*, *ε* be denoted by *α*, *β* ; limitation to *α*, *β*, or *γ*
(*i.e.*, the exclusion of *δ* and *ε*) by *α*, *β*, *γ* ; and so on.

^{174} Thus, the classes being
*S* and *P*, *α* denotes that *S* and *P* are wholly
coincident; *β* that *P* contains *S* and more
besides; *β* that *S* contains *P* and more besides;
*δ* that *S* and *P* overlap each other, but that
each includes something not included by the other; *ε* that
*S* and *P* have nothing whatever in common.

In seeking to express our information by means of the four ordinary
propositional forms, we find that sometimes a single proposition will
suffice for our purpose; thus *α*, *β* is expressed by
*All S is P*. Sometimes we require a combination of propositions;
thus *α* is expressed by the pair of complementary propositions
*All S is P and all P is S*, (since *all S is P* excludes
*γ*, *δ*, *ε*, and *all P is S* further
excludes *β*). Some other cases are more complicated; thus the
fact that we are limited to *α* or *δ* cannot be
expressed more simply than by saying, *Either All S is P and all P
is S*, or else *Some S is P*, *some S is not P, and some P is not
S*.

Let **A** = *All S is P*, **A**_{1} = *All P
is S*, and similarly for the other propositions. Also let
**AA**_{1} = *All S is P and all P is S*, &c. Then
the following is a scheme for all possible cases:— 169

Limitation to |
denoted by |
Limitation to |
denoted by |

α | AA_{1} |
α, β, γ | A or A_{1} |

β | AO_{1} |
α, β, δ | A or IO_{1} |

γ | A_{1}O |
α, β, ε | A or E |

δ | IOO_{1} |
α, γ, δ | A_{1} or IO |

ε | E |
α, γ, ε | A_{1} or E |

α, β | A |
α, δ, ε | AA_{1} or OO_{1} |

α, γ | A_{1} |
β, γ, δ | IO or IO_{1} |

α, δ | AA_{1} or IOO_{1} |
β, γ, ε | AO_{1} or A_{1}O or E |

α, ε | AA_{1} or E |
β, δ, ε | O_{1} |

β, γ | AO_{1} or
A_{1}O |
γ, δ, ε | O |

β, δ | IO_{1} |
α, β, γ, δ | I |

β, ε | AO_{1} or E |
α, β, γ, ε | A or
A_{1} or E |

γ, δ | IO |
α, β,
δ, ε | A or O_{1} |

γ, ε | A_{1}O or E |
α, γ, δ, ε | A_{1}
or O |

δ, ε | OO_{1} |
β, γ, δ, ε | O or O_{1} |

It will be found that any combinations of propositions other than
those given above either involve contradictions or redundancies, or
else give no information because all the five relations that are *à
priori* possible still remain possible.

For example, **AI** is clearly redundant; **AO** is
self-contradictory; **A** or **A**_{1}**O** is
redundant (since the same information is given by **A** or
**A**_{1}); **A** or **O** gives no information
(since it excludes no possible case). The student is recommended to
test other combinations similarly. It must be remembered that
**I**_{1} = **I**, and **E**_{1} = **E**.

170 It should be noticed that if we read the first column downwards and the second column upwards we get pairs of contradictories.

130. *Euler’s
diagrams and the class relations between S, not-S, P,
**not-P*.—In Euler s diagrams, as ordinarily given, there is no
explicit recognition of *not-S* and *not-P*; but it is of
course understood that whatever part of the universe lies outside
*S* is *not-S*, and similarly for *P*, and it may be
thought that no further account of negative terms need be taken.
Further consideration, however, will shew that this is not the case;
and, assuming that *S*, *not-S*, *P*, *not-P* all
represent existing classes, we shall find that *seven*, not five,
determinate class relations between them are possible.

Taking the diagrams given in section 126, the above assumption
clearly requires that in the cases of *α*, *β*, and
*γ*, there should be some part of the universe lying outside
both the circles, since otherwise either *not-S* or *not-P*
or both of them would no longer be contained in the universe. But in
the cases of *δ* and *ε* it is different. *S*,
*not-S*, *P*, *not-P* are now all of them represented
within the circles; and in each of these cases, therefore, the pair of
circles may or may not between them exhaust the universe.

Our results may also be expressed by saying that in the cases of
*α*, *β*, and *γ*, there must be something which
is both *not-S* and *not-P*; whereas in the cases of *δ* and *ε*, there may or may not be something which is both
*not-S* and *not-P*. Euler’s circles, as ordinarily
used, are no doubt a little apt to lead us to overlook the latter of
these alternatives. If, indeed, there were always part of the universe
outside the circles, every proposition, whether its form were
**A**, **E**, **I**, or **O**, would have an inverse and the
same inverse, namely, *Some not-S is **not-P* ; also, every
proposition, including **I**, would have a contrapositive. These
are erroneous results against which we have to be on our guard in the
use of Euler’s fivefold scheme.

We find then that the explicit recognition of *not-S* and
*not-P* practically leaves *α*, *β*, and *γ*
unaffected, but causes *δ* and *ε* each to subdivide
into two cases according as there is or is not anything that is both
*not-S* and *not-P*; and the 171 Eulerian fivefold division has accordingly
to give place to a sevenfold division.

In the diagrammatic representation of these seven relations, the entire universe of discourse may be indicated by a larger circle in which the ordinary Eulerian diagrams (with some slight necessary modifications) are included. We shall then have the following scheme:—

172 It may be
useful to repeat these diagrams with an explicit indication in regard
to each subdivision of the universe as to whether it is *S* or
*not-S*, *P* or *not-P*.175 The scheme will
then appear as follows:—

^{175} We might also represent the
universe of discourse by a long rectangle divided into compartments,
shewing which of the four possible combinations *SP*,
*SPʹ*, *SʹP*, *SʹPʹ* are to be found.
This plan will give the following which precisely correspond, as
numbered, with those in the text:—

(i) | SP |
SʹPʹ |

(ii) | SP |
SʹP |
SʹPʹ |

(iii) | SP |
SPʹ |
SʹPʹ |

(iv) | SP |
SPʹ |
SʹP |
SʹPʹ |

(v) | SP |
SPʹ |
SʹP |

(vi) | SPʹ |
SʹP |
SʹPʹ |

(vii) | SPʹ |
SʹP |

173 Comparing the
above with the five ordinary Eulerian diagrams (which may be
designated *α*, *β* &c. as in section 126), it will
be seen that (i) corresponds to *α*; (ii) to *β*; (iii) to *γ*; (iv) and (v) represent the two cases now yielded by *δ*;
(vi) and (vii) the two yielded by *ε*.

Our seven diagrams might also be arrived at as follows:—Every part of the universe must be either *S* or *Sʹ*,
and also *P* or *Pʹ* ; and hence the mutually exclusive
combinations *SP*, *SPʹ*, *SʹP*,
*SʹPʹ* must between them exhaust the universe. The case
in which these combinations are all to be found is represented by
diagram (iv); if one but one only is absent we obviously have four
cases which are represented respectively by (ii), (iii), (v), and (vi); if only two are to be found it will be seen that we
are limited to the cases represented by (i) and (vii) or we should not
fulfil the condition that neither *S* nor *Sʹ*, *P*
nor *Pʹ*, is to be altogether non-existent; for the same
reason the universe cannot contain less than two of the four
combinations. We thus have the seven cases represented by the
diagrams, and these are shewn to exhaust the possibilities.

174 The four
traditional propositions are related to the new scheme as
follows:—

**A** limits us to (i) or (ii);

**I** to (i), (ii), (iii), (iv), or (v);

**E** to (vi) or (vii);

**O** to (iii), (iv), (v), (vi), or (vii).

Working out the further question how each diagram taken by itself
is to be expressed propositionally we get the following results:

(i) *SaP* and *SʹaPʹ* ;

(ii) *SaP* and *SʹoPʹ* ;

(iii) *SʹaPʹ* and *SoP* ;

(iv) *SoP*, *SoPʹ*, *SʹoP*, and
*SʹoPʹ* ;

(v) *SʹaP* and *SoPʹ* ;

(vi) *SaPʹ* and *SʹoP* ;

(vii) *SaPʹ* and *SʹaP*.

It will be observed that the new scheme is in itself more
symmetrical than Euler’s, and also that it succeeds better in
bringing out the symmetry of the fourfold schedule of propositions.176
**A** and **E** give two alternatives each, **I** and
**O** give five each; whereas with Euler’s scheme **E**
gives only one alternative, **A** two, **O** three, **I**
four, and it might, therefore, seem as if **E** afforded more
definite and unambiguous information than **A**, and **O** than
**I**, which is not really the case. Further, the problem of
expressing each diagram propositionally yields a more symmetrical
result than the corresponding problem in the case of Euler’s
diagrams.

^{176} We have seen that,
similarly, in the case of immediate inferences symmetry can be gained
only by the recognition of negative terms.

This sevenfold scheme of class relations should be compared with the sevenfold scheme of relations between propositions given in section 84.

131. *Lambert’s
diagram and the class-relations between S, not-S, P,
**not-P*.—The following is a compact diagrammatic
representation of the seven possible class-relations between *S*,
*not-S*, *P*, *not-P*, based upon Lambert’s
scheme. 175

In this scheme each line represents the entire universe of discourse, and the first line must be taken in connexion with each of the others in turn. Further explanation will be unnecessary for the student who clearly understands the Lambertian method.

On the same principle and with the aid of dotted lines the four fundamental propositional forms may be represented as follows:

176 In each case
the full extent of a line represents the entire universe of discourse;
any portion of a line that is dotted may be either *S* or
*Sʹ* (or *P* or *Pʹ*, as the case may be).

This last scheme of diagrams is perhaps more useful than any of the
others in shewing at a glance what immediate inferences are obtainable
from each proposition by conversion, contraposition, and inversion (on
the assumption that *S*, *Sʹ*, *P*, and
*Pʹ* all represent existing classes). Thus, from the first
diagram we can read off at a glance *SaP*, *PiS*,
*PʹaSʹ*, *SʹiPʹ* ; from the second
*SeP*, *PeS*, *PʹoSʹ*, *SʹoPʹ* ; from
the third *SiP* and *PiS* ; and from the fourth *SoP*
and *PʹoSʹ*. The last two diagrams are also seen at a
glance to be indeterminate in respect to *Pʹ* and
*Sʹ*, *P* and *Sʹ*, respectively (that is to
say, **I** has no contrapositive and no inverse, **O** has no
converse and no inverse).

132. Illustrate by means of
the Eulerian diagrams (1) the relation between **A** and **E**, (2) the relation between
**I** and **O**, (3) the conversion of **I**, (4) the
contraposition of **O**, (5) the inversion of **E**. [K.]

133. *A* denies that
none but *X* are *Y* ; *B* denies that none but *Y* are *X*. Which of the five class relations between
*X* and *Y* (1) must they agree in rejecting, (2) may they
agree in accepting? [C.]

134. Take all the ordinary propositions connecting any two terms, combine them in pairs so far as is possible without contradiction, and represent each combination diagrammatically. [J.]

135. *Fourfold Implication
of Propositions in Connotation and Denotation*.—In dealing
with the question whether propositions assert a relation between
objects or between attributes or between objects and attributes,
logicians have been apt to commit the fallacy of exclusiveness,
selecting some one of the given alternatives, and treating the others
as necessarily excluded thereby. It follows, however, from the double
aspect of names—in extension and intension—that the
different relations really involve one another, so that all of them
are implied in any categorical proposition whose subject and predicate
are both general names.177 If any one of the relations is selected as
constituting the *meaning* of the proposition, the other
relations are at any rate involved as *implications*.

^{177} In the discussion that
follows we limit ourselves to the traditional scheme of
propositions.

The problem will be made more definite if we confine ourselves to a
consideration of *connotation* and *denotation* in the
strict sense, as distinguished from comprehension and exemplification,
our terms being supposed to be defined intensively.178
Both subject and predicate will then have a denotation determined by
their connotation, and hence our 178 proposition may be considered from four
different points of view, which are not indeed really independent of
one another, but which serve to bring different aspects of the
proposition into prominence. (1) The subject may be read in denotation
and the predicate in connotation; (2) both terms may be read in
denotation; (3) both terms may be read in connotation; (4) the subject
may be read in connotation and the predicate in denotation.

^{178} With extensive definitions
we might similarly work out the relations between the terms of a
proposition in exemplification and comprehension; and with either
intensive or extensive definitions, we might consider them in
denotation and comprehension. The discussion in the text will,
however, be limited to connotation and denotation, except that a
separate section will be devoted to the case in which both subject and
predicate are read in comprehension.

As an example, we may take the proposition, *All men are mortal*.179
According to our point of view, this proposition may be read in any of
the following ways:

(1) The objects denoted by *man* possess the attributes
connoted by *mortal* ;

(2) The objects denoted by *man* are included within the class
of objects denoted by *mortal* ;

(3) The attributes connoted by *man* are accompanied by the
attributes connoted by *mortal* ;

(4) The attributes connoted by *man* indicate the presence of
an object belonging to the class denoted by *mortal*.

^{179} A distinction may perhaps
be drawn between the four following types of propositions; (*a*)
*All men are mortal* ; (*b*) *All men are mortals* ;
(*c*) *Man is mortal* ; (*d*) *Man is a mortal*. Of
these, (*a*) naturally suggests the reading of subject in
denotation and predicate in connotation as *meaning*, the three
other readings being *implications* ; (*b*) is similarly
related to the reading numbered (2) above; (*c*) to (3); and
(*d*) to (4).

It should be specially noticed that a different relation between
subject and predicate is brought out in each of these four modes of
analysing the proposition, the relations being respectively (i)
*possession*, (ii) *inclusion*, (iii) *concomitance*, (iv) *indication*.

It may very reasonably be argued that a certain one of the above
ways of regarding the proposition is (*a*) psychologically the
most prominent in the mind in predication; or (*b*) the most
fundamental in the sense of making explicit that relation which
ultimately determines the other relations; or (*c*) the most
convenient for a given purpose, *e.g.*, for dealing with the
problems of formal logic. We need not, however, select the same mode
of interpretation in each case. There would, for example, be nothing
inconsistent in holding that to read the 179 subject in denotation and the predicate in
connotation is most correct from the psychological standpoint; to read
both terms in connotation the most ultimate, inasmuch as connotation
determines denotation and not *vice versâ*, and to read both
terms in denotation the most serviceable for purposes of logical
manipulation. To say, however, that a certain one of the four readings
alone can be regarded as constituting the import of the proposition to
the exclusion of the others cannot but be erroneous. They are in truth
so much implicated in one another, that the difficulty may rather be
to justify a treatment which distinguishes between them.180

^{180} The true doctrine is
excellently stated by Mrs Ladd Franklin in an article in *Mind*,
October, 1890, pp. 561, 2.

(1) *Subject in denotation, predicate in connotation*.

If we read the subject of a proposition in denotation and the
predicate in connotation, we have what is sometimes called the
*predicative mode* of interpreting the proposition. This way of
regarding propositions most nearly corresponds in the great majority
of cases with the course of ordinary thought;181 that is to say, we
naturally contemplate the subject as a class of objects of which a
certain attribute or complex of attributes is predicated. Such
propositions as *All men are mortal*, *Some violets are
white*, *All diamonds are combustible*, may be taken as
examples. Dr Venn puts the point very clearly with reference to the
last of these three propositions: “If I say that ‘all
diamonds are combustible,’ I am joining together two connotative
terms, each of which, therefore, implies an attribute and denotes a
class; but is there not a broad distinction in respect of the
prominence with which the notion of a class is presented to the mind
in the two cases? As regards the diamond, we think at once, or think
very speedily, of a class of things, the distinctive attributes of the
subject being mainly used to carry the mind on to the contemplation of
the objects referred to by them. But as regards the combustibility,
the attribute itself is the prominent thing … Combustible *things*,
other than the diamond itself, come scarcely, if at all, under 180 contemplation. The
assertion in itself does not cause us to raise a thought whether there
be other combustible things than these in existence”
(*Empirical Logic*, p. 219).

^{181} Though perhaps what is
actually present to the mind is usually rather more complex than what
is represented by any one of the four readings taken by
itself.

Two points may be noticed as serving to confirm the view that
generally speaking the predicative mode of interpreting propositions
is psychologically the most prominent:

(*a*) The most striking difference between a substantive and
an attributive (*i.e.*, an adjective or a participle) from the
logical point of view is that in the former the denotation is usually
more prominent than the connotation, even though it may be ultimately
determined by the connotation, whilst in the latter the connotation is
the more prominent, even though the name must be regarded as the name
of a class of objects if it is entitled to be called a *name* in the
strict logical sense at all. Corresponding to this we find that the
subject of a proposition is almost always a substantive, whereas the
predicate is more often an attributive.

(*b*) It is always the denotation of a term that we quantify,
never the connotation. Whether we talk of *all men* or of *some
men*, the complex of attributes connoted by *man* is taken in its
totality; the distinction of quantity relates entirely to the
denotation of the term. Corresponding to this, we find that we
naturally regard the *quantity* of a proposition as pertaining to
its *subject*, and not to its predicate. It will be shewn in the
following chapter that the doctrine of the quantification of the
predicate has at any rate no psychological justification.

There are, however, numerous exceptions to the statement that the
subject of a proposition is naturally read in denotation and the
predicate in connotation; for example, in the classificatory sciences.
The following propositions may be taken as instances: *All palms are
endogens*, *All daisies are compositae*, *None but solid
bodies are crystals*, *Hindoos are Aryans*, *Tartars are
Turanians*. In such cases as these most of us would naturally think
of a certain class of objects as included in or excluded from another
class rather than as possessing or not possessing certain definite
attributes; in other words, as Dr Venn puts it, “the
class-reference of the predicate is no less definite than that of the
subject” (*Empirical Logic*, p. 220). 181 In the case of such a
proposition as *No plants with opposite leaves are orchids*, the
position is even reversed, that is to say, it is the subject rather
than the predicate that we should more naturally read in connotation.
We may pass on then to other ways of regarding the categorical
proposition.

(2) *Subject in denotation, predicate in denotation*.

If we read both the subject and the predicate of a proposition in
denotation, we have a relation between two classes, and hence this is
called the *class mode* of interpreting the proposition. It must
be particularly observed that the relation between the subject and the
predicate is now one of *inclusion in* or *exclusion from*,
not one of *possession*. It may at once be admitted that the
class mode of interpreting the categorical proposition is neither the
most ultimate, nor—generally speaking—that which we
naturally or spontaneously adopt. It is, however, extremely convenient
for manipulative purposes, and hence is the mode of interpretation
usually selected, either explicitly or implicitly, by the formal
logician. Attention may be specially called to the following points:

(*a*) When subject and predicate are both read in denotation,
they are *homogeneous*.

(*b*) In the *diagrammatic* illustration of propositions
both subject and predicate are necessarily read in denotation, since
it is the denotation—not the connotation—of a term that we
represent by means of a diagram.

(*c*) The predicate of a proposition must be read in
denotation in order to give a meaning to the question whether it is or
is not *distributed*.

(*d*) The predicate as well as the subject must be read in
denotation before such a process as *conversion* is possible.

(*e*) In the treatment of the *syllogism* both subject
and predicate must be read in denotation (or else both in
connotation), since either the middle term (first and fourth figures)
or the major term (second and fourth figures) or the minor term (third
and fourth figures) is subject in one of the propositions in which it
occurs and predicate in the other.

The class mode of interpreting categorical propositions is nevertheless treated by some writers as being positively 182 erroneous. But the arguments used in support of this view will not bear examination.

(i) It is said that to read both subject and predicate in
denotation is psychologically false. It has indeed been pointed out
already that the class mode of interpretation is not that which as a
rule first presents itself to our mind when a proposition is given us;
but we have also seen that there are exceptions to this, as, for
example, in the propositions *All daisies are compositae*. *All
Hindoos are Aryan*, *All Tartars are Turanians*. It is,
therefore, clearly wrong to describe the reading in question as in all
cases psychologically false. On the same shewing, any other reading
would equally be psychologically false, for what is immediately
present to the mind in judgment varies very much in different cases.
Undoubtedly there are many judgments in regard to which we do not
spontaneously adopt the class reading. Still, analysis shews that in
these judgments, as in others, inclusion in or exclusion from a class
is really implicated along with other things, although this relation
may be neither that which first impresses itself upon us nor that
which is most important or characteristic.

(ii) It is asked what we mean by a class, by the class of birds,
for example, when we say *All owls are birds*. “It is
nothing existing in space; the birds of the world are nowhere
collected together so that we can go and pick out the owls from
amongst them. The classification is a mental abstraction of our own,
founded upon the possession of certain definite attributes. The class
is not definite and fixed, and we do not find out whether any
individual belongs to it by going over a list of its members, but by
enquiring whether it possesses the necessary attributes.”182
In so far as this argument applies against reading the predicate in
denotation, it applies equally against reading the subject in
denotation. It is in effect the argument used by Mill (*Logic*,
i. 5, § 3) in order to lead up to his position that the
*ultimate* interpretation of the categorical proposition requires
us to read both subject and predicate in connotation, since denotation
is determined by connotation. But if this be granted, it does not
prove the class reading of the 183 proposition erroneous; it only proves that
in the class reading, the analysis of the import of the proposition
has not been carried as far as it admits of being carried.

^{182} Welton, *Logic*, p. 218.

(iii) It is argued that when we regard a proposition as expressing
the inclusion of one class within another, even then the predicate is
only apparently read in denotation. “On this view, we do not
really assert *P* but ‘inclusion in *P*,’ and
this is therefore the true predicate. For example, in the proposition
‘All owls are birds,’ the real predicate is, on this view,
not ‘birds’ but ‘included in the class
birds.’ But this inclusion is an attribute of the subject, and
the real predicate, therefore, asserts an attribute. It is meaningless
to say ‘Every owl *is* the class birds,’ and it is
false to say ‘The class owls *is* the class
birds.’”183 This argument simply begs the question in
favour of the predicative mode of interpretation. It overlooks the
fact that the precise kind of relation brought out in the analysis of
a proposition will vary according to the way in which we read the
subject and the predicate. An analogous argument might also be used
against the predicative reading itself. Take the proposition,
“All men are mortal.” It is absurd to say that
“Every man *is* the attribute mortality,” or that “The
class men *is* the attribute mortality.”

^{183} Welton, *Logic*, p. 218.

(iv) It is said that a class interpretation of both *S* and
*P* would lead properly to a fivefold, not a fourfold, scheme of
propositions, since there are just five relations possible between any
two classes, as is shewn by the Eulerian diagrams. This contention has
force, however, only upon the assumption that we must have quite
determinate knowledge of the class relation between *S* and
*P* before being able to make any statement on the subject; and
this assumption is neither justifiable in itself nor necessarily
involved in the interpretation in question. It may be added that if a
similar view were taken on the adoption of the predicative mode of
interpretation, we should have a threefold, not a fourfold scheme. For
then the quantity of our subject at any rate would have to be
perfectly determinate, and with *S* and *P* for subject and
predicate, the three possible statements would be—*All S is
P*, *Some S is P and* 184 *some is not*, *No S is P*. The
point here raised will presently be considered further in connexion
with the quantification of the predicate.

(3) *Subject in connotation, predicate in connotation*.

If we read both the subject and the predicate of a proposition in
connotation, we have what may be called the *connotative mode* of
interpreting the proposition. In the proposition *All S is P*,
the relation expressed between the attributes connoted by *S* and
those connoted by *P* is one of
*concomitance*—“the attributes which constitute the
connotation of *S* are always found accompanied by those which
constitute the connotation of *P*.”184 Similarly, in the
case of *Some S is P*,—“the attributes 185 which constitute the
connotation of *S* are sometimes found accompanied by those which
constitute the connotation of *P*”; *No S is
P*,—“the attributes which constitute the connotation
of *S* are never found along with those which constitute the
connotation of *P*”; *Some S is not
P*,—“the attributes which constitute the connotation of
*S* are sometimes found unaccompanied by those which constitute
the connotation of *P*.”

^{184} This is the only possible
reading in connotation, so far as real propositions are concerned, if
the term connotation is used in the strict sense as distinguished both
from subjective intension and from comprehension. Unfortunately
confusion is apt to be introduced into discussions concerning the
intensive rendering of propositions simply because no clear
distinction is drawn between the different points of view which may be
taken when terms are regarded from the intensive side. Hamilton
distinguished between judgments in extension and judgments in
intension, the relation between the subject and the predicate in the
one case being just the reverse of the relation between them in the
other. Thus, taking the proposition *All S is P*, we have in
extension *S is contained under P*, and in intension *S
comprehends P*. On this view the intensive reading of *All men
are mortal* is “mortality is part of humanity” (the
extensive reading being “the class man is part of the class
mortal”). This reading may be accepted if the term intension is
used in the objective sense which we have given to
*comprehension*, so that by *humanity* is meant the totality
of attributes common to all men, and by *mortality* the totality
of attributes common to all mortals. To this point of view we shall
return in the next section. Leaving it for the present on one side, it
is clear that if by *humanity* we mean only what may be called
the distinctive or essential attributes of man, then in order that the
above reading may be correct, the given proposition must be regarded
as analytical. In other words, if *humanity* signifies only those
attributes which are included in the connotation of *man*, then,
if mortality is included in humanity, we shall merely have to analyse
the connotation of the name *man*, in order to obtain our
proposition. Hence on this view it must either be maintained that all
universal affirmative propositions are analytical, or else that some
universal affirmatives cannot be read in intension. But obviously the
first of these alternatives must be rejected, and the second
practically means that the reading in question breaks down so far as
universal affirmatives are concerned.

Hamilton’s reading breaks down even more completely in the
case of particulars and negatives. *The attributes constituting the
intensions of S and P partly coincide* is clearly not equivalent to
*Some S is P* ; for example, the intension (in any sense) of
*Englishman* has something in common with the intension of
*Frenchman*, but it does not follow that *Some Englishmen are
Frenchmen*. Again, from the fact that the intension of *S* has
nothing in common with the intension of *P*, we cannot infer that
*No S is P* ; suppose, for example, that *S* stands for
“ruminant,” and *P* for “cloven-hoofed.”
Compare Venn, *Symbolic Logic*, pp. 391–5.

It will be noticed that in the connotative reading we have always
to take the attributes which constitute the connotation
*collectively*. In other words, by the attributes constituting
the connotation of a term we mean those attributes regarded as a
whole. Thus, *No S is P* does not imply that none of the
attributes connoted by *S* are ever accompanied by any of those
connoted by *P*. This is apparent if we take such a proposition
as *No oxygen is hydrogen*. It follows that when the subject is
read in connotation the quantity of the proposition must appear as a
separate element, being expressed by the word “always” or
“sometimes,” and must not be interpreted as meaning
“all” or “some” of the attributes included in
the connotation of the subject.

It is argued by those who deny the possibility of the connotative
mode of interpreting propositions, that this is not really reading the
subject in connotation at all; *always* and *sometimes* are
said to reduce us to denotation at once. In reply to this, it must of
course be allowed that real propositions affirm no relation between
attributes independently of the objects to which they belong. The
connotative reading implies the denotative, and we must not exaggerate
the nature of the distinction between them. Still the connotative
reading presents the import of the proposition in a new aspect, and
there is at any rate a *prima facie* difference between regarding
one class as included within another, and regarding one attribute as
always accompanied by another, even though a little 186 consideration may shew
that the two things mutually involve one another.185

^{185} Mill attaches great
importance to the connotative mode of interpreting propositions as
compared with the class mode or the predicative mode, on the ground
that it carries the analysis a stage further; and this must be
granted, at any rate so far as we consider the application of the
terms involved to be determined by connotation and not by
exemplification. Mill is, however, sometimes open to the charge of
exaggerating the difference between the various modes of
interpretation. This is apparent, for example, in his rejection of the
*Dictum de omni et nullo* as the axiom of the syllogism, and his
acceptance of the *Nota notae est nota rei ipsius* in its
place.

(4) *Subject in connotation, predicate in denotation*.

Taking the proposition *All S is P*, and reading the subject
in connotation and the predicate in denotation, we have, “The
attributes connoted by *S* are an indication of the presence of
an individual belonging to the class *P*.” This mode of
interpretation is always a possible one, but it must be granted that
only rarely does the import of a proposition naturally present itself
to our minds in this form. There are, however, exceptional cases in
which this reading is not unnatural. The proposition *No plants with
opposite leaves are orchids* has already been given as an example.
Another example is afforded by the proposition *All that glitters is
not gold*. Taking the subject in connotation and the predicate in
denotation we have, *The attribute of glitter does not always
indicate the presence of a gold object* ; and it will be found that
this reading of the proverb serves to bring out its meaning really
better than any of the three other readings which we have been
discussing.

It is worth while noticing here by way of anticipation that on any
view of the existential interpretation of propositions, as discussed
in chapter 8, we shall still have a fourfold reading of categorical
propositions in connotation and denotation. The universal negative and
the particular affirmative may be taken as examples, on the
supposition that the former is interpreted as existentially negative
and the latter as existentially affirmative. The universal negative
yields the following: (1) There is no individual belonging to the
class *S* and possessing the attributes connoted by *P* ; (2)
There is no individual common to the two classes *S* and
*P* ; (3) The attributes 187 connoted by *S* and *P*
respectively are never found conjoined; (4) There is no individual
possessing the attributes connoted by *S* and belonging to the
class *P*. Similarly the particular affirmative yields: (1) There
are individuals belonging to the class *S* and possessing the
attributes connoted by *P* ; (2) There are individuals common to
the two classes *S* and *P* ; (3) The attributes connoted by
*S* and *P* respectively are sometimes found conjoined; (4)
There are individuals possessing the attributes connoted by *S*
and belonging to the class *P*. We see, therefore, that the
question discussed in this section is independent of that which will
be raised in chapter 8; and that for this reason, if for no other, no solution of the general problem raised in the present chapter can
afford a complete solution of the problem of the import of categorical
propositions.

136. *The Reading of
Propositions in Comprehension*.—If, in taking the intensional
standpoint, we consider comprehension instead of connotation, our
problem is to determine what relation is implied in any proposition
between the comprehension of the subject and the comprehension of the
predicate. This question being asked with reference to the universal
affirmative proposition *All S is P*, the solution clearly is
that *the comprehension of S includes the comprehension of P*.
The interpretation in comprehension is thus precisely the reverse of
that in denotation (*the denotation of S is included in the
denotation of P*); and we might be led to think that, taking the
different propositional forms, we should have a scheme in
comprehension, analogous throughout to that in denotation. But this is
not the case, for the simple reason that in our ordinary statements we
do not distributively quantify comprehension in the way in which we do
denotation; in other words, comprehension is always taken in its
totality. Thus, reading an **I** proposition in denotation we
have—*the classes S and P partly coincide* ; and
corresponding to this we should have—*the comprehensions of S
and P partly coincide*. But this is clearly not what we express by
*Some S is P* ; for the partial coincidence of the comprehensions
of *S* and *P* is quite compatible with *No S is P*,
that is to say, the classes *S* and *P* may be mutually
exclusive, and yet some attributes may be common to the whole of
*S* and 188 also
to the whole of *P* ; for example, *No Pembroke undergraduates
are also Trinity undergraduates*. Again, given an **E**
proposition, we have in denotation—*the classes S and P have
no part in common* ; but for the reason just given, it does not
follow that *the comprehension of S and the comprehension of P have
nothing in common*.

It is indeed necessary to obvert **I** and **E** in order to
obtain a correct reading in comprehension. We then have the following
scheme, in which the relation of contradiction between **A** and
**O** and between **E** and **I** is made clearly manifest:

*All S is P*, The comprehension of *S* includes the
comprehension of *P* ;

*No S is P*, The comprehension of *S* includes the
comprehension of *not-P*;

*Some S is P*, The comprehension of *S* does not include
the comprehension of *not-P*;

*Some S is not P*, The comprehension of *S* does not
include the comprehension of *P*.

137. *The employment of
the symbol of Equality in Logic*.—The symbol of equality (=)
is frequently used in logic to express the identity of two classes.
For example,

*Equilateral triangles* = *equiangular triangles* ;

*Exogens* = *dicotyledons* ;

*Men* = *mortal men*.

It is, however, important to recognise that in thus borrowing a
symbol from mathematics we do not retain its meaning unaltered, and
that a so-called *logical equation* is, therefore, something very
different from a mathematical equation. In mathematics the symbol of
equality generally means numerical or quantitative equivalence. But
clearly we do not mean to express mere numerical equality when we
write *equilateral triangles* = *equiangular triangles*.
Whatever this so-called equation signifies, it is certainly something
more than that there are precisely as many triangles with three equal
sides as there are triangles with three equal angles. It is further
clear that we do not intend to express mere similarity. Our meaning is
that the denotations of the terms which are equated are absolutely
identical; in other words, that the class of objects denoted by the
term *equilateral triangle* is absolutely identical with the
class of objects denoted by the term *equiangular triangle*.186
It may, however, be objected that, if this 190 is what our equation comes to, then
inasmuch as a statement of mere identity is empty and meaningless, it
strictly speaking leaves us with nothing at all; it contains no
assertion and can represent no judgment. The answer to this objection
is that whilst we have identity in a certain respect, it is erroneous
to say that we have *mere* identity. We have *identity of
denotation* combined with *diversity of connotation*, and,
therefore, with *diversity of determination* (meaning thereby
diversity in the ways in which the application of the two terms
identified is determined).187 The meaning of
this will be made clearer by the aid of one or two illustrations.
Taking, then, as examples the logical equations already given, we may
analyse their meaning as follows. If out of all triangles we select
those which possess the property of having three equal sides, and if
again out of all triangles we select those which possess the property
of having three equal angles, we shall find that in either case we are
left with precisely the same set of triangles. Thus, each side of our
equation denotes precisely the same class of objects, but the class is
determined or arrived at in two different ways. Similarly, if we
select all plants that are exogenous and again all plants that are
dicotyledonous, our results are precisely the same although our mode
of arriving at them has been different. Once more, if we simply take
the class of objects which possess the attribute of humanity, and
again the class which possess both this attribute and also the
attribute of mortality, the objects selected will be the same; none
will be excluded by our second method of selection although an
additional attribute is taken into account.

^{186} It follows that the
comprehensions (but of course not the connotations) of the terms will
also be identical; this cannot, however, be regarded as the primary
signification of the equation.

^{187} I have practically borrowed
the above mode of expression from Miss Jones, who describes an
affirmative categorical proposition as “a proposition which
asserts identity of application in diversity of signification”
(*General Logic*, p. 20). Miss Jones’s meaning may,
however, be slightly different from that intended in the text, and I
am unable to agree with her general treatment of the import of
categorical propositions, as she does not appear to allow that before
we can regard a proposition as asserting identity of application we
must implicitly, if not explicitly, have quantified the
predicate.

Since the identity primarily signified by a logical equation is an identity in respect of denotation, any equational mode of reading propositions must be regarded as a modification of the 191 “class” mode. What has been said above, however, will make it clear that here as elsewhere denotation is considered not to the exclusion of connotation but as dependent upon it; and we again see how denotative and connotative readings of propositions are really involved in one another, although one side or the other may be made the more prominent according to the point of view which is taken.

Another point to which attention may be called before we pass on to
consider different types of logical equations is that in so far as a
proposition is regarded as expressing an identity between its terms
the distinction between subject and predicate practically disappears.
We have seen that when we have the ordinary logical copula *is*,
propositions cannot always be simply converted, the reason being that
the relation of the subject to the predicate is not the same as the
relation of the predicate to the subject. But when two terms are
connected by the sign of equality, they are similarly, and not
diversely, related to each other; in other words, the relation is
symmetrical. Such an equation, for example, as *S* = *P* can
be read either forwards or backwards without any alteration of
meaning. There can accordingly be no distinction between subject and
predicate except the mere order of statement, and that may be regarded
as for most practical purposes immaterial.

138. *Types of Logical
Equations*188—Jevons (*Principles of Science*,
chapter 3) recognises three types of logical equations, which he calls
respectively *simple identities*, *partial identities*, and
*limited identities*.

^{188} This section may be omitted on a first reading.

*Simple identities* are of the form *S* = *P* ; for
example, *Exogens* = *dicotyledons*. Whilst this is the
simplest case equationally, the information given by the equation
requires two propositions in order that it may be expressed in
ordinary predicative form. Thus, *All S is P* and *All P is
S* ; *All exogens are dicotyledons* and *All dicotyledons are
exogens*. If, however, we are allowed to quantify the predicate as
well as the subject, a single proposition will suffice. Thus, *All S
is all P*, *All exogens are all dicotyledons*. We shall return
presently to a consideration of this type of proposition.

192 *Partial
identities* are of the form *S* = *SP*, and are the
expression equationally of ordinary universal affirmative
propositions. If we take the proposition *All S is P*, it is
clear that we cannot write it *S = P*, since the class *P*,
instead of being coextensive with the class *S*, may include it
and a good deal more besides. Since, however, by the law of identity
*All S is S*, it follows from *All S is P* that *All S is
SP*. We can also pass back from the latter of these propositions to
the former. Hence the two propositions are equivalent. But *All S is
SP* may at once be reduced to the equational form *S* =
*SP*. For this breaks up into the two propositions *All S is
SP* and *All SP is S*, and since the second of these is a mere
formal proposition based on the law of identity, the equation must
necessarily hold good if *All S is SP* is given. To take a
concrete example, the proposition *All men are mortal* becomes
equationally *Men = mortal* *men*. Similarly the universal negative
proposition *SeP* may be expressed in the equational form
*S* = *Sp* (where *p* = *not-P*).

*Limited identities* are of the form *VS* = *VP*,
which may be interpreted “Within the sphere of the class
*V*, all *S* is *P* and all *P* is
*S*,” or “The *S*’s and *P*’s,
which are *V*’s, are identical.” So far as *V*
represents a determinate class, there is little difference between
these limited identities and simple identities. This is shewn by the
fact that Jevons himself gives *Equilateral triangles = equiangular
triangles* as an instance of a simple identity, whereas its proper
place in his classification would appear to be amongst the limited
identities, for its interpretation is that “*within the sphere
of triangles*—all the equilaterals are all the
equiangulars.”

The equation *VS* = *VP* is, however, used by
Boole—and also by Jevons subsequently—as the expression
equationally of the particular proposition, and if it can really
suffice for this, its recognition as a distinct type is justified. If
we take the proposition *Some S is P*, we find that the classes
*S* and *P* are affirmed to have some part in common, but no
indication is given whereby this part can be identified. Boole
accordingly indicates it by the arbitrary symbol *V*. It is then
clear that *All VS is VP* and also that *All VP is VS*, and
we have the above equation.

193 It is no part
of our present purpose to discuss systems of symbolic logic; but it
may be briefly pointed out that the above representation of the
particular proposition is far from satisfactory. In order to justify
it, limitations have to be placed upon the interpretation of *V*
which altogether differentiate it from other class-symbols. Thus, the
equation *VS* = *VP* is consistent with *No S is P*
(and, therefore, cannot be equivalent to *Some S is P*) provided
that no *V* is either *S* or *P*, for in this case we
have *VS* = 0 and *VP* = 0. *V* must,
therefore, be limited by the antecedent condition that it represents
an existing class and a class that contains either *S* or
*P*, and it is in this condition quite as much as in the equation
itself that the real force of the particular proposition is
expressed.189

^{189} Compare Venn, *Symbolic
Logic*, pp. 161, 2.

If particular propositions are true contradictories of universal
propositions, then it would seem to follow that in a system in which
universals are expressed as equalities, particulars should be
expressed as inequalities. This would mean the introduction of the
symbols > and <, related to the corresponding mathematical symbols in
just the same way as the logical symbol of equality is related to the
mathematical symbol of equality; that is to say, *S* > *SP*
would imply logically more than mere numerical inequality, it would
imply that the class *S* includes the whole of the class
*SP* and more besides. Thus interpreted, *S* > *SP*
expresses the particular negative proposition, *Some S is not
P*.190 If we further introduce the symbol 0 as
expressing nonentity, *No S is P* may be written *SP* =
0, and its contradictory, *i.e.*, *Some S is P*, may
be written *SP* > 0. We shall then have the following
scheme (where *p* = *not-P*):

All S is P | expressed by S = SP or by
Sp = 0; |

Some S is not P | ″ ″ S > SP ″ Sp > 0; |

No S is P | ″ ″ SP = 0 ″
S = Sp ; |

Some S is P | ″ ″ SP > 0 ″ S > Sp. |

^{190} Similarly *X* > *Y* expresses the two statements “All *Y* is *X*,
but Some *X* is not *Y*,” just as *X* = *Y*
expresses the two statements “All *Y* is *X* and All
*X* is *Y*.”

194 This scheme, it will be observed, is based on the assumption that particulars are existentially affirmative while universals are existentially negative. This introduces a question which will be discussed in detail in the following chapter. The object of the present section is merely to illustrate the expression of propositions equationally, and the symbolism involved has, therefore, been treated as briefly as has seemed compatible with a clear explanation of its purport. Any more detailed treatment would involve a discussion of problems belonging to symbolic logic.

139. *The expression of
Propositions as Equations*.—There are rare cases in which
propositions fall naturally into what is practically an equational
form; for example, *Civilization and Christianity are
co-extensive*. But, speaking generally, the equational relation, as
implicated in ordinary propositions, is not one that is spontaneously,
or even easily, grasped by the mind. Hence as a psychological account
of the process of judgment the equational rendering may be rejected.
It is, moreover, not desirable that equations should supersede the
generally recognised propositional forms in ordinary logical doctrine,
for such doctrine should not depart more than can be helped from the
forms of ordinary speech. But, on the other hand, the equational
treatment of propositions must not be simply put on one side as
erroneous or unworkable. It has been shewn in the preceding section
that it is at any rate possible to reduce all categorical propositions
to a form in which they express equalities or inequalities; and such
reduction is of the greatest importance in systems of symbolic logic.
Even for purposes of ordinary logical doctrine, the enquiry how far
propositions may be expressed equationally serves to afford a more
complete insight into their full import, or at any rate their full
implication. Hence while ordinary formal logic should not be entirely
based upon an equational reading of propositions, it cannot afford
altogether to neglect this way of regarding them.

We may pass on to consider in somewhat more detail a special equational or semi-equational system—open also to special criticisms—by which Hamilton and others sought to revolutionise ordinary logical doctrine.

195 140. *The
eight propositional forms resulting from the explicit Quantification
of the Predicate*.—We have seen that in the ordinary fourfold
schedule of propositions, the quantity of the predicate is determined
by the quality of the proposition, negatives distributing their
predicates, while affirmatives do not. It seems a plausible view,
however, that by explicit quantification the quantity of the predicate
may be made independent of the quality of the proposition, and Sir
William Hamilton was thus led to recognise eight distinct
propositional forms instead of the customary four:—

All S is all P, | U. |

All S is some P, | A. |

Some S is all P, | Y. |

Some S is some P, | I. |

No S is any P, | E. |

No S is some P, | η. |

Some S is not any P, | O. |

Some S is not some P, | ω. |

The symbols attached to the different propositions in the above schedule are those employed by Archbishop Thomson,191 and they are those now commonly adopted so far as the quantification of the predicate is recognised in modern text-books.

^{191} Thomson himself, however,
ultimately rejects the forms *η* and *ω*.

The symbols used by Hamilton were *Afa*, *Afi*,
*Ifa*, *Ifi*, *Ana*, *Ani*, *Ina*,
*Ini*. Here *f* indicates an affirmative proposition,
*n* a negative; *a* means that the corresponding term is
distributed, *i* that it is undistributed.

For the new forms we might also use the symbols *SuP*,
*SyP*, *SηP*, *SωP*, on
the principle explained in section 62.

141. *Sir William
Hamilton’s fundamental Postulate of Logic*.—The
fundamental postulate of logic, according to Sir William Hamilton, is
“that we be allowed to state explicitly in language all that is
implicitly contained in thought”; and we may briefly consider
the meaning to be attached to this postulate before going on to
discuss the use that is made of it in connexion with the doctrine of
the quantification of the predicate.

196 Giving the
natural interpretation to the phrase “implicitly contained in
thought,” the postulate might at first sight appear to be a
broad statement of the general principle underlying the
logician’s treatment of formal inferences. In all such
inferences the conclusion is implicitly contained in the premisses;
and since logic has to determine what inferences follow legitimately
from given premisses, it may in this sense be said to be part of the
function of logic to make *explicit in language* what is
*implicitly contained in thought*.

It seems clear, however, from the use made of the postulate by
Hamilton and his school that he is not thinking of this, and indeed
that he is not intending any reference to *discursive thought* at
all. His meaning rather is that we should make *explicit in
language* not what is implicit in thought, but what is *explicit
in thought*, or, as it may be otherwise expressed, that we should
make explicit in language all that is really present in thought in the
act of judgment.

Adopting this interpretation, we may come to the conclusion that the postulate is obscurely expressed, but we can have no hesitation in admitting its validity. It is obviously of importance to the logician to clear up ambiguities and ellipses of language. For this reason it is, amongst other things, desirable that we should avoid condensed and elliptical modes of expression. But whether Hamilton’s postulate, as thus interpreted, supports the doctrine of the quantification of the predicate is another question. This point will be considered in the next two sections.

142. *Advantages claimed
for the Quantification of the Predicate*.—Hamilton maintains
that “in thought the predicate is always quantified,” and
hence he makes it follow immediately from the postulate discussed in
the preceding section that “in logic the quantity of the
predicate must be expressed, on demand, in language.” “The
quantity of the predicate,” says Dr Baynes in the authorised
exposition of Hamilton’s doctrine contained in his *New
Analytic of Logical Forms*, “is not expressed in common
language because common language is elliptical. Whatever is not really
necessary to the clear comprehension of what is contained in thought,
is usually elided in 197 expression. But we must distinguish
between the ends which are sought by common language and logic
respectively. Whilst the former seeks to exhibit with clearness the
matter of thought, the latter seeks to exhibit with exactness the form
of thought. Therefore in logic the predicate must always be
quantified.” It is further maintained that the quantification of
the predicate is necessary for intelligible predication.
“Predication is nothing more or less than the expression of the
relation of quantity in which a notion stands to an individual, or two
notions to each other. If this relation were indeterminate—if we
were uncertain whether it was of part, or whole, or none—there
could be no predication.”

Amongst the practical advantages said to result from quantifying the predicate are the reduction of all species of the conversion of propositions to one, namely, simple conversion; and the simplification of the laws of syllogism. As regards the first of these points, it may be observed that if the doctrine of the quantification of the predicate is adopted, the distinction between subject and predicate resolves itself into a difference in order of statement alone. Each propositional form can without any alteration in meaning be read either forwards or backwards, and every proposition may, therefore, rightly be said to be simply convertible.

It is further argued that the new propositional forms resulting
from the quantification of the predicate are required in order to
express relations that cannot otherwise be so simply expressed. Thus,
**U** alone serves to express the fact that two classes are
co-extensive; and even *ω* is said to be needed in logical
divisions, since if we divide (say) Europeans into Englishmen,
Frenchmen, &c., this requires us to think that some Europeans are
not some Europeans (*e.g.*, Englishmen are not Frenchmen).

143. *Objections urged
against the Quantification of the Predicate*.—Those who
reject Hamilton’s doctrine of the quantification of the
predicate deny at the outset the fundamental premiss upon which it is
based, namely, that the predicate of a proposition is always thought
of as a determinate quantity. They go further and deny that it is as a
rule thought of as a 198 quantity, that is, as an aggregate of
objects, at all. We have already in section 135 indicated grounds for the view that, while in the great majority of instances the subject of
a proposition is in ordinary thought naturally interpreted in
denotation, the predicate is naturally interpreted in connotation.
This psychological argument is valid against Hamilton, inasmuch as he
really bases his doctrine upon a psychological consideration; and it
seems unanswerable.

Mill (in his *Examination of Hamilton*, pp. 495-7) puts the
point as follows: “I repeat the appeal which I have already made
to every reader’s consciousness: Does he, when he judges that
all oxen ruminate, advert even in the minutest degree to the question,
whether there is anything else which ruminates? Is this consideration
at all in his thoughts, any more than any other consideration foreign
to the immediate subject? One person may know that there are other
ruminating animals, another may think that there are none, a third may
be without any opinion on the subject: but if they all know what is
meant by ruminating, they all, when they judge that every ox
ruminates, mean exactly the same thing. The mental process they go
through, so far as that one judgment is concerned, is precisely
identical; though some of them may go on further, and add other
judgments to it. The fact, that the proposition ‘Every *A*
is *B*’ only means ‘Every *A* is *some
B*,’ so far from being always present in thought, is not at
first seized without some difficulty by the tyro in logic. It requires
a certain effort of thought to perceive that when we say, ‘All
*A*’s are *B*’s,’ we only identify
*A* with a portion of the class *B*. When the learner is
first told that the proposition ‘All *A*’s are
*B*’s’ can only be converted in the form ‘Some
*B*’s are *A*’s,’ I apprehend that this
strikes him as a new idea; and that the truth of the statement is not
quite obvious to him, until verified by a particular example in which
he already knows that the simple converse would be false, such as,
‘All men are animals, therefore, all animals are men.’ So
far is it from being true that the proposition ‘All
*A*’s are *B*’s’ is spontaneously
quantified in thought as ‘All *A* is some *B*.’”

A word may be added in reply to the argument that if the 199 quantity of the predicate were indeterminate—if we were uncertain whether the reference was to the whole or part or none—there could be no predication. This is perfectly true so long as we are left with all three of these alternatives; but we may have predication which involves the elimination of only one of them, so that there is still indeterminateness as regards the other two. To argue that unless we are definitely limited to one of the three we are left with all of them is practically to confuse contradictory with contrary opposition.

A further objection raised to the doctrine of the quantification of
the predicate is that some of the quantified forms are composite not
simple predications. Thus *All S is all P* is a condensed mode of
expression, which may be analysed into the two propositions *All S
is P* and *All P is S*. Similarly, if we interpret *some*
as exclusive of *all*, a point to which we shall presently
return, *All S is some P* is an exponible proposition resolvable
into *All S is P* and *Some P is not S*. As a rule, however,
the use of exponible forms tends to make the detection of fallacy the
more difficult, and this general consideration applies with undoubted
force to the particular case of the quantification of the predicate.
The bearing of the quantification doctrine upon the syllogism will be
briefly touched upon subsequently, and it will be found that the
problem of discriminating between valid and invalid moods is rendered
more complex and difficult. It may indeed be doubted whether any
logical problem, with the one exception of conversion, is really
simplified by the introduction of quantified predicates.

Even apart from the above objections, the Hamiltonian doctrine of quantification is sufficiently condemned by its want of internal consistency. Its unphilosophical character in this respect will be shewn in the following sections.

144. *The meaning to be
attached to the word “some” in the eight propositional
forms recognised by Sir William Hamilton*.—Professor Baynes,
in his authorised exposition of Sir William Hamilton’s doctrine,
would at the outset lead us to suppose that we have no longer to do
with the indeterminate *some* of the Aristotelian Logic, but that
this word is now to be used in the more definite sense of *some, but
not all*. He argues, as we 200 have seen, that intelligible predication requires an absolutely determinate relation in respect of quantity
between subject and predicate, and that this ought to be clearly
expressed in language. Thus, “if the objects comprised under the
subject be some part, but not the whole, of those comprised under the
predicate, we write *All X is some P*, and similarly with other
forms.”

But if it is true that we know definitely the relative extent of
subject and predicate, and if *some* is used strictly in the
sense of *some but not all*, we should have but *five*
propositional forms instead of eight, namely,—*All S is all
P*, *All S is some P*, *Some S is all P*, *Some S is
some P*,192 *No S is any P*.

^{192} Using *some* in the
sense here indicated, the interpretation of the proposition *Some S
is some P* is not altogether free from ambiguity. The
interpretation I am adopting is to regard it as equivalent to the two
following propositions with unquantified predicates, namely, *Some
but not all S is P* and *Some but not all P is S*. It then
necessarily implies the Hamiltonian propositions *Some S is not any
P* and *No S is some P*.

We have already seen (in section 126) that the only possible relations between two terms in respect of their extension are given by the following five diagrams,—

These correspond respectively to the five propositional forms given above;193 and it is clear that on the view indicated by Dr Baynes the eight forms are redundant.194

^{193} Namely **U**, **A**,
**Y**, **I**, **E**. **O** and η cannot be
interpreted as giving precisely determinate information; **O**
allows an alternative between **Y** and **I**, and η
between **A** and **I**. For the interpretation of ω
see note 2 on page 206.

^{194} Compare Venn, *Symbolic
Logic*, chapter I.

It is altogether doubtful whether writers who have adopted the
eightfold scheme have themselves recognised the pitfalls 201 surrounding the use of
the word *some*. Many passages might be quoted in which they
distinctly adopt the meaning—*some but not all*. Thus,
Thomson (*Laws of Thought*, p. 150) makes **U** and **A**
inconsistent. Bowen (*Logic*, pp. 169, 170) would pass from
**I** to **O** by immediate inference.195 Hamilton himself
agrees with Thomson and Bowen on these points; but he is curiously
indecisive on the general question here raised. He remarks
(*Logic*, II. p. 282) that *some*
“is held to be a definite *some* when the other term is
definite,” *i.e.*, in **A** and **Y**, η and
**O**: but “on the other hand, when both terms are indefinite
or particular, the *some* of each is left wholly
indefinite,” *i.e.*, in **I** and ω.196
This is very confusing, and it would be most difficult to apply the
distinction consistently. Hamilton himself certainly does not so apply
it. For example, on his view it should no longer be the case that two
affirmative premisses necessitate an affirmative conclusion; or that
two negative premisses invalidate a syllogism.197 Thus, the
following should be regarded as valid:

All P is some M, | |

All M is some S, | |

therefore, | Some S is not any P. |

No M is any P, | |

Some S is not any M, | |

therefore, | Some or all S is not any P. |

^{195} “This sort of
inference,” he remarks, “Hamilton would call
*integration*, as its effect is, after determining one part, to
reconstitute the whole by bringing into view the remaining
part.”

^{196} Compare Veitch,
*Institutes of Logic*, pp. 307 to 310, and 367, 8.
“Hamilton would introduce *some only* into the theory of
propositions, without, however, discarding the meaning of *some at
least*. It is not correct to say that Hamilton discarded the
ordinary logical meaning of *some*. He simply supplemented it by
introducing into the propositional forms that of *some
only*.” “*Some*, according to Hamilton, is always
thought as semi-definite (*some only*) where the other term of
the judgment is universal.” Mr Lindsay, however, in expounding
Hamilton’s doctrine (*Appendix to Ueberweg’s System of
Logic*, p. 580) says more decisively,—“Since the
subject must be equal to the predicate, vagueness in the
predesignations must be as far as possible removed. *Some* is
taken as equivalent to *some but not all*.” Spalding
(*Logic*, p. 184) definitely chooses the other alternative. He
remarks that in his own treatise “the received interpretation
*some at least* is steadily adhered to.”

^{197} The anticipation of
syllogistic doctrine which follows is necessary in order to illustrate
the point which we are just now discussing.

202 Such
syllogisms as these, however, are not admitted by Hamilton and
Thomson; and, on the other hand, Thomson admits as valid certain
combinations which on the above interpretation are not valid.
Hamilton’s supreme canon of the categorical syllogism
is:—“What worse relation198 of subject and
predicate subsists between either of two terms and a common third
term, with which one, at least, is positively related; that relation
subsists between the two terms themselves” (*Logic*, II. p. 357). This clearly provides that one
premiss at least shall be affirmative, and that an affirmative
conclusion shall follow from two affirmative premisses. Thomson
(*Laws of Thought*, p. 165) explicitly lays down the same rules;
and his table of valid moods (given on p. 188) is (with the exception
of one obvious misprint) correct and correct only if *some* means
“some, it may be all.”

^{198} The negative relation is
here considered “worse” than the affirmative, and the
particular than the universal.

145. *The use of
“some” in the sense of “some only.”*—Jevons, in reply to the question, “What results would follow if
we were to interpret ‘Some *A*’s are
*B*’s’ as implying that ‘Some other
*A*’s are not *B*’s’?” writes,
“The proposition ‘Some *A*’s are
*B*’s’ is in the form **I**, and according to the
table of opposition **I** is true if **A** is true; but **
A** is the contradictory of **O**, which would be the form of
‘Some other *A*’s are not *B*’s.’
Under such circumstances **A** could never be true at all, because
its truth would involve the truth of its own contradictory, which is
absurd” (*Studies in Deductive Logic*, 151). It is not,
however, the case that we necessarily involve ourselves in
self-contradiction if we use *some* in the sense of *some
only*. What should be pointed out is that, if we use the word in
this sense, the truth of **I** no longer follows from the truth of
**A**; and that, so far from this being the case, these two
propositions are inconsistent with each other.

Taking the five propositional forms, *All S is all P*, *All
S is some P*, *Some S is all P*, *Some S is some P*,
*No S is P*, and interpreting *some* in the sense of *some
only*, it is to be observed that each one of them is inconsistent
with each of the others, whilst at the same time no one is the
contradictory of any 203 one of the others. If, for example, on
this scheme we wish to express the contradictory of **U**, we can
do so only by affirming an alternative between **Y**, **A**,
**I**, and **E**. Nothing of all this appears to have been
noticed by the Hamiltonian writers. Thus, Thomson (*Laws of
Thought*, p. 149) gives a scheme of opposition in which **E**
and **I** appear as contradictories, but **A** and **O** as
contraries.

One of the strongest arguments against the use of *some* in
the sense of *some only* is very well put by Professor Veitch,
himself a disciple of Sir William Hamilton. *Some only*, he
remarks, is not so fundamental as *some at least*. The former
implies the latter; but I can speak of *some at least* without
advancing to the more definite stage of *some only*.
“Before I can speak of *some only*, must I not have formed
two judgments—the one that *some are*, the other that
others of the same class *are not*? …… The *some only* would
thus appear as the composite of two propositions already formed…… It
seems to me that we must, first of all, work out logical principles on
the indefinite meaning of *some at least*…… *Some only* is a
secondary and derivative judgment.” (*Institutes of Logic*, p. 308).

If *some* is used in the sense of *some only*, the
further difficulty arises how we are to express any knowledge that we
may happen to possess about a part of a class when we are in ignorance
in regard to the remainder. Supposing for example, that all the
*S*’s of which I happen to have had experience are
*P*’s, I am not justified in saying either that *all
S’s are P’s* or that *some S’s are
P’s*. The only solution of the difficulty is to say that
*all or some S’s are P’s*. The complexity that this
would introduce is obvious.

146. *The interpretation
of the eight Hamiltonian forms of proposition, “some”
being used in its ordinary logical sense*.199—Taking the
five possible relations between two terms, as illustrated by the
Eulerian diagrams, and denoting them respectively by *α*,
*β*, *γ*, *δ*, *ε*, as in section 126, we
may write against each of the propositional forms the relations which
are compatible with 204 it, on the supposition that *some* is
used in its ordinary logical sense, that is, as exclusive of
*none* but not of *all*:—200

U | α |

A | α, β |

Y | α, γ |

I | α, β, γ, δ |

E | ε |

η | β, δ, ε |

O | γ, δ, ε |

ω | α, β, γ, δ, ε |

^{199} The corresponding
interpretation when *some* is used in the sense of *some
only* is given in notes 1 and 2 on page 200, and in note 2 on page
206.

^{200} If the Hamiltonian writers
had attempted to illustrate their doctrine by means of the Eulerian
diagrams, they would I think either have found it to be unworkable, or
they would have worked it out to a more distinct and consistent
issue.

We have then the following pairs of contradictories—**A**,
**O**; **Y**, η; **I**, **E**. The contradictory
of **U** is obtained by affirming an alternative between η
and **O**.

Without the use of quantified predicates, the same information may be expressed as follows:—

U = SaP, PaS ; |

A = SaP ; |

Y = PaS ; |

I = SiP ; |

E = SeP ; |

η = PoS ; |

O = SoP. |

What information, if any, is given by ω will be discussed in section 149.

147. *The propositions*
**U** *and* **Y**.—It must be admitted that these
propositions are met with in ordinary discourse. 205 We may not indeed find
propositions which are actually written in the form *All S is all
P* ; but we have to all intents and purposes **U**, whenever
there is an unmistakeable affirmation that the subject and the
predicate of a proposition are co-extensive. Thus, all definitions are
practically **U** propositions; so are all affirmative propositions
of which both the subject and the predicate are singular terms.201
Take also such propositions as the following: Christianity and
civilization are co-extensive; Europe, Asia, Africa, America, and
Australia are all the continents;202 The three whom I
have mentioned are all who have ever ascended the mountain by that
route; Common salt is the same thing as sodium chloride.203

^{201} Take the proposition,
“Mr Gladstone is the present Prime Minister.” If any one
denies that this is **U**, then he must deny that the proposition
“Mr Gladstone is an Englishman” is **A**. We have at an
earlier stage discussed the question how far singular propositions may rightly be regarded as constituting a sub-class of universals.

^{202} In this and the example
that follows the predicate is clearly quantified universally; so that
if these are not **U** propositions, they must be **Y**
propositions. But it is equally clear that the subject denotes the
whole of a certain class, however limited that class may be.

^{203} These are all examples of
what Jevons would call *simple identities* as distinguished from
*partial identities*. Compare section 138.

Such propositions as the following, sometimes known as
*exclusive* propositions, may be given as examples of *Y*:
*Only S is P* ; Graduates alone are eligible for the appointment;
Some passengers are the only survivors. These propositions may be
interpreted as being equivalent to the following: *Some S is all
P* ; Some graduates are all who are eligible for the appointment;
Some passengers are all the survivors.204 This is, indeed,
the only way of treating the propositions which will enable us to
retain the original subjects as subjects and the original predicates
as predicates.

^{204} In these propositions,
*some* is to be interpreted in the indefinite sense, and not as
exclusive of *all*.

We cannot then agree with Professor Fowler that the additional
forms “are not merely unusual, but are such as we never do
use” (*Deductive Logic*, p. 31). Still in treating the
syllogism &c. on the traditional lines, it is better to retain the
traditional schedule of propositions. The addition of the forms 206 **U** and **Y**
does not tend towards simplification, but the reverse; and their full
force can be expressed in other ways. On this view, when we meet with
a **U** proposition, *All S is all P*, we may resolve it into
the two **A** propositions, *All S is P* and *All P is S*, which
taken together are equivalent to it; and when we meet with a **Y**
proposition, *Some S is all P* or *S alone is P*, we may
replace it by the **A** proposition *All P is S*, which it
yields by conversion.

148. *The proposition*
η.—This proposition in the form *No S is some P*
is not I think ever found in ordinary use. We may, however, recognise
its possibility; and it must be pointed out that a form of proposition
which we do meet with, namely. *Not only S is P* or *Not S
alone is P*, is practically *η*, provided that we do not
regard this proposition as implying that any *S* is certainly *P*.

Archbishop Thomson remarks that η “has the semblance
only, and not the power of a denial. True though it is, it does not
prevent our making another judgment of the affirmative kind, from the
same terms” (*Laws of Thought*, § 79). This is erroneous;
for although **A** and η may be true together, **U**
and η cannot, and **Y** and η are strictly
contradictories.205 The relation of contradiction in which
**Y** and η stand to each other is perhaps brought out
more clearly if they are written in the forms *Only S is P*,
*Not only S is P*, or *S alone is P*, *Not S alone is
P*. It will be observed, moreover, that η is the converse
of **O**, and *vice versâ*. If, therefore, η has no
power of denial, the same will be true of **O** also. But it
certainly is not true of **O**.

^{205} We are again interpreting
*some* as indefinite. If it means *some at most*, then the
power of denial possessed by η is increased.

149. *The proposition*
ω.—The proposition ω, *Some S is not some
P*, is not inconsistent with any of the other propositional forms,
not even with **U**, *All S is all P*. For example, granting
that “all equilateral triangles are all equiangular
triangles,” still “this equilateral triangle is not that
equiangular triangle,” which is all that ω asserts.
*Some S is not some P* is indeed always true except when both the
subject and the predicate are the name of an individual and the same
individual.206 De 207 Morgan207 (*Syllabus*,
p. 24) observes that its contradictory is—“*S* and
*P* are singular and identical; there is but one *S*, there
is but one *P*, and *S* is *P*.”208 It may be said
without hesitation that the proposition ω is of absolutely no
logical importance.

^{206} *Some* being again
interpreted in its ordinary logical sense. Mr Johnson points out that
if *some* means *some but not all*, we are led to the
paradoxical conclusion that ω is equivalent to **U**. We
may regard a statement involving a reference to *some but not
all* as a statement relating to *some at least*, combined with
a denial of the corresponding statement in which *all* is
substituted for *some*. On this interpretation, *Some S is not
some P* affirms that “*S* and *P* are not
identically one,” but also denies that “some *S* is
not any *P*” and that “some *P* is not *any
S*”; that is, it affirms *SaP* and *PaS*.

^{207} De Morgan in several
passages criticizes with great acuteness the Hamiltonian scheme of
propositions.

^{208} Professor Veitch remarks
that in ω “we assert parts, and that these can be
divided, or that there are parts and parts. If we deny this statement,
we assert that the thing spoken of is indivisible or a unity…… We
may say that there are men and men. We say, as we do every day, there
are politicians and politicians, there are ecclesiastics and
ecclesiastics, there are sermons and sermons. These are but covert
forms of the *some are not some*…… ‘Some vivisection is not
some vivisection’ is true and important; for the one may be with
an anaesthetic, the other without it” (*Institutes of
Logic*, pp. 320, 1). It will be observed that the proposition
*There are politicians and politicians* is here given as a
typical example of ω. The appropriateness of this is denied
by Mr Monck. “Again, can it be said that the proposition
*There are patriots and patriots* is adequately rendered by
*Some patriots are not some patriots*? The latter proposition
simply asserts non-identity: the former is intended to imply also a
certain degree of dissimilarity [*i.e.*, in the characteristics
or consequences of the patriotism of different individuals]. But two
non-identical objects may be perfectly alike” (*Introduction
to Logic*, p. xiv).

150. *Sixfold Schedule of
Propositions obtained by recognising* **Y** *and* η, *in addition to* **A**, **E**, **I**, **O**.209—The schedule of propositions obtained
by adding **Y** and η to the ordinary schedule presents
some interesting features, and is worthy of incidental recognition and
discussion.210 It has been shewn in section 100 that in the
ordinary scheme there are six and only six independent propositions
connecting any two terms, namely, 208 *SaP*, *PaS*, *SeP* (=
*PeS*), *SiP* (= *PiS*), *PoS*, *SoP*. If we
write the second and the last but one of these in forms in which
*S* and *P* are respectively subject and predicate, we have
the schedule which we are now considering, namely,

SaP | = | All S is P ; |

SyP | = | Only S is P ; |

SeP | = | No S is P ; |

SiP | = | Some S is P ; |

SηP | = | Not only S is P ; |

Sop | = | Some S is not P. |

^{209} In this schedule
*some* is interpreted throughout in its ordinary logical sense.
**U** is omitted on account of its composite character; its
inclusion would also destroy the symmetry of the scheme.

^{210} It is not intended that
this sixfold schedule should supersede the fourfold schedule in the
main body of logical doctrine. It is, however, important to remember
that the selection of any one schedule is more or less arbitrary, and
that no schedule should be set up as authoritative to the exclusion of
all others.

It will be observed that the pair of propositions, *SyP* and
*SηP*, are contradictories; so that we now have
three pairs of contradictories. There are of course other additions to
the traditional table of opposition, and some new relations will need
to be recognised, *e.g.*, between *SaP* and *SyP*. With
the help, however, of the discussion contained in section 107, the
reader will have no difficulty in working out the required hexagon of
opposition for himself.

As regards immediate inferences, we cannot in this scheme obtain
any satisfactory obverse of either **Y** or η, the reason
being that they have quantified predicates, and that, therefore, the
negation cannot in these propositions be simply attached to the
predicate. We have, however, the following interesting table of other
immediate inferences:—211

Converse. | Contrapositive. | Inverse. | ||||

SaP | = | PyS | = | PʹaSʹ | = | SʹyPʹ |

SyP | = | PaS | = | PʹySʹ | = | SʹaPʹ |

SeP | = | PeS | = | PʹyS | = | SʹyP |

SiP | = | PiS | = | PʹηS | = | SʹηP |

SηP | = | PoS | = | PʹηSʹ | = | SʹoPʹ |

SoP | = | PηS | = | PʹoSʹ | = | SʹηPʹ |

^{211} It will be observed that
the impracticability of obverting **Y** and η leads to a
certain want of symmetry in the third and fourth columns.

The main points to notice here are (1) that each proposition now admits of conversion, contraposition, and inversion; and (2) that the inferred proposition is in every case equivalent to the original proposition, so that there is not in any of the 209 inferences any loss of logical force. In other words, we obtain in each case a simple converse, a simple contrapositive, and a simple inverse.

151. Explain precisely how
it is that **O** admits of ordinary conversion if the principle of
the quantification of the predicate is adopted, although not
otherwise. [K.]

152. Draw out a table,
corresponding to the ordinary Aristotelian table of opposition, for
the six propositions, **A**, **Y**, **E**, **I**, η, **O** (some being interpreted in the sense of *some at
least*). [K.]

^{212} It will be advisable for
students, on a first reading, to omit this chapter.

153. *Existence and the
Universe of Discourse*.—It has been shew in section 49 that
every judgment involves an objective reference, or—as it may
otherwise be expressed—a reference to some system of reality
distinct from the act of judgment itself. The reference may be to the
total system of reality without limitation, or it may be to some
particular aspect or portion of that system. Whatever it may be, we
may speak of it as the *universe of discourse*.213
The universe of discourse may be limited in various ways; for example,
to physical objects, or to psychical events, or again with reference
to time or space. But in all cases it is a universe of reality in the
sense in which that term has been used in section 49. The nature of
the reference in propositions relating to fictitious objects, for
example, to the characters and occurrences in a play or a novel, may
be specially considered. We may say that in a case of this kind the
universe of discourse consists of a series of statements about persons
and events made by a certain author; and it is clear that such
statements have objective reality, although the persons and events
themselves are fictitious. It follows that, as regards 211 the reference to
reality, such a proposition as “Hamlet killed Polonius”
must be considered elliptical. For the reference is not to real
persons or to the actual course of events in the past history of the
world, as it is when we say “Mary Stuart was beheaded,”
but to a series of descriptions given by Shakespeare in a particular
play. These descriptions have, however, a reality of their own, and
(the different nature of the reference being clearly understood) I am
no more free to say that Hamlet did not kill Polonius (that is, that
Shakespeare did not describe Hamlet as killing Polonius) than I am to
say that Mary Stuart was not beheaded.

^{213} “The universe of
discourse is sometimes limited to a small portion of the actual
universe of things, and is sometimes co-extensive with that
universe” (Boole, *Laws of Thought*, p. 166). On the
conception of a limited universe of discourse, compare also De Morgan,
*Syllabus of a Proposed System of Logic*, §§ 122, 3, and
*Formal Logic*, p. 55; Venn, *Symbolic Logic*, pp. 127, 8;
and Jevons, *Principles of Science*, chapter 3, § 4.

The substance of the above has been expressed by saying that reality is the ultimate subject of every proposition. Every proposition makes an affirmation about a certain universe of discourse, and the universe of discourse (whatever it may be) has some real content. In this sense then every proposition has an existent subject.214 A further question may, however, be raised, namely, whether—using the word “subject” in its ordinary logical signification—all or any propositions should be interpreted as implying the existence (or occurrence) of their subjects within the universe of discourse (or particular portion of reality) to which reference is made. It is mainly with this problem, and the ways in which ordinary logical doctrines are affected by its solution, that we shall be concerned in the present chapter.

^{214} Compare Bradley, *Principles of Logic*, p. 41.

In our discussion of existential import it will not be necessary that we should make any attempt to determine the ultimate nature of reality. The questions at issue are, however, not exactly easy of solution, and various sources of misunderstanding are apt to arise.

There is one sense in which the existence of something
corresponding to the terms employed must be postulated in all
predication. For in order to make use of any term in an intelligible
sense we must mentally attach some meaning to it. Hence there must be
something in the mind corresponding to every term we use. Even in
cases where there cannot be said to be any corresponding mental
product, there must at any rate 212 be some corresponding mental process. This
applies even to such terms as *round square* or *non-human
man* or *root of minus one*. We are not indeed able to form an
image of a round square or an idea of a non-human man, nor can we
evaluate the root of minus one. But we attach a meaning to these
terms, and they must therefore have a mental equivalent of some sort.
In the case of “round square” or “non-human
man” this is not the actual combination in imagination or idea
of “round” with “square” or
“non-human” with “man,” for such combinations
are impossible. But it is the idea of the combination, regarded as a
problem presented for solution, and perhaps involving an unsuccessful
effort to effect the combination in thought. It is apparently of
existence of this kind that some writers are thinking when they
maintain that of necessity every proposition implies logically the
existence of its subject. But our meaning is something quite different
when we speak of existence in the universe of discourse. The nature of
the distinction may be made more clear by the following considerations.

It will be admitted that whatever else is included in the full
implication of a universal proposition, it at least denies the
existence of a certain class of objects. *No S is P* denies the
existence of objects that are both *S* and *P* ; *All S is
P* denies the existence of objects that are *S* without also
being *P*. In these propositions, however, we do not intend to
deny the existence of *SP* (or *SPʹ*) as objects of
thought. For example, in the proposition *No roses are blue* it
is not our intention to deny that we can form an idea of *blue
roses* ; nor in the proposition *All ruminant animals are
cloven-hoofed* is it our intention to deny that *ruminant animals
without cloven hoofs* can exist as objects of thought. These
illustrations may help us to understand more clearly what is meant by
existence in the universe of discourse. *The universe of discourse
in the case of the proposition No S is P is the universe*
(*whatever it may be*) *in which the existence of SP is
denied*. The universe of discourse in the case of a universal
affirmative proposition may be defined similarly. As regards
particulars it may be best to seek an interpretation through the
universals by which the particulars 213 are contradicted. Thus, the universe of
discourse in the case of the proposition *Some S is P* may be
defined as the universe (whatever it may be) in which the existence of
*SP* would be understood to be denied in the corresponding
universal negative. The proposition *Some S is not P* may be
dealt with similarly.

The question whether a categorical proposition is to be interpreted
as formally implying that its terms are the names of existing things
may then be interpreted as follows: *Given a categorical proposition
with S and P as subject and predicate, is the existence of S or of P
formally implied in that sphere* (*whatever it may be*) *in
which the existence of SP* (*or SPʹ*) *is denied by the
proposition* (*or by its contradictory*)*?*

The question may be somewhat differently expressed as follows. Such
a proposition as *No S is P* denies the existence of a certain
complex of attributes, namely, *SP*. But with rare exceptions,
*S* itself signifies a certain complex of attributes; and so does
*P*. Does the proposition affirm the existence of these latter
complexes in the same sense as that in which it denies the existence
of the former complex?

No general criterion can be laid down for determining what is
actually the universe of discourse in any particular case. It may,
however, be said that knowledge as to what is the universe referred to
is involved in understanding the meaning of any given proposition; and
cases in which there can be any practical doubt are exceptional.215
Thus, in the propositions *No roses are blue*, *All men are
mortal*, *All ruminant animals are cloven-hoofed*, the
reference clearly is to the actual physical universe; in *The wrath
of the Olympian gods is very terrible* to the universe of the Greek
mythology;216 in *Fairies are able to assume different
forms* to the universe of folk-lore;217 in *Two straight
lines cannot enclose a space* to the universe of spatial
intuitions.

^{215} It must at the same time be
admitted that controversies sometimes turn upon an unrecognised want
of agreement between the controversialists as to the universe of
discourse to which reference is made.

^{216} The universe of the Greek
mythology does not consist of gods, heroes, centaurs, &c., but of
accounts of such beings currently accepted in ancient Greece, and
handed down to us by Homer and other authors. As regards the reference
to reality, therefore, such a proposition as *The wrath of the
Olympian gods is very terrible* is elliptical in a sense already
explained.

^{217} Here again there is an
ellipsis. The universe of folk-lore does not consist of fairies,
elves, &c., but of descriptions of them, based on popular beliefs,
and conventionally accepted when such beings are referred to. Of
course for anyone who really believed in the existence of fairies
there would be no ellipsis, and the universe of discourse would be
different.

214 With respect
to the existential import of propositions the following questions
offer themselves for consideration:

(1) Is the problem one with which logic, and more particularly
formal logic, is properly concerned?

(2) How should the propositions belonging to the traditional
schedule be interpreted as regards their existential implications?

(3) Can we formulate a schedule of propositions which directly
affirm or deny existence, and how will such a schedule be related to
the traditional schedule?

(4) How are ordinary logical doctrines affected by the answer given
to the second of these questions?

It is clear that the first and fourth of these questions are connected, since if the fourth admits of any positive answer at all, the first is thereby answered in the affirmative. Since, however, the first question blocks the way and seems to demand an answer before we carry the discussion further, it will be well to deal with it briefly at the outset.

The second and third questions are also closely connected together.

Between the second and fourth questions an important distinction must be drawn. The second question is one of interpretation, and within certain limits the answer to it is a matter of convention. Hence a given solution may be preferred on grounds that would not justify the rejection of other solutions as altogether erroneous, although they may be considered inconvenient or unsuitable. But the answer to the fourth question is not similarly a matter of convention. On the basis of any given interpretation of propositional forms, the manner in which logical doctrines are affected can admit of only one correct solution.

It is to be observed further that the fourth question can be dealt with hypothetically, that is to say, we can work out the consequences of interpretations which we have no intention of 215 adopting; and it is desirable that we should work out such consequences before deciding upon the adoption of any given interpretation. Hence we propose to deal with the fourth question before discussing the second. The third question may conveniently be taken after the first.

154. *Formal Logic and the
Existential Import of Propositions*.—We have then, in the
first place, briefly to consider the question whether the problem of
existential import is one with which logic has any proper concern. It
may be urged that formal logic, at any rate, cannot from its very
nature be concerned with questions relating to existence in any other
sphere than that of thought. The function of the formal logician, it
may be said, is to distinguish between that which is self-consistent
and that which is self-contradictory; it is his business to
distinguish between what can and what cannot exist in the world of
thought. But beyond this he cannot go. Any considerations relating to
objective existence are beyond the scope of formal logic.

We may meet the above argument by clearly defining our position. It is of course no function of logic to determine whether or not certain classes actually exist in any given universe of discourse, any more than it is the function of logic to determine whether given propositions are true or false. But it does not follow that logic has, therefore, no concern with any questions relating to objective existence. For, just as, certain propositions being given true, logic determines what other propositions will as a consequence also be true, so given an assertion or a set of assertions to the effect that certain combinations do or do not exist in a given universe of discourse, it can determine what other assertions about existence in the same universe of discourse follow therefrom.218 As a matter of fact, the premisses in any argument necessarily contain certain implications in regard to existence in the particular universe of 216 discourse to which reference is made, and the same is true of the conclusion; it is accordingly essential that the logician should make sure that the latter implications are clearly warranted by the former.

^{218} The latter part of this
statement is indeed nothing more than a repetition of the former part
from a rather different point of view. The doctrine that the
conclusions reached by the aid of formal logic can never do more than
relate to what is merely conceivable is a very mischievous error. The
material truth of the conclusion of a formal reasoning is only limited
by the material truth of the premisses.

Without at present going into any detail we may very briefly
indicate one or two existential questions that cannot be altogether
excluded from consideration in formal logic. Universal propositions,
as we have seen, assert non-existence in some sphere of reality; and
it is not possible to bring out their full import without calling
attention to this fact. Again, the proposition *All S is P* at
least involves that if there are any *S*’s in the universe
of discourse, there must also be some *P*’s, while it does
not seem necessarily to involve that if there are any *P*’s
there must be some *S*’s. But now convert the proposition.
The result is *Some P is S*, and this does involve that if there
are any *P*’s there must be some *S*’s.219
How then 217 can the
process of conversion be shewn to be valid without some assumption
which will serve to justify this latter implication? Similarly, in
passing from *All S is P* to *Some not-S is not-P*, it must
at least be assumed that if *S* does not constitute the entire
universe of discourse, neither does *P* do so. It is indeed quite
impossible to justify the process of inversion in any case without
having some regard to the existential interpretation of the
propositions concerned.220

^{219} Dr Wolf denies this. His
argument is, however, based mainly on the misinterpretation of a
single concrete example. “Let us,” he says, “take a
concrete example. *Some things that children fear are ghosts*.
Does this proposition imply that if there is anything that children
fear then there are also ghosts? Surely one may legitimately make such
an assertion while believing that there are things that children fear,
and yet absolutely disbelieving in the existence of ghosts. In fact
the above proposition might very well be used in conjunction with an
express denial of the existence of ghosts in order to prove that,
while some things that children fear are real, they are also afraid of
things that do not exist, but are merely imaginary” (*Studies
in Logic*, p. 144). Any speciousness that this argument may possess
arises from the ambiguity of the words “thing” and
“real.” It is clear that in order to make the proposition
in question intelligible the word “things” must be
interpreted to mean “things, real or imaginary.” Moreover
“imaginary things” have a reality of their own, though it
is not a physical, material reality. Ghosts, therefore, do exist in
the universe of discourse to which reference is made. The objects
denoted by the predicate of the proposition have in fact just the same
kind of existence as certain of the objects denoted by the subject.
Looking at the matter from a slightly different point of view, it is
clear that if by “things” in the subject we mean things
having material existence, then unless ghosts have a similar existence
the proposition is not true.

Bearing in mind the constant ambiguity of language, and the ways in
which verbal forms may fail to represent adequately the judgments they
are intended to express, it would in any case be unsatisfactory to
allow a question of the kind we are here discussing to be decided by a
single concrete example. Dr Wolf’s view is that *Some S is P* does
not imply that if there are any *S*’s there are also some
*P*’s. Suppose then that there are some *S*’s
and that there are no *P*’s. It follows that there are
*S*’s but not a single one of them is *P*. What in
these circumstances the proposition *Some S is P* can mean it is
difficult to understand.

So far as Dr Wolf’s argument is independent of the above
concrete example, it appears to depend upon an identification of the
proposition *Some S is P* with the proposition *S may be P*.
The latter is a modal form, and is undoubtedly consistent with the
existence of *S* and the non-existence of *P*. But I venture
to think that the identification of the two forms runs entirely
counter to the current use of language. I am quite prepared to admit
that if *All S is P* is interpreted as an unconditional
universal, meaning *S as such is P*, its true contradictory is
*S may be P*, not *Some S is P*. But this is just because I
do not think that *Some S is P* would be understood to express
merely the abstract compatibility of *S* and *P*. Certainly
Dr Wolf’s own concrete example, referred to above, cannot bear this
interpretation. For some further observations on modals in connexion
with existential import, see sections 160 and 163.

^{220} Jevons remarks that he does
not see how there can be in deductive logic any question about
existence, and observes, with reference to the opposite view taken by
De Morgan, that “this is one of the few points in which it is
possible to suspect him of unsoundness “ (*Studies in
Deductive Logic*, p. 141). It is, however, impossible to attach any
meaning to Jevons’s own “Criterion of Consistency,”
unless it has some reference to “existence.” “It is
assumed as a necessary law that every term must have its negative.
Thence arises what I propose to call the *Criterion of
Consistency*, stated as follows:—*Any two or more
propositions are contradictory when, and only when, after all possible
substitutions are made, they occasion the total disappearance of any
term, positive or negative, from the Logical Alphabet*” (p.
181). What can this mean but that although we may deny the existence
of the combination *AB*, we cannot without contradiction deny the
existence of *A* itself, or *not-A*, or *B*, or
*not-B*? This assumption regarding the existential implication of
propositions runs through the whole of Jevons’s equational
logic. The following passage, for example, is taken almost at random:
“There remain four combinations, *ABC*, *aBC*,
*abC*, *abc*. But these do not stand on the same logical
footing, because if we were to remove *ABC*, there would be no
such thing as *A* left; and if we were to remove *abc* there
would be no such thing as *c* left. Now it is the criterion or
condition of logical consistency that every separate term and its
negative shall remain. Hence there must exist some things which are
described by *ABC*, and other things described by
*abc*” (p. 216).

218 155. *The
Existential Formulation of Propositions*.—We may define an
*existential proposition* as one that directly affirms or denies
existence (or occurrence) in the universe of discourse (or portion of
reality) to which reference is made. Such propositions are of course
met with in ordinary forms of speech: for example, *God exists*,
*It rains*, *There are white hares*, *It does not
rain*, *Unicorns are non-existent*. *There is no rose
without a thorn*. Sometimes the affirmation or denial of existence
takes a less simple form, but is none the less direct: for example,
*The assassination of Caesar is an historical event*,
*D’Artagnan is not an imaginary person*, *The centaur is a
fiction of the poets*, *The large copper butterfly is extinct*.

In the formal expression of existential propositions it will be
convenient to make use of certain symbols described in the preceding
chapter. Thus, the affirmation of the existence of *S* may be
written in the form *S* > 0, and the denial of the existence of
*S* in the form *S* = 0. We shall then have an existential schedule
of propositions if we reduce our statements to one or other of these
forms or to a conjunctive or disjunctive combination of them. The
relation between the traditional schedule and an existential schedule
of this kind will be discussed in the next section but one.

It may here be pointed out that since the universe of discourse is
itself assumed to be real and hence cannot be entirely emptied of
content, any denial of existence involves also an affirmation of
existence. For if we deny the existence of *S*, we thereby
implicitly affirm the existence of *not-S*, since by the law of
excluded middle everything in the universe of discourse must be either
*S* or *not-S*. It follows that every proposition contains
directly or indirectly an affirmation of existence.221

^{221} In an article in
Baldwin’s *Dictionary of Philosophy and Psychology*, Mrs
Ladd Franklin points out that the proposition *All S is P* is
equivalent to the proposition *Everything is P or not-S*, and
hence necessarily implies the existence of either *P* or
*not-S*. Write *x* for *not-S* and *y* for
*P*, so that the original proposition becomes *All but x is
y* ; it then implies, as its minimum existential import, the
existence of *either x or y*.

156. *Various Suppositions
concerning the Existential Import of Categorical
Propositions*.—Several different views may be 219 taken as to what
implication with regard to existence, if any, is involved in
categorical propositions of the traditional type. The following may be
formulated for special discussion:—222

^{222} The suppositions that
follow are not intended to be exhaustive. We might, for instance,
regard propositions as implying the existence both of their subjects
and their predicates, but not of the contradictories of these; or we
might regard universals as always implying the existence of their
subjects, but particulars as not necessarily implying the existence of
theirs (see note 3 on p. 241); or affirmatives as always implying the existence of their subjects, but negatives as not necessarily implying
the existence of theirs. This last supposition represents the view of
Ueberweg. Still another view is taken by Lewis Carroll, who regards
all categorical propositions, except universal negatives, as implying
the existence of their subjects. “In every proposition beginning
with *some* or *all*, the actual existence of the subject is
asserted. If, for instance, I say ‘all misers are
selfish,’ I mean that misers *actually exist*. If I wished
to avoid making this assertion, and merely to state the *law*
that miserliness necessarily involves *selfishness*, I should say
‘no misers are unselfish,’ which does not assert that any
misers exist at all, but merely that, if any *did* exist, they
would be selfish” (*Game of Logic*, p. 19). It would take
too much space, however, to give a separate discussion to suppositions
other than those mentioned in the text.

(1) It may be held that every categorical proposition should be
interpreted as implying the existence both of objects denoted by the
terms directly involved and also of objects denoted by their
contradictories; that, for example, *All S is P* should be
regarded as implying the existence of *S*, *not-S*,
*P*, *not-P*. This view is implied in Jevons’s
Criterion of Consistency mentioned in the note on page 217. It is also
practically adopted by De Morgan.223

^{223} “By the
*universe* (of a proposition) is meant the collection of all
objects which are contemplated as objects about which assertion or
denial may take place. *Let every name which belongs to the whole
universe be excluded as needless*: this must be particularly
remembered. Let every object which has not the name *X* (*of
which there are always some*) be conceived as therefore marked with
the name *x* meaning *not-X*” (*Syllabus*, pp.
12, 13). Compare, also, De Morgan’s *Formal Logic*, p.
55.

(2) It may be held that every proposition should be interpreted as implying simply the existence of its subject. This is Mill’s view (as regards real propositions); for he holds that we cannot give information about a non-existent subject.224 This is no doubt the view that, at any rate on a first 220 consideration of the subject, appears to be at once the most reasonable and the most simple.

^{224} “An accidental or
non-essential affirmation does imply the real existence of the
subject, because in the case of a non-existent subject there is
nothing for the proposition to assert” (*Logic*, I. 6, § 2).

(3) It may be held that we should not regard propositions as
necessarily implying the existence either of their subjects or of
their predicates. On this view, the full implication of *All S is
P* may be expressed by saying that it denies the existence of
anything that is at the same time *S* and *not-P*. Similarly
*No S is P* implies the existence neither of *S* nor of
*P*, but merely denies the existence of anything that is both
*S* and *P*. *Some S is P* (or *is not P*) may be
read *Some S, if there is any S, is P* (or *is not P*). Here
we neither affirm nor deny the existence of any class absolutely;225
the sum total of what we affirm is that *if any S* exists, then
something which is both *S* and *P* (or *S* and
*not-P*) also exists. On this interpretation, therefore, particular propositions have a hypothetical and not a purely
categorical character.

^{225} Jevons lays down the
*dictum* that “we cannot make any statement except a truism
without implying that certain combinations of terms are contradictory
and excluded from thought” (*Principles of Science*, 2nd
edition, p. 32). This is true of universals (though somewhat loosely
expressed), but it does not seem to be true of particular
propositions, whatever view may be taken of them.

(4) It may be held that universal propositions should not be
interpreted as implying the existence of their subjects, but that
particular propositions should be interpreted as doing so.226
On this view *All S is P* merely denies the existence of anything
that is both *S* and *not-P*; *No S is P* denies the
existence of anything that is both *S* and *P* ; *Some S is
P* affirms the existence of something that is both *S* and
*P* ; *Some S is not P* affirms the existence of something
that is both *S* and *not-P*. Thus, *universals* are
interpreted as having existentially a *negative* force, while
*particulars* have an *affirmative* force. This hypothesis
will be found to lead to certain paradoxical results, but it will also
be shewn to lead to a more satisfactory and symmetrical treatment of
logical problems than is otherwise possible.227

^{226} Dr Venn advocates this
doctrine with special reference to the operations of symbolic logic;
but there is no reason why it should not be extended to ordinary
formal logic.

^{227} The hypothesis in question
has been already provisionally adopted in the scheme of logical
equivalences given in section 108, and also in the symbolic scheme of propositions given on page 193.

221 157.
*Reduction of the traditional forms of proposition to the form of
Existential Propositions*.—Without at present attempting to
decide between the different possible suppositions as to the
existential import of the traditional forms of proposition, we may
enquire how on the different suppositions they may be reduced to
existential form. It will be assumed throughout that both the
traditional forms and the existential forms are interpreted
assertorically. In the case of each of the traditional forms it will
suffice to deal with the two fundamental suppositions, namely, that it
does and that it does not imply the existence of its subject.

*The universal affirmative*. (1) If *SaP* is interpreted
as not carrying with it any existential implication in regard to its
separate terms, it is equivalent to the existential proposition
*SPʹ* = 0. Dr Wolf denies this on the ground that *SaP*
contains further the implication “If there are any
*S*’s, they must all be *P*’s”; and hence
that, while on the supposition in question *SPʹ* = 0 is an
*inference* from *SaP*, it is *not equivalent* to it.
It is of course a very elementary truth that inferences are not always
the exact equivalents of their premisses. But in the above argument Dr
Wolf has apparently overlooked the fact that *SPʹ* = 0,
equally with *SaP*, contains the implication “If there are
any *S*’s they are all *P*’s.”228
By the law of excluded middle, every *S* (if there are any
*S*’s) must be *P* or *not P*, and since
*SPʹ* = 0, the above inference clearly follows.
*SPʹ* = 0 carries with it in fact the two implications *If
S* > 0 *then **P* > 0, *If P* > 0 *then **Sʹ* > 0.
These may also be written in the forms *Either **S* = 0 *or **P*
> 0, *Either **Pʹ* = 0 *or **Sʹ* > 0.

^{228} Dr Wolf perhaps draws a
distinction between the proposition “If there are any
*S*’s they must all be *P*’s” and the
proposition “If there are any *S*’s they are all
*P*’s,” giving to the former an apodeictic, and to
the latter a merely assertoric, force. But if so, then the former is
implied by *All S is P*, only if this proposition is apodeictic,
not if it is merely assertoric. The argument is in this case
irrelevant so far as the position which I take is concerned, since it
is only the assertoric *SaP* that I regard as equivalent to
*SPʹ* = 0. Dr Wolf can hardly maintain that all propositions
of the form *All S is P* are apodeictic. His whole treatment of
the subject with which we are now dealing appears, however, to be
valid only if it relates to a modal schedule of propositions. At the
same time he nowhere clearly indicates a limitation of this kind, and
many of the doctrines which he criticises are intended by those who
adopt them to apply only to an assertoric schedule.

222 (2) If
*SaP* is interpreted as implying the existence of *S*, then
it may be expressed existentially *S* > 0 and *SPʹ* = 0.
These existential forms carry with them the implications *P* > 0,
*Either **Pʹ* = 0 *or **Sʹ* > 0.

*The universal negative*. Taking the same two suppositions the
corresponding existentials will be:—

(1) *SP* = 0 (carrying with it the implications *Either
**S* = 0 *or **Pʹ* > 0, *Either **P* = 0 *or **Sʹ* >
0);

(2) *S* > 0 *and* *SP* = 0 (with the implications
*Pʹ* > 0, *Either **P* = 0 *or **Sʹ* > 0).

These results need no separate discussion.

*The particular affirmative*. (1) On the supposition that
*SiP* does not carry with it any implication as to the separate
existence of its terms, it can be expressed existentially *Either
**S* = 0 *or **SP* > 0. It might also be written in the form *If **S* > 0 *then **SP* > 0. Complications resulting from the
introduction of considerations of modality will, however, be more
easily avoided if the hypothetical form is not made use of.

(2) On the supposition that the existence of *S* is implied,
*SiP* is reducible to the form *SP* > 0.

*The particular negative*. Here the corresponding results are
(1) *Either **S* = 0 *or **SPʹ* > 0; (2) *SPʹ* > 0.

We may sum up our results with reference to the third and fourth of the suppositions formulated in the preceding section.

*Let no proposition be interpreted as implying the existence of
its separate terms*. Then corresponding to the traditional schedule
we have the following existential schedule:—

A,—SPʹ = 0; |

E,—SP = 0; |

I,—Either S = 0 or SP > 0; |

O,—Either S = 0 or SPʹ > 0. |

This represents what may be regarded as the *minimum*
existential import of each of the traditional propositions
(interpreted assertorically).

It must be remembered that *SPʹ* = 0 carries with it the
implications *Either **S* = 0 *or **P* > 0, *Either **Pʹ*
= 0 *or **Sʹ* > 0.

*Let particulars be interpreted as implying, while universals are
not interpreted as implying, the existence of their subjects*.
223 We then
have:—

A,—SPʹ = 0; |

E,—SP = 0; |

I,—SP > 0; |

O,—SPʹ > 0. |

158. *Immediate Inferences
and the Existential Import of Propositions*.—It has been
already suggested that before coming to any decision in regard to the
existential import of propositions, it will be well to enquire how
certain logical doctrines are affected by the different existential
assumptions upon which we may proceed. This discussion will as far as
possible be kept distinct from the enquiry as to which of the
assumptions ought normally to be adopted. The latter question is of a
highly controversial nature, but the logical consequences of the
various suppositions ought to be capable of demonstration, so as to
leave no room for differences of opinion.

We shall in the present section enquire how far different hypotheses regarding the existential import of propositions affect the validity of obversion and conversion and the other immediate inferences based upon these. In the next section we shall consider inferences connected with the square of opposition.

We may take in order the suppositions formulated in section 156.

(1) *Let every proposition he understood to imply the existence
of both its subject and its predicate and also of their
contradictories*.

It is clear that on this hypothesis the validity of conversion,
obversion, contraposition, and inversion will not be affected by
existential considerations. The terms of the original proposition
together with their contradictories being in each case identical with
the terms of the inferred proposition together with their
contradictories, the latter cannot possibly contain any existential
implication that is not already contained in the original
proposition.229

^{229} The reader may be reminded
that in our first working out of these immediate inferences we
provisionally assumed, apart from any implication contained in the
propositions themselves, that the terms involved and also their
contradictories represented existing classes.

224 (2) *Let
every proposition he understood to imply simply the existence of its
subject*.

(*a*) The validity of obversion is not affected.

(*b*) The conversion of **A** is valid, and also that of
**I**. If *All S is P* and *Some S is P* imply directly
the existence of *S*, then they clearly imply indirectly the
existence of *P* ; and this is all that is required in order that
their conversion may be legitimate. The conversion of **E** is not
valid; for *No S is P* implies neither directly nor indirectly
the existence of *P*, whilst its converse does imply this.

(*c*) The contraposition of **E** is valid, and also that
of **O**. *No S is P* and *Some S is not P* both imply on
our present supposition the existence of *S*, and since by the
law of excluded middle every *S* is either *P* or
*not-P*, it follows that they imply indirectly the existence of
*not-P*. The contraposition of **A** is not valid; for it
involves the conversion of **E**, which we have already seen not to
be valid.230

(*d*) The process of inversion is not valid; for it involves
in the case of both **A** and **E** the conversion of an
**E** proposition.231

If along with an **E** proposition we are specially given the
information that *P* exists, or if this is implied in some other
proposition given us at the same time, then the **E** proposition
may of course be converted. In corresponding circumstances the
contraposition and inversion of **A** and the inversion of **E**
may be valid.232 Or again, given simply *No S is P*, we
may infer *Either P is non-existent or no P is S* ; and similarly
in other cases.

^{230} Or we might argue directly
that the contraposition of **A** is not valid, since *All S is
P* does not imply the existence of *not-P*, whilst its
contrapositive does imply this.

^{231} Or again we might argue
directly from the fact that neither *All S is P* nor *No S is
P* implies the existence of *not-S*.

^{232} For example, given (*α*) *No S is P*, (*β*) *All R is P*, we may under our
present supposition convert (*α*), since (*β*) implies
indirectly the existence of *P* ; and we may contraposit (*β*), since (*α*) implies indirectly the existence of
*not-P*. It will also he found that, given these two propositions
together, they both admit of inversion.

(3) *Let no proposition he understood to imply the existence
either of its subject or of its predicate*.

225 Having now got
rid of the implication of the existence either of subject or predicate
in the case of all propositions, we might naturally suppose that in no
case in which we make an immediate inference need we trouble ourselves
with any question of existence at all. As already indicated, however,
this conclusion would be erroneous.

(*а*) The process of obversion is still valid. Take, for
example, the obversion of *No S is P*. The obverse *All S is
not-P* implies that if there is any *S* there is also some
*not-P*. But this is necessarily implied in the proposition *No
S is P* itself. If there is any *S* it is by the law of
excluded middle either *P* or *not-P*; therefore, given that
*No S is P*, it follows immediately that if there is any *S*
there is some *not-P*.

(*b*) The conversion of **E** is valid. Since *No S is
P* denies the existence of anything that is both *S* and
*P*, it implies that if there is any *S* there is
some *not-P* and that if there is any *P* there is some
*not-S* ; and these are the only implications with regard to
existence involved in its converse. The conversion of **A**,
however, is not valid; nor is that of **I**. For *Some P is S*
implies that if there is any *P* there is also some *S* ; but
this is not implied either in *All S is P* or in *Some S is
P*.

(*c*) That the contraposition of **A** is valid follows
from the fact that the obversion of **A** and the conversion of
**E** are both valid.233 That the
contraposition of **E** and that of **O** are invalid follows
from the fact that the conversion of **A** and that of **I** are
both invalid.

(*d*) That inversion is invalid follows similarly.

On our present supposition then the following are valid: the
obversion and contraposition of **A**, the obversion of **I**,
the obversion and conversion of **E**, the obversion of **O**;
the following are invalid: the conversion and inversion of **A**,
the conversion of **I**, the contraposition and inversion of
**E**, the contraposition of **O**.234

^{233} Or we might argue directly
as follows; since the proposition *All S is P* denies the
existence of anything that is both *S* and *not-P*, it
implies that if there is any *S* there is some *P* and that
if there is any *not-P* there is some *not-S* ; and these are the only implications with regard to existence involved in its
contrapositive.

^{234} Dr Wolf holds in opposition
to the view here expressed that on the supposition in question all the
ordinary immediate inferences remain valid. This conclusion is based
on the doctrine that *Some S is P* does not imply that if there
is any *S* there is also some *P*. “*All S is P*
and *Some S is P*, it is true, do not imply that ‘if there
is any *P* there is also some *S*.’ But then *Some P
is S* does not necessarily imply that either. There can, therefore,
be no objection, on that score, against inferring, by conversion,
*Some P is S* from *All S is P* or *Some S is P*. With
the vindication of conversion all the remaining supposed illegitimate
inferences connected with it are also vindicated. We may, therefore,
conclude that to let no propositional form as such necessarily imply
the existence of either its subject or its predicate in no way affects
the validity of any of the traditional inferences of logic”
(*Studies in Logic*, p. 147). I have dealt with Dr Wolf’s
position in the note on page 216; and it is unnecessary to repeat the argument here. If importance is attached to concrete examples, I may
suggest, as an example for conversion, *All blue roses are blue*
(a formal proposition which must be regarded as valid on the
existential supposition under discussion); and, as an example for
inversion, *All human actions are foreseen by the Deity*. There
are, moreover, certain difficulties connected with syllogistic and
more complex reasonings that need a brief separate discussion, even
when the case of conversion has been disposed of.

226 (4) *Let
particulars be understood to imply, while universals are not
understood to imply, the existence of their subjects*.

(*a*) The validity of obversion is again obviously
unaffected.235

(*b*) The conversion of **E** is valid, and also that of
**I**, but not that of **A**.236

(*c*) The contraposition of **A** is valid, and also that
of **O**, but not that of **E**.

(*d*) The process of inversion is not valid.

These results are obvious; and the final outcome is—as might
have been anticipated—that we may infer a universal from a
universal, or a particular from a particular, but not a particular
from a universal.237

227 An important
point to notice is that in the immediate inferences which remain valid
on this supposition (namely, obversion, simple conversion, and simple
contraposition) there is no loss of logical force; while at the best
the reverse would be the case in those that are no longer valid
(namely, conversion *per accidens*, contraposition *per
accidens*, and inversion).

^{235} Obversion thus remains
valid on all the suppositions which have been specially discussed
above. If, however, affirmatives are interpreted as implying the
existence of their subjects while negatives are not so interpreted,
then of course we cannot pass by obversion from **E** to **A**,
or from **O** to **I**.

^{236} But from the two
propositions, *All S is P*, *Some R is S*, we can infer
*Some P is S* ; and similarly in other cases.

^{237} On the assumption, however,
that the universe of discourse can never be entirely emptied of
content, *Something is P* may be inferred from *Everything is
P*, and *Something is not P* may be inferred from *Nothing
is P*. Again, as is shewn by Dr Venn (*Symbolic Logic*, pp.
142–9), the three universals *All S is P*, *No not-S is
P*, *All not-S is P*, together establish the particular
*Some S is P*. Any universe of discourse contains *à priori*
four classes—(1) *SP*, (2) *S not-P*, (3) *not-S
P*, (4) *not-S not-P*. *All S is P* negatives (2); *No
not-S is P* negatives (3); *All not-S is P* negatives (4).
Given these three propositions, therefore, we are able to infer that
there is some *SP*, for this is all that we have left in the
universe of discourse. As already pointed out, the assumption that the
universe of discourse can never be entirely emptied of content is a
necessary assumption, since it is an essential condition of a
significant judgment that it relate to reality. If the universe of
discourse is entirely emptied of content we must either fail to
satisfy this condition, or else unconsciously transcend the assumed
universe of discourse and refer to some other and wider one in which
the former is affirmed not to exist.

159. *The Doctrine of
Opposition and the Existential Import of Propositions*.—The
ordinary doctrine of opposition, in its application to the traditional
schedule of propositions, is as follows: (*a*) The truth of
*Some S is P* follows from that of *All S is P*, and the
truth of *Some S is not P* from that of *No S is P*
(doctrine of subalternation); (*b*) *All S is P* and *Some
S is not P* cannot both be true and they cannot both be false,
similarly for *Some S is P* and *No S is P* (doctrine of
contradiction); (*c*) *All S is P* and *No S is P* cannot both
be true but they may both be false (doctrine of contrariety);
(*d*) *Some S is P* and *Some S is not P* may both be
true but they cannot both be false (doctrine of sub-contrariety). We
will now examine how far these several doctrines hold good under
various suppositions respecting the existential import of
propositions.238

^{238} Of course the doctrine of
contradiction always holds good in the sense that a pair of real
contradictories cannot both be true or both false; and similarly with
the other doctrines. The doctrines that we have to consider are not
these, but whether *SaP* and *SoP* are really
contradictories irrespective of the existential interpretation of the
propositions, whether *SaP* and *SeP* are really contraries,
and so on.

It should be added that, throughout the discussion, the propositions are supposed to be interpreted assertorically, as has always been the custom with the traditional schedule. The necessity for this proviso will from time to time be pointed out.

(1) *Let every proposition be interpreted as implying the*
228 *existence both
of its subject and of its predicate and also of their
contradictories*.239

^{239}It would be quite a
different problem if we were to assume the existence of *S* and
*P* independently of the affirmation of the given proposition. A
failure to distinguish between these problems is probably responsible
for a good deal of the confusion and misunderstanding that has arisen
in connexion with the present discussion. But it is clearly one thing
to say (*a*) “All *S* is *P* and *S* is
assumed to exist,” and another to say (*b*) “all
*S* is *P*,” *meaning thereby* “*S*
exists and is always *P*.” In case (*a*) it is futile
to go on to make the supposition that *S* is non-existent; in
case (*b*), on the other hand, there is nothing to prevent our
making the supposition, and we find that, if it holds good, the given
proposition is false.

On this supposition, if either the subject or the predicate of a
proposition is the name of a class which is unrepresented in the
universe of discourse or which exhausts that universe, then that
proposition is false; for it implies what is inconsistent with fact.
It follows that a pair of contradictories as usually stated, and also
a pair of sub-contraries, may both be false. For example, *All S is
P* and *Some S is not P* both imply the existence of *S*
in the universe of discourse. In the case then in which *S* does
not exist in that universe, these propositions would both be false.

If a concrete illustration is desired, we may take the
propositions, *None of the answers to the question shewed
originality*, *Some of the answers to the question shewed
originality*, and assume that each of these propositions includes
as part of its implication the actual occurrence of its subject in the
universe of discourse. Then our position is that if there were no
answers to the question at all, the truth of both the propositions
must be denied. The fact of there having been no answers does not
render the propositions meaningless; but it renders them false, their
full import being assumed to be, respectively, *There were answers
to the question but none of them shewed originality*, *There were
answers to the question and some of them shewed originality*.

We must not of course say that under our present supposition true
contradictories cannot be found; for this is always possible. The true
contradictory of *All S is P* is *Either some S is not P, or
else either S or not-S or P or not-P is non-existent*. Similarly in
other cases. The ordinary doctrines of subalternation and contrariety
remain unaffected.

229 (2) *Let
every proposition be interpreted as implying the existence of its
subject*.

For reasons similar to those stated above, the ordinary doctrines
of contradiction and sub-contrariety again fail to hold good. The true
contradictory of *All S is P* now becomes *Either some S is not
P, or S is non-existent*. The ordinary doctrines of subalternation
and contrariety again remain unaffected.

(3) *Let no proposition be interpreted as implying the existence
either of its subject or of its predicate*.

(*a*) The ordinary doctrine of subalternation holds good.

(*b*) The ordinary doctrine of contradiction does not hold
good. *All S is P*, for example, merely denies the existence of
any *S*’s that are not *P*’s; *Some S is not
P* merely asserts that *if* there are any *S*’s
some of them are not *P*’s. In the case in which *S*
does not exist in the universe of discourse we cannot affirm the
falsity of either of these propositions.240

230 (*c*) The
ordinary doctrine of contrariety does not hold good. For if there is
no implication of the existence of the subject in universal
propositions we are not actually precluded from asserting together two
propositions that are ordinarily given as contraries. *All S is
P* merely denies that there are any *S not-P*’s, *No S
is P* that there are any *SP*’s. We may, therefore,
without inconsistency affirm both *All S is P* and *No S is
P* ; but this is virtually to deny the existence of *S*.241

(*d*) The ordinary doctrine of sub-contrariety remains
unaffected.

^{240} Dr Wolf (*Studies in
Logic*, p. 132) denies the validity of this reasoning. He admits
apparently that the existential propositions *SPʹ* = 0 and
*Either **S* = 0 *or **SPʹ* > 0 are not contradictories; but
he denies that on the supposition under discussion *SaP* and
*SPʹ* = 0 are equivalent. His main ground for taking this
view is that *SaP* carries with it the implication *If there
are any S’s they are all P’s*, while *SPʹ* = 0
does not carry with it any such implication. This position has been
already criticized in section 157. Dr Wolf relies partly upon concrete examples, but in so doing he complicates the discussion by introducing
modal forms of expression. Thus for the proposition “Some
successful candidates do not receive scholarships,” we find
substituted in the course of his argument “If there are any
successful candidates then some of them do not (or *need not*)
receive scholarships,” and the insertion of the words in
brackets yields a proposition which, although an inference from the
original proposition, is not really equivalent to it, unless the
original proposition is itself interpreted modally. Later on Dr Wolf
explicitly alters the whole problem by assuming that what is under
consideration is a modal schedule of propositions. Thus he goes on to
say, “What *SaP* and *SeP* really express severally is
the *necessity* and the *impossibility* of *S* being
*P*”; and for the purpose of contradicting *SaP* and
*SeP*, “*SiP* and *SoP* need mean no more than
*S may be P* and *S need not be P*.” The question how
far *SaP* and *SeP* should be interpreted modally is
discussed elsewhere. All I would point out here is that it is a
distinct question from that raised in the text, which is a question
relating to the traditional schedule of propositions interpreted
assertorically. The whole question of existential import is indeed one
that cannot be discussed to any purpose until the character of the
schedule of propositions under consideration has been defined. From
the mixing up of schedules and interpretations nothing but confusion
can result. In the following section the opposition of modals will be
briefly considered in connexion with their existential import.

^{241} Of course on the view under
consideration we ought not to continue to speak of these two
propositions as contraries.

(4) *Let particulars be interpreted as implying, while universals
are not interpreted as implying, the existence of their subjects*.

(*a*) The ordinary doctrine of subalternation does not hold
good. *Some S is P*, for example, implies the existence of
*S*, while this is not implied by *All S is P*.

(*b*) The ordinary doctrine of contradiction holds good.
*All S is P* denies that there is any *S* that is
*not-P*; *Some S is not P* affirms that there is some
*S* that is *not-P*. It is clear that these propositions
cannot both be true; it is also clear that they cannot both be false.
Similarly for *No S is P* and *Some S is P*.

(*c*) The ordinary doctrine of contrariety does not hold good.
*All S is P* and *No S is P* are not inconsistent with one
another, but the force of asserting both of them is to deny that there
are any *S*’s.242 This follows just
as in the case of our third supposition.243

231 (*d*) The
ordinary doctrine of sub-contrariety does not hold good.244
*Some S is P* and *Some S is not P* are both false in the
case in which *S* does not exist in the universe of discourse.

^{242} If, however, we are given
*No S is P* and also *Some S is P*, then we are able to
infer that *All S is P* is false. The second of these
propositions affirms the existence of *S*, and therefore destroys
the hypothesis on which alone the first and third can be treated as
compatible.

^{243} The above doctrine has been
criticized on the ground that it practically amounts to saying that
neither of the given propositions has any meaning whatever, but that
each is a mere sham and pretence of predication; and a request is made
for concrete examples. The following example may perhaps suffice to
illustrate the particular point now at issue: “An honest miller
has a golden thumb”; “Well, I am sure that no miller,
honest or otherwise, has a golden thumb.” These two propositions
are in the form of what would ordinarily be called contraries; but
taken together they may quite naturally be interpreted as meaning that
no such person can be found as an honest miller. The former
proposition would indeed probably be intended to be supplemented by
the latter or by some proposition involving the latter, and so to
carry inferentially the denial of the existence of its subject.

Another example is contained in the following quotation from Mrs
Ladd Franklin: “*All x is y*, *No x is y*, assert
together that *x* is neither *y* nor *not-y*, and hence
that there is no *x*. It is common among logicians to say that
two such propositions are incompatible; but that is not true, they are
simply together incompatible with the existence of *x*. When the
schoolboy has proved that the meeting point of two lines is not on the
right of a certain transversal and that it is not on the left of it,
we do not tell him that his propositions are incompatible and that one
or other of them must be false, but we allow him to draw the natural
conclusion that there is no meeting point, or that the lines are
parallel” (*Mind*, 1890, p. 77 *n.*).

Dr Wolf (*Studies in Logic*, p. 140), criticizing Mrs Ladd
Franklin’s concrete example, maintains that the two propositions
given by her are sub-contraries (**I** and **O**), not
contraries (**A** and **E**). A moment’s consideration
will, however, shew that this is not the case since neither of the
propositions is particular. At the same time it is true that a little
manipulation is required to bring them to the forms **A** and
**E**. There is also the assumption that “on the right”
and “on the left” exhaust the possibilities and are
therefore contradictory terms. Granting this assumption, the two
propositions may be expressed symbolically in the forms *No S is
P*, *No S is not P*, and it then needs only the obversion of
one of them to bring them to the forms **A** and **E**.

^{244} It may be worth observing
that, given (*b*), (*d*) might be deduced from (*c*) or
*vice versâ*.

The relation between contradictories is by far the most important relation with which we are concerned in dealing with the opposition of propositions, and it will be observed that the last of the above suppositions is the only one under which the ordinary doctrine of contradiction holds good.

160. *The Opposition of
Modal Propositions considered in connexion with their Existential
Import*.—The propositions discussed in the preceding sections
have been the propositions belonging to the traditional schedule
interpreted assertorically. Turning now to the corresponding modal
schedule, we may briefly consider how the doctrine of opposition is
affected, if at all, on the supposition that the propositions included
in the schedule are not interpreted as implying the existence of their
232 subjects. We find
that on this supposition *S as such is P* and *S need not be
P* are true contradictories.

*S as such is P* (interpreted as not necessarily implying the
existence of *S*) does more than deny the actual occurrence of
the conjunction *S not-P*, it denies the possibility of such a
conjunction; and all that is necessary in order to contradict this is
to affirm the possibility of the conjunction. This is done by the
proposition *S need not be P* (also interpreted as not
necessarily implying the existence of *S*). On the same
supposition, *S as such is P*, *S as such is other than P*,
are true contraries.

Here, however, another problem suggests itself. Leaving on one side
the question as to any implication of *actuality*, are modal
propositions to be interpreted as containing any implication in regard
to the *possibility* of their antecedents? And, further, how does
our answer to this question affect the opposition of modals? The
consideration of this problem may be deferred until we come to deal
with the opposition of conditional propositions (see section 176).

161. *Jevons’s
Criterion of Consistency*.—In passing to the explicit
discussion of the existential import of categorical propositions, we
may consider first the Criterion of Consistency, which is laid down by
Jevons (following De Morgan):—Any two or more propositions are
contradictory when, and only when, after all possible substitutions
are made, they occasion the total disappearance of any term, positive
or negative, from the Logical Alphabet. The criterion amounts to this,
that every proposition must be understood to imply the existence of
things denoted by every simple term contained in it, and also of
things denoted by the contradictories of such terms. If, for example,
we have the proposition *All S is P*, this implies that among the
members of the universe of discourse are to be found *S*’s
and *P*’s, *not-S*’s, and *not-P*’s.
In defence of this doctrine Jevons appears to rely mainly upon the
psychological law of relativity, namely, that we cannot think at all
without separating what we think about from other things. Hence if
either a term or its contradictory represents nonentity, that term
cannot be either subject or predicate in a significant 233 proposition.245
It is clear, however, that this psychological argument falls away as
soon as it is allowed that we may be confining ourselves to a limited
universe of discourse, or indeed if we confine ourselves to any
universe less extensive than that which covers the whole realm of the
conceivable. Of course the more limited the universe to which our
proposition is supposed to relate the more easily may *S* or
*P* either exhaust it or be absent from it; but with very complex
subjects and predicates the contradictory of one or both of our terms
may easily exhaust even an extended universe. Take, for example, the
proposition, *No satisfactory solution of the problem of squaring
the circle has ever been published by Mr A.* Here the subject is
non-existent; and it may happen also that Mr A. has never published
anything at all.246 Further, if I am not allowed to negative
*X*, why should I be allowed to negative *AB*? There is
nothing to prevent *X* from representing a class formed by taking
the part common to two other classes. In certain combinations indeed
it may be convenient to substitute *X* for *AB*, or *vice
versâ*. It would appear then that what is contradictory when we use
a certain set of symbols may not be contradictory when we use another
set of symbols. This argument has a special bearing on the complex
propositions which are usually relegated to symbolic logic, but to
which Jevons’s criterion is intended particularly to apply.

^{245} This point is put somewhat
tentatively in a passage in Jevons’s *Principles of
Science* (chapter 6, § 5) where he remarks: “If *A* were
identical with ‘*B* or not-*B*,’ its negative
not-*A* would be non-existent. This result would generally be an
absurd one, and I see much reason to think that in a strictly logical
point of view it would always be absurd. In all probability we ought
to assume as a fundamental logical axiom that every term has its
negative in thought. We cannot think at all without separating what we
think about from other things, and these things necessarily form the
negative notion. If so, it follows that any term of the form
‘*B* or not-*B*’ is just as self-contradictory
as one of the form ‘*B* and not-*B*’.”

^{246} Other examples will be
given in the following section.

No doubt Jevons’s criterion is sometimes a convenient assumption to make; provisionally, for example, in working out the doctrine of immediate inferences on the traditional lines. But it is an assumption that should always be explicitly referred to when made; and it ought not to be regarded as having an 234 axiomatic and binding force, so as to make it necessary to base the whole of logic upon it.

162. *The Existential
Import of the Propositions included in the Traditional
Schedule*.—We may now turn to the consideration of the
question whether the propositions *SaP*, *SeP*, *SiP*,
*SoP* should or should not be interpreted as implying the
existence of their subjects in the universe of discourse to which
reference is made. In this section it will be assumed that the import
of all the propositions under discussion is assertoric, not modal.

A brief reference may be made to two sources of misunderstanding to
which attention has already been called.

(*а*) All propositions contain affirmations relating to some
system of reality; and by analysis every proposition may be made to
yield an “ultimate subject” which is real, namely, the
system of reality to which the proposition relates. This system of
reality is what we mean by the universe of discourse; and, as we have
seen, the universe of discourse can never be entirely emptied of
content. It must then be understood that if we decide that certain
propositional forms are not to be interpreted as containing as part of
their import the affirmation of the existence of their subjects, it is
far from being thereby intended that propositions falling into these
forms contain no affirmation relating to reality.247

(*b*) We must put on one side a very summary solution of our
problem, which, if it were correct, would render any further
discussion needless. How, it may be asked, can we possibly speak about
anything and at the same time exclude it from the universe of
discourse? This question suggests a certain ambiguity which may attach
to the phrase *universe of discourse*, but which can hardly
remain an ambiguity after the explanations already given. The answer
is that we can certainly think and speak about a thing *with
reference to* a given universe of discourse without implying, or
even believing in, its existence in that universe. Suppose, for
example, that I say there are no such things as unicorns. If this
statement is to be accepted, it must be interpreted literally (not
elliptically); and it is clear that the universe of discourse referred
to is the material 235 universe.248 I speak then of
unicorns *with reference to* the material universe, but deny that
such creatures are to be found (or exist) in it.

^{247} Compare Sigwart, *Logic*, i. p. 97 *n*.

^{248} It is hardly necessary to
point out that ideas of unicorns exist in imagination, and that
statements about unicorns are to be met with in fairy tales.

The question we have to discuss is one of the *interpretation of
propositional forms*,249 and the solution
will therefore be to some extent a matter of convention. We shall be
guided in our solution partly by the ordinary usage of language, and
partly by considerations of logical convenience and suitability.

As regards the ordinary usage of language there can be no doubt
that we seldom do as a matter of fact make predications about
non-existent subjects. For such predications would in general have
little utility or interest for us. “The practical exigencies of
life,” as Dr Venn remarks, “confine most of our
discussions to what does exist, rather than to what might exist” (*Symbolic Logic*, p. 131). We must, however, consider
whether there are not exceptional cases; and if we can find any in
which it is clear that the speaker would not necessarily intend to
imply the existence of the subject, we may draw the conclusion that
the propositional form of which he makes use is not in popular usage
uniformly intended to convey such an implication.

*Universal Affirmatives*. If a universal affirmative
proposition is obtained by a process of exhaustive enumeration
(*e.g.*, *All the Apostles were Jews*, *All the books on
that shelf are bound in morocco*), or if it is obtained by
empirical generalisation based on the examination of individual
instances (*e.g.*, *All ruminant animals are
cloven-hoofed*), then it is clear that the existence of the subject
is a presupposition of the affirmation. We may, however, note certain
other classes of cases in which such a presupposition is not
necessary.

(*a*) We may affirm an abstract connexion of attributes, based
on considerations of a deductive character or at any rate not obtained
by direct generalisation from observed instances of the subject, and
the existence of the subject is then not essential. For example,
*The impact of two perfectly elastic* 236 *bodies leads to no diminution of kinetic
energy* ; *Every body, not compelled by impressed forces to change
its state, continues in a state of rest or of uniform motion in a
straight line*.

It may perhaps be said that all propositions falling within this
category will be really apodeictic, and that our present discussion
has been limited to assertoric propositions. There is some force in
this criticism. It is, however, to be remembered that the assertoric
*SaP* can be inferred from the apodeictic *SaP*, so that if
we can have the latter without any implication as to the existence of
*S* we may have the former also, unless indeed we decide to
differentiate between them in regard to their existential implication.
The examples that we have given are moreover expressed in ordinary
assertoric form, and not in any distinctive apodeictic form, such as
*S as such is P*, *It is inherent in the nature of S to be
P*.

(*b*) The proposition *SaP* may express a rule laid down,
and remaining in force, without any actual instance of its application
having arisen. For example, *All candidates arriving five minutes
late are fined one shilling*, *All candidates who stammer are
excused reading aloud*, *All trespassers are prosecuted*.

If it is argued that, in such cases as these,250 the propositions
ought properly to be written in the conditional and not in the
categorical form (*e.g.*, *If any candidate arrives five
minutes late, that candidate is fined one shilling*), the reply is
that this is to misunderstand the point just now at issue, which is
whether we meet with propositions in ordinary discourse which are
categorical in form and yet are hypothetical so far as the existence
of their subjects is concerned. It is of course open to us to decide
that for logical purposes we will so interpret categorical
propositions that in such cases as the above the categorical form can
no longer be used. But for the present we are merely discussing
popular usage.

^{250} This argument might be used
with, reference to cases coming under (*a*) or (*c*) as well
as with reference to those coming under (*b*).

(*c*) Assertions in regard to possible future events are
sometimes thrown into the form *SaP*. For example, *Who steals
my purse steals trash*, *Those who pass this examination an*
237 *lucky men*.
The first of these propositions would not be invalidated supposing my
purse never to be stolen, and the latter, as Dr Venn remarks,251
would be tacitly supplemented by the clause “if any such there
be.”

^{251} *Symbolic Logic*, p. 132.

(*d*) There are cases in which the intended implication of a
proposition of the form *All S is P* is to deny that there are
any *S*’s; for example, *An honest miller has a golden
thumb*, *All the carts that come to Crowland are shod with
silver*.252

^{252} Both these propositions are
naturally to be interpreted as containing an indirect denial of the
existence of their subjects. “Crowland is situated in such
moorish rotten ground in the Fens, that scarce a horse, much less a
cart, can come to it” (Bohn’s *Handbook of Proverbs*,
p. 211). It would appear, however, that this proverb has now lost its
force, inasmuch as “since the draining, in summer time, carts
may go thither.”

*Universal Negatives*. It is still easier to find instances
from common speech in which universal negative propositions, that is,
propositions of the form *No S is P*, are not to be regarded as
necessarily implying the existence of their subjects.

(*a*) There are again cases in which the proposition is
reached by a process of abstract reasoning about a subject the actual
existence or occurrence of which is not presupposed; for example, *A
planet moving in a hyperbolic orbit can never return to any position
it once occupied*.253

^{253} This example is taken from
Dixon, *Essay on Reasoning*, p. 62.

(*b*) The import of the proposition may be distinctly to
imply, if not definitely to affirm, the non-existence of the subject;
for example, *No ghosts have troubled me*, *No unicorns have
ever been seen*.254

^{254} The universe of discourse
must here be taken to be the material universe. With reference to this
example, however, a critic writes, “But surely the universe of
imagination is the only one applicable; for unicorns have long been
known not to belong to the actual material universe.” The
universe of imagination may be required in order to sustain the
position that the subject of the proposition exists in the universe of
discourse; but any person making the statement would certainly not be
referring to the world of imagination or the universe of heraldry, for
the simple reason that in either of these cases the proposition (which
must then be interpreted elliptically) would obviously not be true. On
the other hand, we can quite well suppose the statement made with
reference to the material universe: “Whether unicorns exist or
not, at any rate they have never been seen.” Again, to take
another example of a similar kind where the reference is also to the
phenomenal universe, we can quite well suppose the statement made:
“Whether there are ghosts or not, at any rate none have ever
troubled me.” In order to avoid misapprehension, it is important
to distinguish the above examples from such (elliptical) propositions
as the following: “The wrath of the Homeric gods is very
terrible,” “Fairies are able to assume different
forms.” In each of these cases, the subject of the proposition
(properly interpreted) exists in the particular universe to which
reference is made. See notes 2 and 3 on page 213.

238 (*c*) A denial of the conjunction ABC may be expressed in the
form *No AB is C* without any intention of thereby affirming the
conjunction *AB* ; for example, *No satisfactory solution of the
problem of squaring the circle has been published*, *No woman
candidate for the Theological Tripos has been educated at Newnham
College*, *No Advanced Student in Law is on the boards of Trinity
College*.255

^{255} “As an instance of a
possibly non-existent subject of a negative proposition, take the
following: ‘No person condemned for witchcraft in the reign of
Queen Anne was executed.’” (Venn, *Symbolic Logic*,
p. 132.)

*Particulars*. In the case of particular propositions, it is
far less easy to give examples, such as might be met with in ordinary
discourse, in which there is no implication of the existence of the
subjects of the propositions. There may be exceptions, but at any rate
the cases are exceedingly rare in which in ordinary speech we
predicate anything of a non-existent subject without doing so
universally. The main reason for this is, as Dr Venn points out, that
“an assertion confined to ‘some’ of a class
generally rests upon observation or testimony rather than on reasoning
or imagination, and therefore almost necessarily postulates existent
data, though the nature of this observation and consequent existence
is, as already remarked, a perfectly open question. ‘Some
twining plants turn from left to right,’ ‘Some griffins
have long claws,’ both imply that we have looked in the right
quarters to assure ourselves of the fact. In one case I may have
observed in my own garden, and in the other on crests or in the works
of the poets, but according to the appropriate tests of verification,
we are in each case talking of what *is*.”256 If we look at the
question 239 from the
other side, we find that when our primary object is to affirm the
existence of a class of objects, our assertion very naturally takes
the form of a particular proposition. If, for example, we desire to
affirm the existence of black swans, we say *Some swans are
black*. The existential implication of a proposition of this kind
in ordinary discourse is one of its most fundamental characteristics.

^{256} *Symbolic Logic*, p.
131. Again, in such a proposition as “Some sea-serpents are not
half a mile long” (meaning *your so-called* sea-serpents),
the subject of the proposition exists in the universe to which
reference is made, namely, the universe which may be described as the
universe of travellers’ tales. We are here regarding the
proposition as elliptical in a sense that has been already
explained.

On the whole it cannot be said that the usages of ordinary speech
afford a decisive solution of the problem under discussion. It has,
however, been shewn (1) that we seldom or never make statements about
non-existent subjects in the form *Some S is P* or the form
*Some S is not P* ; (2) that, although it is also true that we do
not as a rule do so in the form *All S is P* or the form *No S
is P*, still there are several classes of cases in which the use of
these latter forms is not to be understood as necessarily carrying
with it the implication that *S* is existent. Hence we should be
departing very little from ordinary usage if we were to decide to
interpret particulars as implying the existence of their subjects, but
universals as not doing so (that is, as not doing so by their bare form).

I do not, however, regard this solution as necessitated by popular
usage. It is, for instance, still open to anyone to adopt the
convention that, for logical purposes, the categorical form shall only
be used when the implication of the existence of the subject is
intended. On this interpretation, the conditional or hypothetical form
must be adopted whenever the existence of the subject is left an open
question. Thus, if we are doubtful about the existence of *S*
(or, at any rate, do not wish to affirm its existence), we must be
careful to say, *If there is any S, then all S is P*, instead of
simply *All S is P* ; in other words, the hypothetical character
of the proposition so far as the existence of its subject is concerned
must be made explicit.

The problem then not being decided by considerations of popular usage alone, we must go on to enquire how the question is affected by considerations of logical convenience and suitability. Here again there is no one solution that is inevitable. Reasons can, however, be urged in favour of interpreting particulars as implying, but universals as not implying, the existence of their 240 subjects;257 and this, as we have seen, is a solution that derives some sanction from popular usage.

^{257} On this view whenever it is
desired specially to affirm the existence in the universe of discourse
of the subject of a universal proposition, a separate statement to
this effect must be made. For example, *There are S’s, and
all of them are P’s*. If, on the other hand, it is ever
desired to affirm a particular proposition without implying the
existence of the subject, then recourse must be had to the
hypothetical or conditional form of statement. Thus, if we do not
intend to imply the existence of *S*, instead of writing *Some
S’s are P’s*, we must write, *If there are any
S’s, then in some such cases they are also
P’s*.

(1) A consideration of the manner in which the validity of
immediate inferences is affected by the existential import of
propositions affords reasons for the adoption of this
interpretation.258 The most important immediate inferences are
simple conversion (*i.e.*, the conversion of **E** and of
**I**) and simple contraposition (*i.e.*, the contraposition
of **A** and of **O**). If, however, universals are regarded as
implying the existence of their subjects, then, as shewn in section
158, neither the conversion of **E** nor the contraposition of
**A** is valid, irrespective of some farther assumption; whereas,
if universals are not regarded as implying the existence of their
subjects, then both these operations are legitimate without
qualification. On the other hand, the conversion of **I** and the
contraposition of **O** are valid only if particulars *do* imply the
existence of their subjects.259

^{258} It has been objected that
to base our view of the existential import of propositions upon the
validity or invalidity of immediate inferences is to argue in a
circle. “Whether,” it is said, “the immediate
inferences are valid or not must be a consequence of the view taken of
the existential import of the proposition and should not, therefore,
be made a portion of the ground on which that view is based.”
This objection involves a confusion between different points of view
from which the problem of the relation between the existential import
of propositions and the validity of logical operations may be
regarded. In section 158 the logical consequences of various
assumptions were worked out without any attempt being made to decide
between these assumptions. Our point of view is now different; we are
investigating the grounds on which one of the assumptions may be
preferred to the others, and there is no reason why the consequences
previously deduced should not form part of our data for deciding this
question. The argument contains nothing that is of the nature of a
*circulus in probando*.

^{259} Thus, the table of
equivalences given in section 106 is valid on the interpretation with which we are now dealing. The dependence of the table given in section
108 upon the same supposition is still more obvious. It has been
already pointed out that the remaining immediate inferences based on
conversion and obversion are of much less importance; see page
227.

241 Turning to
immediate inferences of another kind, it is clear that if universal
propositions formally imply the existence of their subjects, we cannot
legitimately pass from *All X is Y* to *All AX is Y*.260
For it is possible that there may be *X*’s and yet no
*AX*’s, and in this case the former proposition may be
true, while the latter will certainly be false. Again, given that *A
is X*, *B is Y*, *C is Z*, we cannot infer that
*ABC* is *XYZ*. Such restrictions as these would constitute
an almost insurmountable bar to progress in inference as soon as we
have to do with complex propositions.261

^{260} It will be observed further
that upon the same assumption we cannot even affirm the formal
validity of the proposition *All X is X*. For *X* might be
non-existent, and the proposition would then be false.

^{261} Hence Mrs Ladd Franklin is
led to the conclusion that “no consistent logic of universal
propositions is possible except with the convention that they do not
imply the existence of their terms” (*Mind*, 1890, p. 88).

(2) We may next consider the existential import of propositions
with reference to the doctrine of opposition. It has been shewn in
section 159 that if particulars are interpreted as implying the
existence of their subjects, while universals are not so interpreted,
then **A** and **O**, **E** and **I**, are true
contradictories; but that this is not the case under any of the other
suppositions discussed in the same section.262 There can,
however, be no doubt that one of the most important functions of
particular propositions is to contradict the universal propositions of
opposite quality; and hence we have a strong argument in favour of a
view of the existential import of propositions which will leave the
ordinary doctrine of contradiction unaffected.

^{262} **A** and **O**,
**E** and **I**, will also be true contradictories if universals
are interpreted as implying the existence of their subjects, while
particulars are not so interpreted. It would be interesting, if space
permitted, to work out the results of this supposition in detail. If
the student does this for himself, he will find that this is the *only*
supposition, under which the ordinary doctrine of opposition holds
good throughout. All other considerations, however, are opposed to its
adoption. It altogether conflicts with popular usage; it renders the
processes of simple conversion and simple contraposition illegitimate;
and whilst making universals double judgments, it destroys the
categorical character of particulars altogether. In regard to this
last point see page 220.

As regards the doctrines of subalternation, contrariety, and
subcontrariety, our results (namely, that **I** does not follow
from **A**, or **O** from **E**, that **A** and **E**
may both be true, and that **I** 242 and **O** may both be false) are no
doubt paradoxical. But this objection is far more than counterbalanced
by the fact that the doctrine of contradiction is saved. For as
compared with the relation between contradictories, these other
relations are of little importance. We may specially consider the
relation between **A** and **I**. *Some S is P* cannot now
without qualification be inferred from *All S is P*, since the
former of these propositions implies the existence of *S*, while
the latter does not. But as a matter of fact this is an inference
which we never have occasion to make. If their existential import is
the same why should we ever lay down a particular proposition when the
corresponding universal is at our service? On the other hand, the view
that we are advocating gives *Some S is P* a status relatively to
*All S is P* as well as relatively to *No S is P* which it
could not otherwise possess; and similarly for *Some S is not P*.
Our result as regards the relation between *SaP* and *SiP*
has been described as equivalent to saying “that a statement of
partial knowledge carries more real information than a statement of
full knowledge; since if we only possess limited information, and so
can only assert *SiP*, we thereby affirm the existence of
*S* ; but if we have sufficient knowledge to speak of *all S*
(*S* remaining the same) the statement of that full knowledge
immediately casts a doubt upon that existence.” This way of
putting it is, however, misleading if not positively erroneous. On the
view in question it is incorrect to say simply that *SiP* and
*SaP* give “partial” and “full” knowledge
respectively, for *SiP* while giving less knowledge than
*SaP* in one direction gives more in another. In other words, the
knowledge which is “full” relatively to *SiP* is not
expressed by *SaP* by itself, but by *SaP* together with the
statement that there are such things as *S*.263

^{263} The position taken above in
regard to subalternation is very well expressed by Mrs Ladd Franklin.
“Nothing of course is now illogical that was ever logical
before. It is merely a question of what convention in regard to the
existence of terms we adopt before we admit the warm-blooded sentences
of real life into the iron moulds of logical manipulation. With the
old convention (which was never explicitly stated) subalternation ran
thus: *No x’s are y’s* (and we hereby mean to imply
that there are *x’s*, whatever *x* may be),
therefore, *Some x’s are non-y’s*. With the new
convention the requirement is simply that if it is known that there
are *x’s* (as it is known, of course, in by far the greater
number of sentences that it interests us to form) that fact must be
expressly stated. The argument then is: *No x’s are
y’s*, *There are x’s*, therefore, *There are
x’s which are non-y’s*.”

243 (3) There is
one further point of importance to be noted, and that is, that the
interpretation of **A**, **E**, **I**, **O** propositions
under consideration is the only interpretation according to which each
one of these propositions is resolved into a *single categorical
statement*. For if **A** and **E** imply the existence of
their subjects they express *double*, not single, judgments,
being equivalent respectively to the statements: *There are
S’s, but there are no SPʹ’s* ; *There are
S’s, but there are no SP’s* ; whereas on the
interpretation here proposed they simply express respectively the
single judgments: *There are no SPʹ’s* ; *There are no
SP’s*. On the other hand, if **I** and **O** do not
imply the existence of their subjects, instead of expressing
categorical judgments, they express somewhat complex hypothetical
ones, being equivalent respectively to the statement: *If there are
any S’s then there are some SP’s* ; *If there are any
S’s then there are some SPʹ’s* ; whereas on our
interpretation they express respectively the categorical judgments:
*There are SP’s* ; *There are SPʹ’s*.264

On the whole, there is a strong cumulative argument in favour of interpreting particulars, but not universals, as implying formally the existence of their subjects.265 This solution 244 is to be regarded as partly of the nature of a convention. We arrive, however, at the conclusion that no other solution can equally well suffice as the basis of a scientific treatment of the traditional schedule of propositions, so long, at any rate, as the propositions included in the schedule are regarded as assertoric and not modal.

^{265} We may briefly discuss in a
note one or two objections to this view which have not yet been
explicitly considered.

(*а*) Mill argues that a synthetical proposition necessarily
implies “the real existence of the subject, because in the case
of a non-existent subject there is nothing for the proposition to
assert” (*Logic*, i. 6, § 2). In answer to this it is
sufficient to point out that a non-existent thing will be described as
possessing attributes which are separately attributes of existing
things, although that particular combination of them may not anywhere
be found, and if we know (as we may do) that certain of these
attributes are always accompanied by other attributes we may predicate
the latter of the non-existent thing, thereby obtaining a real
proposition which does not involve the actual existence of its
subject. As an argument *ad hominem* it may further be pointed
out that Mill inclines to deny the existence of perfect straight lines
or perfect circles. Would he therefore affirm that we can make no real
assertions about such things?

(*b*) Mr Welton repeats several times that a proposition which
relates to a non-existent subject must be a mere jumble of words, a
predication in appearance only. “That the meaning of a universal
proposition can be expressed as a denial is true, but this is not its
primary import. And this denial itself must rest upon what the
proposition affirms. Unless *SaP* implies the existence of
*S*, and asserts that it possesses *P*, we have no data for
denying the existence of *SPʹ*. For if *S* is
non-existent the denial that it is non-*P* can have no
intelligible meaning” (*Logic*, p. 241). The examples which
we have already given are sufficient to dispose of this objection; but
it may be worth while to add a further argument. According to Mr
Welton, an **E** proposition implies the existence of its subject
but not of its predicate. We cannot then infer *PeS* from
*SeP* because we have no assurance of the existence of *P*.
But in accordance with the position taken by Mr Welton, we ought to go
further and say that *PeS* must be a mere jumble of words unless
we are assured of the existence of *P*. It is impossible,
however, to regard *PeS* as a mere unmeaning jumble of words, a
predication in appearance only, when *SeP* is a significant and
true proposition. *PeS* may be false, or it may be an unnatural
form of statement, but it cannot be meaningless if *SeP* has a
meaning. Take, for example, the propositions—*No woman is now
hanged for theft in England*, *No person now hanged for theft in
England is a woman*. The second of these propositions is false if
it is taken to imply that there are at the present time persons who
are hanged for theft in England, but how it can possibly be regarded
as meaningless I cannot understand.

(*c*) Miss Jones argues that if *some* carries with it an
implication of existence, when used with a subject-term, it must do so
equally when used with a predicate-term; but the predicate of an
**A** proposition being undistributed is practically qualified by
*some* ; hence, if *Some S is P* implies the existence of
*S* and therefore of *P*, *All S is P* must imply the
existence of *P* and therefore of *S*. In reply to this
argument it may be pointed out, first, that a distinction may fairly
be drawn without any risk of confusion between a term explicitly
quantified by the word *some* and a term which we can shew to be
undistributed but which is not explicitly quantified at all; and,
secondly, that the position which we have taken is based upon a
consideration of the import of propositions as a whole, not upon the
force of signs of quantity considered in the abstract. The irrelevancy
of the argument will be apparent if it is taken in connexion with the
reasons which we have urged for holding that particulars should be
interpreted as implying the existence of their subjects.

163. *The Existential
Import of Modal Propositions*.—Of apodeictic propositions it
may be said still more emphatically than of assertoric universals that
they do not necessarily imply the existence of their subjects. For
they assert a necessary relation between attributes, the ground of
which is frequently 245 to be sought in abstract reasoning rather
than in concrete experiences. And the same is true of the denial of
apodeictic propositions. We may on abstract grounds assert the
possibility of a certain concomitance (or non-concomitance) of
attributes without having had actual experience of that concomitance
(or non-concomitance), and without intending to imply its actuality.
Hence we should not interpret the proposition *S may be P*, any
more than the proposition *S must be P*, as by its bare form
affirming the existence of *S*.

It has been shewn that in order that the propositions *All S is
P* and *Some S is not P* may be true contradictories, one or
other of them must be interpreted as implying the existence of
*S*. It follows, however, from what has been said above that the
same condition need not be fulfilled in order that *S must be P*
and *S need not be P* may be true contradictories.266

^{266} It is because Dr Wolf
identifies the ordinary particular proposition with the problematic
proposition that he is led to the conclusion that *SaP* and
*SoP* are true contradictories although neither of them is
interpreted as implying the existence of *S*.

But to this it has to be added that, in order that these two
propositions may be true contradictories, one or other of them must be
interpreted as implying the *possible existence* of *S*.
This line of thought has been suggested in section 160, and it will be pursued farther in sections 176 and 179.

164. The *particular*
judgment has, from different stand-points, been identified (*a*)
with the *existential* judgment, (*b*) with the *problematic*
judgment, (*c*) with the *narrative* judgment. Comment on each of
these views. [C.]

The student may find that to write a detailed answer to this
question will help to clear up his views respecting the particular
proposition. No detailed answer will here be given; but attention may
be called to one or two points.

(*a*) Two kinds of existential judgments may be distinguished.

(i) Those which affirm existence indefinitely, that is, 246 somewhere in the
universe of discourse; for example, *There are white hares*,
*There is a devil*.

(ii) Those which affirm existence with reference to some definite
time and place; for example, *It rains*, *I am hungry*.

The particular may perhaps be identified with (i), hardly with (ii).

(*b*) We may be justified in affirming the problematical *S
may be P*, when we cannot affirm the particular *Some S is P*.
There are reasons for interpreting the latter judgment existentially
as regards its subject, which do not apply to the former judgment.

(*c*) The narrative judgment need not have the indefinite
character of the particular. We may, however, hold that the two kinds
of judgment have this in common that there are grounds for
interpreting both existentially as regards their subjects.

165. Discuss the relation
between the propositions *All S is P* and *All not-S is P*.

This is an interesting case to notice in connexion with the discussion raised in sections 158 and 159.

We have

*SaP* = *SePʹ* = *PʹeS* ;*SʹaP* =
*SʹePʹ* = *PʹeSʹ* = *PʹaS*.

The given propositions come out, therefore, as contraries.

On the view that we ought not to enter into any discussion
concerning existence in connexion with immediate inference, we must, I
suppose, rest content with this statement of the case. It seems,
however, sufficiently curious to demand further investigation and
explanation. We may as before take different suppositions with regard
to the existential import of propositions.

(1) If every proposition implies the existence of both subject and
predicate and their contradictories, then it is at once clear that the
two propositions cannot both be true together; for between them they
deny the existence of *not-P*.

(2) On the view that propositions imply simply the existence of
their subjects, it has been shewn in section 158 that we are not
justified in passing from *All not-S is P* to *All not-P is
S* unless we are given independently the existence of *not-P*.
But it will be observed that in the case before us the given
propositions make this impossible. Since *all S is P* and *all
not-S is P*, and everything is either *S* or *not-S* by
the law of excluded middle, it follows that 247 nothing is *not-P*. In order,
therefore, to reduce the given propositions to such a form that they
appear as contraries (and consequently267 as inconsistent
with each other) we have to assume the very thing that taken together
they really deny.

(3) and (4). On the view that at any rate universal propositions do
not imply the existence of their subjects, we have found in section
159 that the propositions *No not-P is S*, *All not-P is S*,
are not necessarily inconsistent, for they may express the fact that
*P* constitutes the entire universe of discourse. But this fact
is just what is given us by the propositions in their original form.

Under each hypothesis, then, the result obtained is satisfactorily
accounted for and explained.

^{267} It will be remembered that
under suppositions (1) and (2) the ordinary doctrine of contrariety
holds good.

166. “The boy is in
the garden.”

“The centaur is a creation of the
poets.”

“A square circle is a contradiction.”

Discuss the above propositions as illustrating different functions
of the verb “to be”; or as bearing upon the logical
conception of different universes of discourse or of different kinds
of existence. [C.]

167. Discuss the existential
import of singular propositions.

“The King of Utopia did not die on Tuesday last.”
Examine carefully the meaning to be attached to the denial of this
proposition. [K.]

168. Some logicians hold
that from *All S is P* we may infer *Some not-S is not-P*.
Take as an illustration, *All human actions are foreseen by the
Deity*. [C.]

169. Discuss the validity of the following inference:—All trespassers will be prosecuted, No trespassers have been prosecuted, therefore, There have been no trespassers. [C.]

170. On the assumption that
particulars are interpreted as implying while universals are not
interpreted as implying the existence of their subjects in the
universe of discourse, examine (stating your reasons) the validity of
the following inferences; *All S is P* and *Some R is not S*
therefore, *Some not-S is not P* ; *All S is P* and *Some R
is not P*, therefore, *Some not-S is* 248 *not P* ; *All S is P* and
*Some R is S*, it is, therefore, false that *No P is S* ;
*All S is P* and *Some R is P*, it is, therefore, false that
*No P is S*. [K.]

171. Discuss the formal
validity of the following arguments, (i) on the supposition that all
categorical propositions are to be interpreted as implying the
existence of their subjects in the universe of discourse, (ii) on the
supposition that no categorical propositions are to be so interpreted:

(*a*) All *P* is *Q*, therefore, All *AP* is
*AQ* ;

(*b*) All *AP* is *AQ*, therefore, Some
*P* is *Q*. [K.]

172. Work out the doctrine of Opposition and the doctrine of Immediate Inferences on the hypothesis that universals are to be interpreted as implying, while particulars are not to be interpreted as implying, the existence of their subjects in the universe of discourse. [K.]

173. *The distinction
between Conditional Propositions and Hypothetical Propositions*268—Propositions commonly written in the
form *If A is B, C is D* belong to two very different types. For
they may be the expression either of simple judgments or of compound
judgments (as distinguished in section 55).

^{268} For the distinction
indicated in the present section I was in the first instance indebted
to an essay, written in 1884, by Mr W. E. Johnson. This essay has not
been published in its original form; but the substance of it has been
included in some papers on *The Logical Calculus* by Mr Johnson
which appeared in *Mind* in 1892.

In the first place, *A being B* and *C being D* may be
two events or two combinations of properties, concerning which it is
affirmed that whenever or wherever the first occurs the second will
occur also. For example, *If an import duty is a source of revenue,
it does not afford protection* ; *If a child is spoilt, his
parents suffer* ; *If a straight line falling upon two other
straight lines makes the alternate angles equal to one another, the
two straight lines are parallel to one another* ; *If a lighted
match is applied to gunpowder, there will be an explosion* ;
*Where the carcase is, there shall the eagles be gathered
together*. What is affirmed in all such cases as these is a
connexion between phenomena; it may be either a co-inherence of
attributes in a common subject, or a relation in time or space between
certain occurrences. Propositions belonging to this type may be called
distinctively *conditional*.

But again, *A is B* and *C is D* may be two propositions
of independent import, the relation between which cannot be 250 directly resolved into
any time or space relation or into an affirmation of the co-inherence
of attributes in a common subject. In other words, a relation may be
affirmed between the truth of two judgments as holding good once and
for all without distinction of place or time or circumstance. For
example, *If it be a sin to covet honour, I am the most offending
soul alive* ; *If patience is a virtue, there are painful
virtues* ; *If there is a righteous God, the wicked will not
escape their just punishment* ; *If virtue is involuntary, so is
vice* ; *If the earth is immoveable, the sun moves round the
earth*. Propositions belonging to this type may be called
*hypothetical* as distinguished from conditional, or they may be
spoken of still more distinctively as *true hypotheticals* or
*pure hypotheticals*.269

^{269} The above distinction has
been adopted in some recent treatises on Logic, but it must be borne
in mind that most logicians use the terms *conditional* and
*hypothetical* as synonymous or else draw a distinction between
them different from the above.

The parts of the conditional and also of the true hypothetical are
called the *antecedent* and the *consequent*. Thus, in the
proposition *If A is B, C is D*, the antecedent is *A is B*,
the consequent is *C is D*.

It is impossible formally to distinguish between conditionals and
hypotheticals so long as we keep to the expression *If A is B, C is
D*, since this may be either the one or the other. The following
forms, however, are unmistakeably conditional: *Whenever A is B, C
is D* ; *In all cases in which A is B, C is D* ; *If any P is
Q then that P is R*.270 The form *If A is true then C is true*
is, on the other hand, distinctively hypothetical. *A* and
*C* here stand for *propositions* or *judgments*, not
terms, and the words “is true” are introduced in order to
make this explicit. It is quite sufficient, however, to write the true
hypothetical in the form *If A then C*.

^{270} Conditionals can generally
be reduced to the last of these three forms without much difficulty,
and such reduction is sometimes useful. A consideration of the
concrete examples already given will, however, shew that a certain
amount of manipulation may be required in order to effect the
reduction. The following are examples: *If any child is spoilt, then
that child will have suffering parents* ; *If any two straight
lines are such that another straight line falling upon them makes the
alternate angles equal to one another, then those two straight lines
are parallel to one another*.

251 Since a
conditional proposition usually contains a reference to some
concurrence in time or space, the *if* of the antecedent may as a
rule be replaced either by *when* or by *where*, as the case
may be, without any change in the significance of the proposition; but
the same cannot be said in the case of the true hypothetical. This
consideration will often suffice to resolve any doubt that may arise
in concrete cases as to the particular type to which any given
proposition belongs. Another and more fundamental criterion may be
found in the answer to the question whether or not the antecedent and
consequent are propositions of independent import, whose meaning will
not be impaired if they are considered apart from one another. If the
answer is in the affirmative, then the proposition is hypothetical.
Thus, taking examples of hypotheticals already given, we find that the
antecedents, *It is a sin to covet honour*, *Patience is a
virtue*, *Virtue is involuntary*, and the consequents, *I am
the most offending soul alive*, *There are painful virtues*,
*Vice is involuntary*, all retain their full meaning though
separated from one another. If, on the other hand, the consequent
necessarily refers us back to the antecedent in order that it may be
fully intelligible, then the proposition is conditional. Thus, taking
by itself the consequent in the first conditional given on page 249, namely, *it does not afford protection*, we are at once led to
ask what is here meant by *it*. The answer is—*that
import duty*. But *what* import duty? An adequate answer can
be given only by introducing into the consequent the whole of the
antecedent,—*an import duty which is a source of revenue does
not afford protection*. We now have the full force of our original
conditional proposition in the form of a single categorical. It will
be found that if other conditionals are treated in the same way, they
resolve themselves similarly into categoricals of the form *All PQ
is R*.271 252 The problem of the reduction of
conditionals and hypotheticals to categorical form will be considered
in more detail later on in this chapter, and it will be shewn that
whilst such reduction is always possible, and generally simple and
natural, in the case of conditionals, it is not possible at all (with
terms corresponding to the original antecedent and consequent) in the
case of hypotheticals.272

^{271} As another example, we may
take the conditional proposition, *If the weather is dry, the
British root-crops are light*. Here it may at first sight appear
that the consequent is a proposition of independent import. The
proposition, *The British root-crops are light*, is, however, a
judgment incompletely stated. For it contains a time-reference that
needs to be made explicit. The conditional really means, *If in any
year the weather is dry, the British root-crops in that year are
light* ; and this is equivalent to the categorical, *Any year in
which the weather is dry is a year in which the British root-crops are
light*. By looking at the conditional in this way, we see the
necessity of referring back to the antecedent in order that the
consequent may be fully expressed.

^{272} The question may be raised
whether a proposition of the form, *If this P is Q, it is R*, is
properly to be described as a singular conditional or as a
hypothetical. The answer is that a proposition of this form affords a
kind of junction between the conditional and the hypothetical; it is
derivable from the conditional, *If any P is Q, it is R* ; but it
is itself hypothetical. The antecedent and the consequent are
propositions of independent import; and the proposition as a whole is
not directly reducible (as is the conditional, *If any P is Q, it is
P*) to categorical form. Thus, the proposition, *If any P is Q,
it is R*, may *prima facie* be reduced to the form *Any P
that is Q is R* ; but the proposition, *If this P is Q, it is
R*, certainly cannot be identified with the singular categorical,
*This P which is Q is R*.

174. *The Import of
Conditional Propositions*.—It is sometimes held that the real
*differentia* of all propositions of the form *If A is B, C is
D* is “to express human doubt.” Clearly, however, there
is no intention to express doubt as regards the relation between the
antecedent and the consequent; and the doubt must, therefore, be
supposed to relate to the actual occurrence of the antecedent. But so
far at any rate as *conditionals* are concerned, the doubt which they
may thus imply must be considered incidental rather than the
fundamental or differentiating characteristic belonging to them. The
*if* of the conditional may, as we have seen, usually be replaced
by *when* without altering the significance of the proposition,
and in this case the element of doubt is no more prominent than in the
categorical proposition. From the *material* standpoint, conditionals
may or may not involve the actual occurrence of their antecedents.
Whenever the connexion between the antecedent and the consequent can
be inferred from the nature of the antecedent independently of
specific experience (and this may be the more usual case), then the
actual happening of the 253 antecedent is not involved; but if our
knowledge of the connexion does depend on specific experience (as it
sometimes may), then such actual happening is materially involved. For
example, the statement, “If we descend into the earth, the
temperature increases at a nearly uniform rate of 1° Fahr. for every
fifty feet of descent down to almost a mile,” is based upon
knowledge gained by actual descents into the earth having been made,
and apart from such experience the truth of the statement would not
have been known.

The question of main importance in regard to the import of
conditional propositions is whether such propositions are to be
interpreted as modal or as merely assertoric. Confining ourselves for
the present to the universal affirmative, that is, to the form *If
any P is Q then it is R*, are we affirming a necessary relation
between *P* being *Q* and its being *R*, or are we
merely affirming that it so happens that every *P* that is
*Q* is also *R*? This is really in another form the
distinction already drawn between unconditionally universal
propositions and empirically universal propositions, and our answer
must again be that the same form of words may express the one judgment
or the other. There can be no doubt that the proposition, *If the
angles at the base of a triangle are equal to one another, that
triangle is isosceles*, is intended to be interpreted modally as
expressing a necessary connexion, while the proposition, *If any
book is taken down from that shelf, it will be found to be a
novel*, would be intended to be interpreted merely assertorically.

In ordinary discourse conditionals are as a rule modal; but this is
not universally the case. Unless, therefore, we are prepared to depart
from ordinary usage (and there is a good deal to be said for such
departure), we must recognise both *assertoric conditionals* and
*modal conditionals*, and this distinction must be borne in mind
in all that follows. We shall find that practically the same problem
arises in regard to true hypotheticals, and we shall have to consider
it further in that connexion.

175. *Conditional
Propositions and Categorical Propositions*.—We may go on to
consider what is the essential nature of the distinction between
conditional propositions and 254 categorical propositions, and in
particular whether the distinction is one of verbal form only or one
that corresponds to a real distinction between judgments.

If a vital distinction is to be drawn between the two forms, it must be on one or other of the two following grounds, namely, either (i) that the categorical is to be interpreted assertorically while the conditional is to be interpreted modally, or (ii) that the categorical is to be interpreted as implying the existence of its subject while the conditional is not to be interpreted as implying the occurrence of its antecedent.

(i) There is much to be said for adopting a convention by which the
categorical form would be interpreted assertorically and the
conditional form modally. The adoption of this convention would,
however, necessitate some modification of the forms of ordinary
speech, for, as we have already seen, the proposition *All S is
P* is in current use sometimes apodeictic, while the proposition
*If any S is P then it is Q* may (though perhaps rarely) be
merely assertoric. Whether the one form or the other is used really
depends a good deal on linguistic considerations. Consider, for
instance, the propositions, *All isosceles triangles have the angles
at their base equal to one another*, *If the angles at the base
of a triangle are equal to one another, that triangle is
isosceles*. These propositions fall naturally into the categorical
and conditional forms respectively, simply because there happens to be
no single adjective (like “isosceles”) which connotes
“having two equal angles.” It is clear, however, that the
use of the one form rather than the other is not intended to imply any
fundamental difference in the character of the relation asserted. If
either of the propositions in its ordinary use is apodeictic, so is
the other; if either is merely assertoric, so is the other.

It is to be added that if we adopt the convention under
consideration then the universal categorical is inferable from the
universal conditional, but not *vice versâ* ; while, on the other
hand, the problematic conditional (which corresponds to the
particular) is inferable from the particular categorical, but not
*vice versâ*. Thus, *All PQ is R* is subaltern to *If any
P is Q it* 255
*is R*, while *If any P is Q it may be R* is subaltern to
*Some PQ is R*.

(ii) We may pass on to consider whether categoricals and conditionals are to be differentiated in respect of their existential import.

We have seen in section 163 that if categoricals are interpreted modally they are not to be regarded as necessarily implying the existence of their subjects; and certainly conditionals, interpreted modally, are not to be regarded as necessarily implying the occurrence of their antecedents. Hence if both propositional forms are interpreted modally, we have no differentiation as regards their existential import.

It further seems clear that, so far as universal are concerned, a conditional proposition—even though interpreted as merely assertoric—is not to be regarded as necessarily implying the actual occurrence of its antecedent. Hence, whether, on the assertoric interpretation of both, the two forms are to be existentially differentiated depends upon our existential interpretation of the categorical.

(*a*) If a universal categorical is interpreted as necessarily
implying the actual existence of its subject, then we have a marked
distinction between the two forms.273 *If any P is Q
then it is also R* cannot be resolved into *All PQ is R*,
since the latter implies the existence of *PQ*, while the former
does not.

^{273} This is Ueberweg’s
view, “The categorical judgment, in distinction from the
hypothetical, always includes the pre-supposition of the existence of
the subject” (*Logic*, § 122).

(*b*) If, on the other hand, universal categoricals are not
interpreted as necessarily implying the existence of their subjects,
then universal conditionals and universal categoricals (both being
interpreted assertorically) may be resolved into one another. We may
say indifferently *All S is P* or *If anything is S it is
P* ; *If ever A is B then on all such occasions C is B* or
*All occasions of A being B are occasions of C being D*.

Particular conditionals, so far as they are merely assertoric, are almost without exception based upon specific experience. Hence they may not unreasonably be interpreted as implying the occurrence of their antecedents, as, for example, in the 256 proposition, “Sometimes when Parliament meets, it is opened by the Sovereign in person.” The existential interpretation of categoricals for which a preference was expressed in the preceding chapter may therefore be adopted for conditionals also, so far as they are merely assertoric; and the two forms become mutually interchangeable.

On the whole, except in so far as we adopt the convention indicated under (i) above, there seems no reason for drawing a vital distinction between judgments according as they are expressed in the conditional or the categorical form.274 Many of the conditionals of ordinary discourse are indeed so obviously equivalent to categoricals that they hardly seem to require a separate consideration.275 At the same time, as we have seen, some statements fall more naturally into the one form and some into the other. The more complex the subject-term, the greater is the probability that the natural form of the proposition will be conditional.

^{274} It has been argued that,
starting from the categorical form, we cannot pass to the conditional,
if the subject of the proposition is a simple term. The basis of this
argument is that the antecedent of a conditional requires two terms,
and that in the case supposed these are not provided by the
categorical. Thus, Miss Jones (*Elements of Logic*, p. 112) takes
the example, “All lions are quadrupeds.” It will not do,
she says, to reduce this to the form, “If any creatures are
lions, they are quadrupeds,” since this involves the
introduction of a new term, and passing back again to the categorical
form, we should have “All creatures which are lions are
quadrupeds,” a proposition not equivalent to our original
proposition. If, however, “creature” is regarded as part
of the connotation of “lion,” there is no reason for
refusing to allow that the two propositions are equivalent to one
another. Similarly, in any concrete instance, by taking some part of
the connotation of the subject of our categorical proposition, we can
obtain the additional term required for its reduction to the
conditional form. Where we are dealing with purely symbolic
expressions, and this particular solution of the difficulty is not
open to us, we may have recourse to the all-embracing term
“anything,” such a proposition as *All S is P* being
reduced to the form *If anything is S it is P*.

^{275} The examples given at the
commencement of section 173 are reducible to the following
categoricals: *Import duties which are sources of revenue do not
afford protection* ; *All spoilt children have suffering
parents* ; *All pairs of straight lines which are such that
another straight line falling upon them makes the alternate angles
equal to one another are parallel* ; *All occasions of the
application of a lighted match to gunpowder are occasions of an
explosion* ; *Any place where there is a carcase is a place where
the eagles will gather together*.

176. *The Opposition of
Conditional Propositions*.—This question needs a separate
discussion according as conditionals are interpreted (*a*)
assertorically, or (*b*) modally.

257 (*a*) If
conditionals are interpreted assertorically, then the ordinary
distinctions both of quality and of quantity can be applied to them in
just the same way as to categoricals. We may regard the quality of a
conditional as determined by the quality of its consequent; thus, the
proposition *If any P is Q then that P is not R* may be treated
as negative.276 As regards quantity, conditionals are to be
regarded as universal or particular, according as the consequent is
affirmed to accompany the antecedent in all or merely in some cases.

^{276} The negative force of this
proposition would be more clearly brought out if it were written in
the form *If any P is Q then it is not the case that it is also
R*. The categorical equivalent is *No PQ is R*.

We have then the four types included in the ordinary four-fold schedule:—

*If any P is Q, it is also R* ; **A**

*If any P is Q, it is not also R* ; **E**
*Sometimes if a P is Q, it is also R* ; **I**
*Sometimes if a P is Q, it is not also R*. **O**

These propositions constitute the ordinary square of opposition,
and if conditionals are assimilated to categoricals so far as their
existential import is concerned, then the opposition of conditionals
on the assertoric interpretation seems to require no separate
discussion.277 It may, however, be pointed out that there
is more danger of contradictories being confused with contraries in
the case of conditionals than in the case of categoricals. *If A is
B then C is not D* is very liable to be given as the contradictory
of *If A is B then C is D*. But it is clear on consideration that
both these propositions may be false. For example, the two
statements—If the Times says one thing, the Westminster Gazette
says another; If the Times says one thing, the Westminster Gazette
says the same, *i.e.*, does not say another—might be, and
as a matter of fact are, both false; the two papers are sometimes in
agreement and sometimes not.

^{277} The four propositions are
precisely equivalent to the four categoricals,—*All PQ is
R*, *No PQ is R*, *Some PQ is R*, *Some PQ is not
R*.

(*b*) On the modal interpretation, the distinction between
258 apodeictic and
problematic takes the place of that between universal and particular;
and if we maintain the distinction between affirmative and negative,
we have the four following propositions corresponding to the ordinary
square of opposition:

*If any P is Q, that P must be R* ; **A**_{m}

*If any P is Q, that P cannot be R* ; **E**_{m}

*If any P is Q, that P may be R* ; **I**_{m}

*If any P is Q, that P need not be R*. **O**_{m}

It will be convenient to have distinctive symbols to denote modal propositions, and those that we have here introduced will serve to bring out the analogies between modals and the ordinary assertoric forms.

In the above schedule, subject to a certain condition mentioned
below, **A**_{m} and **O**_{m},
and also **E**_{m} and **I**_{m},
are contradictories according to the definition given in section 84;
**A**_{m} and **E**_{m} are
contraries; **A**_{m} and
**I**_{m}, and also **E**_{m} and
**O**_{m}, are subalterns; and
**I**_{m} and **O**_{m} are
subcontraries.

The condition referred to relates to the interpretation of the
propositions as regards the implication of the possibility of their
antecedents. Thus, in order that **A**_{m} and
**O**_{m} (or **E**_{m} and
**I**_{m}) may be true contradictories it is
necessary that apodeictic and problematic propositions shall be
interpreted *differently* in this respect. If, for example,
**A**_{m} is interpreted as not implying the
possibility of its antecedent then its full import is to deny the
possibility of the combination *P* and *Q* without *R*.
Its contradictory must affirm this possibility.
**O**_{m} will not, however, do this unless it is
interpreted as implying the possibility of the combination *P*, *Q*.

It is necessary to call attention to this complication, but hardly necessary to work out in detail the results which follow from the various principles of interpretation that might be adopted. If the student will do this for himself, he will find that the results correspond broadly with those obtained in section 159.278

^{278} In connexion with the
problem of opposition we may touch briefly on the relation between the
apodeictic proposition *If any P is Q that P must be R* and the
assertoric proposition *Some PQ is not R*. These propositions are
not contradictories, for they may both be false. They cannot, however,
both be true; and the latter, if it can be established, affords a
valid ground for the denial of the former. Mr Bosanquet appears not to
admit this, but to maintain, in opposition to it, that the enumerative
particular is of no value as overthrowing the abstract universal.
“When we have said that *If* (*i.e.*, *in so far
as*) *a man is good, he is wise*, it is idle to reply that
*Some good men are not wise*. This is to attach an abstract
principle with unanalysed examples. What we must say in order to deny
the above-mentioned abstract judgment is something of this kind:
*If* or *Though a man is good, yet it does not follow that he
is wise*, that is, *Though a man is good, yet he need not be
wise*” (*Logic*, i. p. 316). But surely if we find that
some good men are not wise, we are justified in saying that though a
man is good yet he need not be wise. Of course the converse does not
hold. We might be able to shew that wisdom does not necessarily
accompany goodness by some other method than that of producing
instances. But if we can produce undoubted instances, that amply
suffices to confute the apodeictic conditional.

259 177.
*Immediate Inferences from Conditional Propositions*.—In a
conditional proposition the antecedent and the consequent correspond
respectively to the subject and the predicate of a categorical
proposition. In conversion, therefore, the old consequent must be the
new antecedent, and in contraposition the negation of the old
consequent must be the new antecedent.

(*a*) On the assertoric interpretation, the analogy with
categoricals is so close that it is unnecessary to treat immediate
inferences from conditionals in any detail. One or two examples may
suffice. Taking the **A** proposition, *If any P is Q then it is
R*, we have for its converse *Sometimes if a P is R it is also
Q*, and for its contrapositive *If any P is not R then it is not
Q*. Taking the **E** proposition *If any P is Q then it is not
R*, we have for its converse *If any P is R then it is not Q*,
and for its contrapositive *Sometimes if a P is not R it is Q*.
The validity of these inferences is of course affected by the
existential interpretation of the propositions just as in the case of
the categoricals. It will be noticed that in some immediate inferences
(for example, the contraposition of **A**) the conditional form has
an advantage over the ordinary categorical form inasmuch as it avoids
the use of negative terms, the employment of which is so strongly
objected to by Sigwart and some other logicians.279

(*b*) If conditionals are interpreted modally, then the
apodeictic form takes the place of the universal, and the 260 problematic takes the
place of the particular. On this basis, the converse of *If any P is
Q that P must be R* would be *If any P is R that P may be Q*,
and the contrapositive would be *If any P is not R that P cannot be Q*.

Are these inferences legitimate? On the interpretation that a modal
proposition implies nothing as to the possibility of its antecedent,
then our answer must be in the affirmative, as regards the
contraposition of **A**_{m}. The full import both of
the original proposition and of the contrapositive is to deny the
possibility of the combination *P* and *Q* without *R*.
On the same interpretation, however, the conversion of
**A**_{m} is not valid. For the converse implies
that if *PR* is possible then *PQ* is possible, while the
possibility of *PR* combined with the impossibility of *PQ*
is compatible with the truth of the original proposition. It can be
shewn similarly that, while the conversion of
**E**_{m} is valid, its contraposition is invalid.

If we were to vary the interpretation, the results would be different.

The correspondence between the results shewn above and our results
respecting the conversion and contraposition of the assertoric
**A** and **E** propositions, on the interpretation that no
proposition implies the existence of its subject (see page 225), is
obvious. The truth is that the interpretation of modals in respect to
the *possibility* of their antecedents gives rise to problems
precisely analogous to those arising out of the interpretation of
assertoric propositions in respect to the *actuality* of their
subjects. It is unnecessary that we should work out the different
cases in detail.

Amongst immediate inferences from a conditional proposition, its
reduction to categorical form, so far as this is valid, is generally
included. This is a case of what has been called *change of
relation*, meaning thereby an immediate inference in which we pass
from a given proposition to another which belongs to a different
category in the division of propositions according to relation (see
section 54). The more convenient term *transversion* is used by
Miss Jones for this process.

How far conditionals can be inferred from categoricals and *vice
versâ* depends on their interpretation. If both types of 261 propositions are
interpreted assertorically or both modally, and if they are
interpreted similarly as regards the implication of the existence (or
possibility) of their subjects (or antecedents), then the validity of
passing from either type to the other cannot be called in question.
Some doubt may, however, be raised as to whether in this case we have
an inference at all or merely a verbal change. This is a distinction
to which attention will be called later on.

If conditionals are interpreted modally and categoricals
assertorically then (apart from any complications that may arise from
existential implications) **A** can be inferred from
**A**_{m} or **E** from
**E**_{m}, but not *vice versâ*. On the other
hand, **I**_{m} can be inferred from **I**, or
**O**_{m} from **O**, but not *vice versâ*.

We have another case of transversion when we pass from conditional to disjunctive, or from disjunctive to conditional. The consideration of this case must be deferred until we have discussed disjunctives.

178. *The Import of
Hypothetical Propositions*.—The pure hypothetical may be
written symbolically in the form *If A is true then C is true*,
or more briefly, *If A then C*, where *A* and *C* stand
for propositions of independent import. It is clear that this
proposition affirms nothing as regards the truth or falsity of either
*A* or *C* taken separately. We may indeed frame the
proposition, knowing that *C* is false, with the express object
of showing that *A* is false also. What we have is of course a
judgment not about either *A* or *C* taken separately, but
about *A* and *C* in relation to one another.

The main question at issue in regard to the import of the
hypothetical proposition is whether it is merely assertoric or is
modal. The contrast may be simply put by asking whether, when we say
*If A then C*, our intention is merely *to deny the actuality
of the conjunction of A true with C false* or is *to declare this
conjunction to be an impossibility*.

The contrast between these two interpretations can be brought out
most clearly by asking how the proposition *If A then C* is to be
contradicted. If our intention is merely to deny the actuality of the
conjunction of *A* true with *C* false, then the
contradictory must assert the actuality of this conjunction; if 262 our intention is to
deny the possibility of the conjunction, then the contradictory will
merely assert its possibility. In other words, on the assertoric
interpretation the contradictory will be *A is true but C is
false* ;280 on the modal interpretation it will be *If
A is true C may be false*.281

^{280} We may look at it in this
way. Let *AC* denote the truth of both *A* and *C*,
*ACʹ* the truth of *A* and the falsity of *C*, and
so on. Then there are four *à priori* possibilities, namely,
*AC*, *ACʹ*, *AʹC*, *AʹCʹ*, one or
other of which must hold good, but any pair of which are mutually
inconsistent. The proposition *If A then C* merely excludes
*ACʹ*, and still leaves *AC*, *AʹC*,
*AʹCʹ*, as possible alternatives. In denying it,
therefore, we must definitely affirm *ACʹ*, and exclude the
three other alternatives. Hence the contradictory as above
stated.

^{281} A certain assumption is
necessary, in order that this result may be correct. The opposition of
hypotheticals on the modal interpretation will be discussed in more
detail in section 179.

Hypotheticals intended to be interpreted assertorically are to be
met with in ordinary discourse, but they are unusual. There appear to
be two cases: (*a*) When we know that one or other of two
propositions is true but do not know (or do not remember) which, we
may express our knowledge in the form of a hypothetical, *If X is
not true then Y is true*, and such hypothetical will be merely
assertoric. For example, *If the flowers I planted in this bed were
not pansies they were violets*. Here the intention is merely to
deny the actuality of the flowers being neither pansies nor violets.
(*b*) We may deny a proposition emphatically by a hypothetical in
which the proposition in question is combined as antecedent with a
manifestly false consequent; and such hypothetical will again be
merely assertoric. For example, *If what you say is true, I’m a
Dutchman* ; *If that boy comes back, I’ll eat my head*
(vide *Oliver Twist*); *I’m hanged if I know what you
mean*. In these examples the intention is to deny the actuality
(not the possibility) of the conjunctions,—What you say is true
and *I am not a Dutchman* ; That boy will come back and *I shall
not eat my head* ; *I am not hanged* and I know what you mean;
and since the elements of the conjunctions printed in italics are
admittedly true, the force of the propositions is to deny the truth of
the other elements, that is to say, to affirm,—*What you say
is not true*, *That boy will not come back*, *I do not know
what you mean*. Similarly 263 we may sometimes employ the hypothetical
form of expression as an emphatic way of declaring the truth of the
consequent (an antecedent being chosen which is admittedly true); for
example, *If he cannot act, at any rate he can sing*. Here once
more the hypothetical is merely assertoric.

It cannot, however, be maintained that any of the above are typical
hypotheticals; and the claim that our natural interpretation of
hypotheticals is ordinarily modal may be justified on the ground that
we do not usually consider it to be necessary to affirm the antecedent
in order to be able to deny a hypothetical. We have seen that, in order
to deny the assertoric hypothetical *If A then C*, we must affirm
*A* and deny *C* ; but we should usually regard it as
sufficient for denial if we can shew that there is no necessary
connexion between the truth of *A* and that of *C*, whether
*A* is actually true or not.

We shall then in the main be in agreement with ordinary usage if we interpret hypotheticals modally, and the adoption of such an interpretation will also give hypotheticals a more distinctive character. In what follows the hypothetical form will accordingly be regarded as modal, except in so far as an explicit statement is made to the contrary.282

^{282} *Either C is true or A is
not true* is usually regarded as the disjunctive equivalent of the
hypothetical *If A is true then C is true*. The relation between
these two propositions will be discussed further later on. It is,
however, desirable to point out at once that, if the equivalence is to
hold good, both the propositions must be interpreted assertorically or
both modally. There is a good deal to be said for differentiating the
two forms by regarding the hypothetical as modal and the disjunctive
as merely assertoric. This method of treatment is explicitly adopted
by Mr McColl. He writes (using the symbolism, *a* : *b* for
*If a then b*, *a* + *b* for *a or b*,
*aʹ* for the denial of *a*)—“The
expression *a* : *b* may be read *a implies b* or *If
a is true, b must be true*. The statement *a* : *b*
implies *aʹ* + *b*. But it may be asked are not the
two statements really equivalent; ought we not therefore to write
*a* : *b* = *aʹ* + *b*? Now if the two
statements are really equivalent their denials will also be
equivalent. Let us see if this will be the case, taking as concrete
examples: ‘If he persists in his extravagance he will be
ruined’; ‘He will either discontinue his extravagance or
he will be ruined.’ The denial of *a* : *b* is
(*a* : *b*)ʹ and this denial may be read—‘He may persist in his extravagance without necessarily being
ruined.’ The denial of *aʹ* + *b* is
*abʹ* which may be read—‘He will persist in his
extravagance and he will not be ruined.’ Now it is quite evident
that the second denial is a much stronger and more positive statement
than the first. The first only asserts the *possibility* of the
combination *abʹ* ; the second asserts the *certainty*
of the same combination. The denials of the statements *a* :
*b* and *aʹ* + *b* having thus been proved to be
not equivalent, it follows that the statements *a* : *b* and
*aʹ* + *b* are themselves not equivalent, and that,
though *aʹ* + *b* is a necessary consequence of
*a* : *b*, yet *a* : *b* is not a necessary
consequence of *aʹ* + *b*” (see *Mind*,
1880, pp. 50 to 54; one or two slight verbal changes have been made in
this quotation).

264 Some writers
who adopt the modal interpretation of hypotheticals speak of the
consequent as being an *inference* from the antecedent. There are
no doubt some hypotheticals to which this description accurately
applies. Thus, we may have hypotheticals which are *formal* in
the sense in which that term has been used in section 31, the
consequent being, for instance, an immediate inference from the
antecedent, or being the conclusion of a syllogism of which the
premisses constitute the antecedent. The following are
examples,—*If all isosceles triangles have the angles at the
base equal to one another, then no triangle the angles at whose base
are unequal can be isosceles* ; *If all men are mortal and the
Pope is a man, then the Pope must be mortal*.

But more usually the consequent of a hypothetical proposition
cannot be inferred from the antecedent alone. The aid is required of
suppressed premisses which are taken for granted, the premiss which
alone is expressed being perhaps the only one as to the truth of which
any doubt is regarded as admissible. It would, therefore, be better to
speak of the consequent as being the *necessary consequence* of
the antecedent, than as being an *inference* from it. When we speak of
*C* as being an inference from *A*, there is a suggestion
that *A* affords the complete justification of *C*, whereas
when we speak of it as a necessary consequence, this suggestion is at
any rate less prominent.283

^{283} Miss Jones (*General
Logic*, p. 45) divides hypotheticals into *formal* or
*self-contained hypotheticals* and *referential
hypotheticals*. In the former, “the consequent is an
inference from the antecedent alone”; in the latter, “the
consequent is inferred not from the antecedent alone, but from the
antecedent taken in conjunction with some other unexpressed
proposition or propositions.”

179. *The Opposition of
Hypothetical Propositions*.—Regarding hypotheticals as always
affirming a necessary consequence, it may reasonably be held that they
do not admit of distinctions of *quality*. Sigwart accordingly
lays it down that all hypotheticals are affirmative. “Passing to
hypothetical judgments 265 containing negations, we find that the
form ‘If *A* is, *B* is not’ represents the
negation of a proposition as the necessary consequence of an
affirmation, thus affirming that the hypotheses *A* and *B*
are incompatible.”284 The force of this
argument must be admitted. There is, however, some convenience in
distinguishing between hypotheticals according as they lead up, in the
consequent, to an affirmation or a denial; and in the formal treatment
of hypotheticals, we shall be better able to preserve an analogy with
categoricals and conditionals if we denote the proposition *If X is
true then Y is true* by the symbol **A**_{m}, and
the proposition *If X is true then Y is not true* by the symbol
**E**_{m}.

^{284} *Logic*, i. p. 226.

Whether or not we decide thus to recognise distinctions of quality
in the case of hypotheticals, we certainly cannot recognise
distinctions of *quantity*. The antecedent of a hypothetical is
not an event which may recur an indefinite number of times, but a
proposition which is simply true or false. We have already seen that
the same proposition cannot be sometimes true and sometimes false,
since propositions referring to different times are different
propositions.285

^{285} This, as Mr Johnson has
pointed out, must be taken in connexion with the recognition of
propositions as involving *multiple quantification*. “Thus
we may indicate a series of propositions involving single, double,
triple … quantification, which may reach any order of multiplicity:
(1) ‘All luxuries are taxed’; (2) ‘In some countries
all luxuries are taxed’; (3) ‘At some periods it is true
that in all countries all luxuries are taxed’.… with respect to
each of the types of proposition (1), (2), (3).… I contend that, when
made explicit with respect to time or place, etc., it is absurd to
speak of them as sometimes true and sometimes false”
(*Mind*, 1892, p. 30 *n.*).

Do not distinctions of *modality*, however, take the place of
distinctions of quantity? Up to this point, we have practically
confined our attention to the *apodeictic* hypothetical, *If A
then C*. This proposition is denied by the proposition *If A is
true still C need not be true* (that is to say, *The truth of C
is not a necessary consequence of the truth of A*). Can this latter
proposition be described as a *problematic* hypothetical? Clearly
it is not a hypothetical at all if we begin by defining a hypothetical
as the affirmation of a necessary consequence. There seems, however,
no need for this limitation. We may define a 266 hypothetical as a proposition which
starting from the hypothesis of the truth (or falsity) of a given
proposition affirms (or denies) that the truth (or falsity) of another
proposition is a necessary consequence thereof. But, whether or not we
adopt this definition, there can be no doubt that the proposition
*If A then possibly C* appropriately finds a place in the same
schedule of propositions as *If A then necessarily C*. In such a
schedule we have the four forms,—

*If A is true then C is true* ; **A**_{m}

*If A is true then C is not true* ; **E**_{m}

*If A is true still C may be true* ; **I**_{m}

*If A is true still C need not be true*. **O**_{m}

These four propositions correspond to those included in the
ordinary square of opposition; and, if we start with the assumption
that *A* is possibly true,286 the ordinary
relations of opposition hold good between them.
**A**_{m} and **O**_{m},
**E**_{m} and **I**_{m} are pairs
of contradictories; **A**_{m} and
**E**_{m} are contraries;
**A**_{m} and **I**_{m},
**E**_{m} and **O**_{m}, are pairs
of subalterns; **I**_{m}and
**O**_{m} are subcontraries.

^{286} By this is meant that we
start with the assumption that *A* is possibly true
*independently of the affirmation of any one of the propositions in
question*. The reader must particularly notice that this assumption
is quite different from the assumption that each of the propositional
forms implies as part of its import that *A* is possibly true;
otherwise the results reached in this paragraph may appear to be
inconsistent with those reached in the following paragraph.

If, however, it is *not* assumed that *A* is possibly
true, then the problem is more complicated, since the character of the
relations is affected by the manner in which the propositions are
interpreted in respect to the possibility of their antecedents. The
results are substantially the same as in the case of modal
conditionals (section 176), and correspond with those obtained in
section 159, where the analogous problem in regard to categoricals
(assertorically interpreted) is discussed. Thus, in order that
**A**_{m}, and **O**_{m},
**E**_{m} and **I**_{m}, may be
contradictories, apodeictic and problematic propositions must be
interpreted *differently* as regards the implication or non-implication
of the possible truth of their antecedents; while, on the other hand,
in order that **A**_{m} and
**I**_{m}, **E**_{m} and
**O**_{m}, may be subalterns, 267 problematic
propositions must not be interpreted as implying the possible truth of
their antecedents unless apodeictic propositions are interpreted
*similarly* in this respect. If we interpret neither apodeictic
nor problematic hypotheticals as implying the possible truth of their
antecedents, then the contradictory of *If A, then C* may be
expressed in the form *Possibly A, but not C* (or, as it may also
be formulated, *A is possibly true, and if it is true, still C need
not be true*).

It would occupy too much space to discuss in detail all the
problems that might be raised in this connexion. The principles
involved have been sufficiently indicated; and the reader will find no
difficulty in working out other cases for himself. We may, however,
touch briefly on the relation between the propositions *If A then
C* and *If A then not C*, shewing in particular that on no
supposition are they true contradictories.

If these two propositions are interpreted assertorically, then so
far from being contradictories, they are subcontraries. For, supposing
*A* happens not to be true, then it cannot be said that either of
them is false: the statement *If A then C* merely excludes
*ACʹ*, and *If A then Cʹ* merely excludes *AC* ;
hence two possibilities are left, *AʹC* or
*AʹCʹ*, neither of which is inconsistent with either of
the propositions.287 On the other hand, the propositions cannot
both be false, since this would mean the truth of both *ACʹ*
and *AC*.

^{287} The validity of the above
result will perhaps be more clearly seen by substituting for the
hypotheticals their (assertoric) disjunctive equivalents, namely,
*Either A is not true or C is true*, *Either A is not true or C
is not true*. As a concrete example, we may take the propositions,
“If this pen is not cross-nibbed, it is corroded by the
ink,” “If this pen is not cross-nibbed, it is not corroded
by the ink.” Supposing that the pen happens to be cross-nibbed,
we cannot regard either of these propositions as false. It will be
observed that their disjunctive equivalents are, “This pen is
either cross-nibbed or corroded by the ink,” “This pen is
either cross-nibbed or not corroded by the ink.” Take again the
propositions, “If the sun moves round the earth, some
astronomers are fallible.” “If the sun moves round the
earth, all astronomers are infallible.” The truth of the first
of these propositions will not be denied, and on the interpretation of
hypotheticals with which we are here concerned the second cannot be
said to be false. It may be taken as an emphatic way of denying that
the sun does move round the earth.

Returning to the modal interpretation of the propositions, then if
interpreted as implying the possible truth of their 268 antecedents, they are
contraries. They cannot both be true, but may both be false. It may be
that neither the truth nor the falsity of *C* is a necessary
consequence of the truth of *A*.288

^{288} It has been argued that
*If A then C* must have for its contradictory *If A then not
C*, since the consequent must either follow or not follow from the
antecedent. But to say that *C* does not follow from *A* is
obviously not the same thing as to say that *not-C* follows from
*A*.

Once more, if interpreted modally but not as implying the possible truth of their antecedents, the propositions may both be true as well as both false. This case is realised when we establish the impossibility of the truth of a proposition by shewing that, if it were true, inconsistent results would follow.

180. *Immediate Inferences
from Hypothetical Propositions*.—The most important immediate
inference from the proposition *If A then C* is *If Cʹ then
Aʹ*. This inference is analogous to *contraposition* in
the case of categoricals, and may without any risk of confusion be
called by the same name. We may accordingly define the term
*contraposition* as applied to hypotheticals as *a process of
immediate inference by which we obtain a new hypothetical having for
its antecedent the contradictory of the old consequent, and for its
consequent the contradictory of the old antecedent*. If we
recognise distinctions of quality in hypotheticals, then (as regards
apodeictic hypotheticals) this process is valid in the case of
affirmatives only. It will be observed that from the contrapositive we
can pass back to the original proposition; and from this it follows
that the original proposition and its contrapositive are
equivalents.289 The following are examples: “If
patience is a virtue, there are painful virtues” = “If
there are no painful virtues, patience is not a virtue”;
“If there is a righteous God, the wicked will not escape their
just punishment” = “If the wicked escape their just
punishment, there is no righteous God.”

^{289} This holds good whether we
adopt the assertoric or the modal interpretation. On the former
interpretation, the import of both the propositions *If A then C*
and *If Cʹ then Aʹ* is to negative *ACʹ* ; on the
latter interpretation, the import of both is to deny the possibility
of the conjunction *ACʹ*.

From the negative hypothetical *If A is true then C is not
true* we can infer *If C is true then A is not true*. This is
analogous to *conversion* in the case of categoricals.

269 From the
affirmative *If A then C*, we may obtain by conversion *If C
then possibly A* ; but this is only on the interpretation that both
propositions imply the possibility of the truth of their
antecedents.290 The reader will notice that to pass from
*If A then C* to *If C then A* would be to commit a fallacy
analogous to simply converting a categorical **A** proposition; and
this is perhaps the most dangerous fallacy to be guarded against in
the use of hypotheticals.291

^{290} Compare section 158. The
various results obtained in section 158 may be applied *mutatis
mutandis* to modal hypotheticals. The reader may consider for
himself the contraposition of **E**_{m}.

^{291} On the assertoric
interpretation *If A then C* merely negatives *ACʹ*,
while *If C then A* merely negatives *AʹC*, and hence it
is clear that neither of these propositions involves the other; on the
modal interpretation the result is the same, for the truth of *C*
may be a necessary consequence of the truth of *A*, while the
converse does not hold good.

A consideration of immediate inferences enables us to shew from
another point of view that *If A then C* and *If A then
Cʹ* are not true contradictories. For the contrapositives *If
A then Cʹ*, *If C then Aʹ*, are equivalent to one
another; and whenever two propositions are equivalent, their
contradictories must also be equivalent. But *If A then C* is not
equivalent to *If C then A*.

If distinctions of quality are admitted, then the process of
*obversion* is applicable to hypotheticals. For example, *If A
is true then C is not true* = *If A is true then Cʹ is
true*. It is nearly always more natural and more convenient to take
hypotheticals in their affirmative rather than in their negative form;
and hence in the case of hypotheticals more importance attaches to the
process of *contraposition* than to that of *conversion*.

If the falsity of *C* is assumed to be possible, then we may
pass by inversion from *If A then C* to *It is possible for
both A and C not to be true* ; or, putting the same thing in a
different way, we may by inversion pass from *If A then C* to
*If the falsity of C is possible then the falsity of both A and C is
possible*.292 It is of course a fallacy to argue from
*If A then C* to *If Aʹ then Cʹ*.

^{292} The inversion of
**E**_{m} may be worked out similarly. Here, as
elsewhere, the process of inversion, although of little or no
practical importance, raises problems that are of considerable
theoretical interest.

Turning to problematic hypotheticals, we find that from the
proposition *If A is true C may be true*, we obtain by conversion
*If C is true A may be true* ; and from the proposition *If A
is* 270 *true C
need not be true* we obtain by contraposition *If C is true A
need not be true*. Here the analogy with categoricals is again very
close.

181. *Hypothetical
Propositions and Categorical Propositions*.—A true
hypothetical proposition has been defined as a proposition expressing
a relation between two other propositions of independent import, not
between two terms; and it follows that a true hypothetical is not,
like a conditional, easily reducible to categorical form. So far as we
can obtain an equivalent categorical, its subject and predicate will
not correspond with the antecedent and consequent of the hypothetical.
Thus, the proposition *If A then C* may, according to our
interpretation of it, be expressed in one or other of the following
forms; *A is a proposition the truth of which is incompatible with
the falsity of C* ; *A is a proposition from the truth of which
the truth of C necessarily follows*. It will be observed that,
apart from the fact that these propositions are not of the ordinary
categorical type,293 the predicate is not in either of them
equivalent to the consequent of the hypothetical.294 No doubt a
hypothetical proposition may be based on a categorical proposition of
the ordinary type. But that is quite a different thing from saying
that the two propositions are equivalent to one another.

^{293} Since they are
*compound*, not simple, propositions. The expression of compound
propositions in categorical form is not convenient, and it is better
to reserve the hypothetical and disjunctive forms for such
propositions, the categorical and conditional forms being used for
simple propositions.

^{294} Amongst other differences
the contrapositives of both these propositions differ from the
contrapositive of the hypothetical. For, on either interpretation of
the hypothetical, its contrapositive is *If C is not true then A is
not true*, whilst the contrapositives of the above propositions are
respectively,—*A proposition whose truth is compatible with
the falsity of the proposition C is not the proposition A*, *A
proposition from which the proposition C is not a necessary
consequence is not the proposition A*.

The relation between hypothetical and disjunctive propositions will be discussed in the following chapter.

182. *Alleged Reciprocal
Character of Conditional and Hypothetical Judgments*.—Mr
Bosanquet argues that the hypothetical judgment (and under this
designation he would include the conditional as well as what we have
called the true 271
hypothetical) “when ideally complete must be a reciprocal
judgment. *If A is B, it is C* must justify the inference *If A
is C, it is B*. We are of course in the habit of dealing with
hypothetical judgments which will not admit of any such conversion,
and the rules of logic accept this limitation … If in actual fact … *AB* is found to involve *AC* while *AC* does not
involve *AB*, it is plain that what was relevant to *AC* was
not really *AB* but some element *αβ* within it … Apart from time on the one hand and irrelevant elements on the other,
I cannot see how the relation of conditioning differs from that of
being conditioned … In other words, if there is nothing in *A*
beyond what is necessary to *B*, then *B* involves *A*
just as much as *A* involves *B*. But if *A* contains
irrelevant elements, then of course the relation becomes one-sided … The relation of Ground is thus essentially reciprocal, and it is only
because the ‘grounds’ alleged in every-day life are
burdened with irrelevant matter or confused with causation in time,
that we consider the Hypothetical Judgment to be in its nature not
reversible” (*Logic*, I. pp. 261–3).

The question here raised is analogous to that of the possibility of plurality of causes which is discussed in inductive logic. It may perhaps be described as a wider aspect of the same question. So long as a given consequence has a plurality of grounds, it is clear that the hypothetical proposition affirming it to be a consequence of a particular one of these grounds cannot admit of simple conversion, for the converted proposition would hold good only if the ground in question were the sole ground.

Mr Bosanquet urges that the relation between ground and consequence
will become reciprocal by the elimination from the antecedent of all
irrelevant elements. It should be added that we can also secure
reciprocity by the expansion of the consequent so that what follows
from the antecedent is fully expressed. Thus, if we have the
hypothetical *If A then γ*, which is not reciprocal, it
is possible that *A* may be capable of analysis into *αβ*, and *γ* of expansion into *γδ*,
so that either of the hypotheticals *If α then
γ*, *If αβ then γδ*, is reciprocal. In the former case we have a more exact
statement of the ground, all extraneous 272 elements being eliminated; in the latter
case we have a more complete statement of the consequence. Sometimes,
moreover, the latter of these alternatives may be practicable while
the former is not.

This may be tested by reference to a formal hypothetical. The
proposition *If all S is M and all M is P, then all S is P* is
not reciprocal. We may make it so by expanding the consequent so that
the proposition becomes *If all S is M and all M is P, then whatever
is either S or M is P and is also M or not S*. But how in this case
it would be possible to eliminate the irrelevant from the antecedent
it is difficult to see. Our object is to eliminate *M* from the
consequent, and if in advance we were to eliminate it from the
antecedent the whole force of the proposition would be lost. And the
same is true of non-formal hypotheticals, at any rate in many cases.
Instances of reciprocal conditionals may be given without difficulty,
for example, *If any triangle is equilateral, it is equiangular*.
Such propositions are practically **U** propositions. We may also
find instances of pure hypotheticals that are reciprocal; but, on the
whole, while agreeing with a good deal that Mr Bosanquet says on the
subject, I am disposed to demur to his view that the reciprocal
hypothetical represents an ideal at which we should always aim. We
have seen that there are two possible ways of securing reciprocity,
whether or not they are always practicable; but the expansion of the
consequent would generally speaking be extremely cumbrous and worse
than useless, while the elimination from the antecedent of everything
not absolutely essential for the realisation of the consequent would
sometimes empty the judgment of all practical content for a given
purpose. With reference to the case where *AB* involves
*AC*, while *AC* does not involve *AB*, Mr Bosanquet
himself notes the objection,—“But may not the irrelevant
element be just the element which made *AB* into *AB* as
distinct from *AC*, so that by abstracting from it *AB* is
reduced to *AC*, and the judgment is made a tautology, that is,
destroyed?” (p. 261). This argument, although somewhat
overstated, deserves consideration. The point upon which I should be
inclined to lay stress is that in criticising a judgment we ought to
have regard 273 to
the special object with which it has been framed. Our object may be to
connect *AC* with *AB*, including whatever may be irrelevant
in *AB*. Consider the argument,—*If anything is P it is
Q*, *If anything is Q it is R*, therefore, *If anything is P
it is R*. It is clear that if we compare the conclusion with the
second premiss, the antecedent of the conclusion contains
irrelevancies from which the antecedent of the premiss is free. Yet
the conclusion may be of the greatest value to us while the premiss is
by itself of no value. If our aim were always to get down to first
principles, there would be a good deal to be said for Mr
Bosanquet’s view, though it might still present some
difficulties; but there is no reason why we should identify the
conditional or the hypothetical proposition with the expression of
first principles.

It is to be added that, if Mr Bosanquet’s view is sound, we
ought to say equally that the **A** categorical proposition is
imperfect, and that in categoricals the **U** proposition is the
ideal at which we should aim. In categoricals, however, we clearly
distinguish between **A** and **U**; and so far as we give
prominence to the reciprocal modal, whether conditional or
hypothetical, we ought to recognise its distinctive character. We may
at the same time assign to it the distinctive symbol
**U**_{m}.

183. Give the contrapositive
of the following proposition: If either no *P* is *R* or no
*Q* is *R*, then nothing that is both *P* and *Q*
is *R*. [K.]

184. There are three men in
a house, Allen, Brown, and Carr, who may go in and out, provided that
(1) they never go out all at once, and that (2) Allen never goes out
without Brown.

Can Carr ever go out? [LEWIS CARROLL.]

185. There are two
propositions, *A* and *B*.

Let it be granted that

If *A* is true, *B* is true. (i)

Let there be another proposition *C*, such that

If *C* is true, then if *A* is true *B* is not true. (ii)

274 (ii) amounts to this,—

If *C* is true, then (i) is not true.

But, *ex hypothesi*, (i) is true.

Therefore, *C* cannot be true; for the assumption of *C* involves an absurdity.

Examine this argument. [LEWIS CARROLL.]

[If the problem in section 184 is regarded as a problem in conditionals, this is the corresponding problem in hypotheticals.]

186. Assuming that rain
never falls in Upper Egypt, are the following genuine pairs of
contradictories?

(*a*) The occurrence of rain in Upper Egypt is always
succeeded by an earthquake; the occurrence of rain in Upper Egypt is
sometimes not succeeded by an earthquake.

(*b*) If it is true that it rained in Upper Egypt on the 1st
of July, it is also true that an earthquake followed on the same day;
if it is true that it rained in Upper Egypt on the 1st of July, it is
not also true that an earthquake followed on the same day.

If the above are not true contradictories, suggest what should be
substituted. [B.]

187. Give the contrapositive
and the contradictory of each of the following propositions:

(1) If any nation prospers under a Protective System, its citizens
reject all arguments in favour of free-trade;

(2) If any nation prospers under a Protective System, we ought to
reject all arguments in favour of free-trade. [J.]

188. Examine the logical
relation between the two following propositions; and enquire whether
it is logically possible to hold (*a*) that both are true,
(*b*) that both are false: (i) If volitions are undetermined,
then punishments cannot rightly be inflicted; (ii) If punishments can
rightly be inflicted, then volitions are undetermined. [J.]

189. *The terms
Disjunctive and Alternative as applied to
Propositions.*—Propositions of the form *Either X or Y is
true* are ordinarily called *disjunctive*. It has been pointed
out, however, that two propositions are really *dis*joined when
it is denied that they are both true rather than when it is asserted
that one or other of them is true; and the term *alternative*, as
suggested by Miss Jones (*Elements of Logic*, p. 115), is
obviously appropriate to express the latter assertion. We should then
use the terms *conjunctive*, *disjunctive*,
*alternative*, *remotive*, for the four following
combinations respectively: *X and Y are both true*, *X and Y
are not both true*, *Either X or Y is true*,295
*Neither X nor Y is true*.

^{295} Some writers indeed regard
the proposition *Either X or Y is true* as expressing a relation
between *X* and *Y* which is disjunctive in the above sense
as well as alternative; but the disjunctive character of this
proposition as regards *X* and *Y* is at any rate open to
dispute, whilst its alternative character is unquestionable (see
section 191).

Whilst, however, the name *alternative* is preferable to
*disjunctive* for the proposition *Either X or Y is true*,
the latter name has such an established position in logical
nomenclature that it seems inadvisable altogether to discontinue its
use in the old sense. It may be pointed out further that an
alternative contains a veiled disjunction (namely, between
*not-X* and *not-Y*) even in the stricter sense; for the
statement that *Either X or Y is true* is equivalent to the
statement that *Not-X and not-Y are not both true*. Hence,
although generally using the term *alternative*, I shall not
entirely discard the term *disjunctive* as synonymous with it.

276 190. *Two
types of Alternative Propositions*.—In the case of
propositions which are ordinarily described as simply disjunctive a
distinction must be drawn similar to that drawn in the preceding
chapter between conditionals and true hypotheticals. For the
alternatives may be events or combinations of properties one or other
of which it is affirmed will (always or sometimes) occur, *e.g.*,
*Every blood vessel is either a vein or an artery*, *Every
prosperous nation has either abundant natural resources or a good
government* ; or they may be propositions of independent import
whose truth or falsity cannot be affected by varying conditions of
time, space, or circumstance, and which must therefore be simply true
or false, *e.g.*, *Either there is a future life or many
cruelties go unpunished*, *Either it is no sin to covet honour or
I am the most offending soul alive*.

Any proposition belonging to the first of the above types may be
brought under the symbolic form *All* (*or some*) *S is
either P or Q*, and may, therefore, be regarded as an ordinary
categorical proposition with an *alternative term* as predicate.
It is usual and for some reasons convenient to defer the discussion of
the import of alternative terms until propositions of this type are
being dealt with. Such propositions might otherwise be dismissed after
a very brief consideration.296

^{296} It should be particularly
observed that although the proposition *Every S is P or Q* may be
said to state an alternative, it cannot be resolved into a true
alternative combination of *propositions*. Such a resolution is,
however, possible if the proposition (while remaining affirmative and
still having an alternative predicate) is *singular* or
*particular*: for example, *This S is P or* *Q* = *This* *S
is P or this S is Q* ; *Some S is P or* *Q* = *Some* *S is P or
some S is Q*.

Corresponding to this, we may note that an affirmative categorical
proposition with a *conjunctive* predicate is equivalent to a
conjunction of propositions if it is singular or universal, but not if
it is particular. Thus, *This S is P and Q* = *This S is P and
this S is Q* ; *All S is P and* *Q* = *All* *S is P and all S is
Q*. From the proposition *Some S is P and Q* we may indeed
infer *Some S is P and some S is Q* ; but we cannot pass back from
this conclusion to the premiss, and hence the two are not equivalent
to one another.

It may be added that a negative categorical proposition with an
alternative predicate cannot be said to state an alternative at all,
since to deny an alternation is the same thing as to affirm a
conjunction. Thus the proposition *No S is either P or Q* can
only be resolved into a *conjunctive* synthesis of propositions,
namely, *No S is P and no S is Q*.

277 Alternative
propositions of the second type are compound (as defined in section
55). They contain an alternative combination of propositions of
independent import: and they have for their typical symbolic form
*Either X is true or Y is true*, or more briefly, *Either X or
Y*, where *X* and *Y* are symbols representing
*propositions* (not terms). So far as it is necessary to give
them a distinctive name, they have a claim to be called *true*
alternative propositions, since they involve a true alternative
synthesis of *propositions*, and not merely an alternative
synthesis of terms.

It will be convenient to speak of *P* and *Q* as the
*alternants* of the alternative term *P or Q*, and of
*X* and *Y* as the *alternants* of the alternative
proposition *Either X or Y*.

191. *The Import of
Disjunctive* (*Alternative*) *Propositions*.—The two
main questions that arise in regard to the import of alternative
propositions are (1) whether the alternants of such propositions are
necessarily to be regarded as mutually exclusive, (2) whether the
propositions are to be interpreted as assertoric or modal.

(1) We ask then, in the first place, whether in an alternative
proposition the alternants are to be interpreted as formally exclusive
of one another; in other words, whether in the proposition *All S is
either A or B* it is necessarily (or formally) implied that no
*S* is both *A* and *B*,297 and whether in
the proposition *X is true or Y is true* it is necessarily (or
formally) implied that *X* and *Y* are not both true. It is
desirable to notice at the outset that the question is one of the
interpretation of a propositional form, and one that does not arise
except in connexion with the expression of judgments in language.
Hence the solution will be, at any rate partly, a matter of
convention.

^{297} This is an alternative
proposition of the first type, and the same question is raised by
asking whether the term *A* or *B* includes *AB* under
its denotation or excludes it; in other words, whether the denotation
of *A or B* is represented by the shaded portion of the first or
of the second of the following diagrams:

278 The following
considerations may help to make this point clearer. Let *X* and
*Y* represent two judgments. Then the following are two possible
states of mind in which we may be with regard to *X* and *Y*:

(*a*) we may know that one or other of them is true, and that
they are not both true;

(*b*) we may know that one or other of them is true, but may
be ignorant as to whether they are or are not both true.

Now whichever interpretation (exclusive or non-exclusive) of the
propositional form *X or Y* is adopted, there will be no
difficulty in expressing alternatively either state of mind. On the
exclusive interpretation, (*a*) will be expressed in the form
*X or Y*, (*b*) in the form *XY or XYʹ or XʹY*
(*Xʹ* representing the falsity of *X*, and *Yʹ*
the falsity of *Y*). On the non-exclusive interpretation,
(*a*) will be expressed in the form *XʹY or XYʹ*,
(*b*) in the form *X or Y*. There can, therefore, be no
intrinsic ground based on the nature of judgment itself why *X or
Y* *must* be interpreted in one of the two ways to the
exclusion of the other.

As then we are dealing with a question of the interpretation of a
certain form of expression, we must look for our solution partly in
the usages of ordinary language. We ask, therefore, whether in
ordinary speech we intend that the alternants in an alternative
proposition should necessarily be understood as excluding one
another?298 A very few instances will enable us to
decide in the negative. Take, for example, the proposition, “He
has either used bad text-books or he has been badly taught.” No
one would naturally understand this to exclude the possibility of a
combination of bad teaching and the use of bad text-books. Or suppose
it laid down as a 279
condition of eligibility for some appointment that every candidate
must be a member either of the University of Oxford, or of the
University of Cambridge, or of the University of London. Would anyone
regard this as implying the ineligibility of persons who happened to
be members of more than one of these Universities? Jevons (*Pure
Logic*, p. 68) instances the following proposition: “A peer
is either a duke, or a marquis, or an earl, or a viscount, or a
baron.” We do not consider this statement incorrect because many
peers as a matter of fact possess two or more titles. Take, again, the
proposition, “Either the witness is perjured or the prisoner is
guilty.” The import of this proposition, as it would naturally
be interpreted, is that the evidence given by the witness is
sufficient, supposing it is true, to establish the guilt of the
prisoner; but clearly there is no implication that the falsity of this
particular piece of evidence would suffice to establish the
prisoner’s innocence.

^{298} There are no doubt many
cases in which as a matter of fact we understand alternants to be
mutually exclusive. But this is not conclusive as shewing that even in
these cases the mutual exclusiveness is *intended to be
expressed* by the alternative proposition. For it will generally
speaking be found that in such cases the fact that the alternants
exclude one another is a matter of common knowledge quite
independently of the alternative proposition; as, for example, in the
proposition, *He was first or second in the race*. This point is
further touched upon in Part III, Chapter
6.

But it may be urged that this does not definitely settle the question of the best way of interpreting alternative propositions. Granted that in common speech the alternants may or may not be mutually exclusive, it may nevertheless be argued that in the use of language for logical purposes we should be more precise, and that an alternative statement should accordingly not be admitted as a recognised logical proposition except on the condition that the alternants mutually exclude one another.

We may admit that the argument from the ordinary use of speech is not final. But at any rate the burden of proof lies with those who advocate a divergence from the usage of everyday language; for it will not be denied that, other things being equal, the less logical forms diverge from those of ordinary speech the better. Moreover, condensed forms of expression do not conduce to clearness, or even ultimately to conciseness.299 280 For where our information is meagre, a condensed form is likely to express more than we intend, and in order to keep within the mark we must indicate additional alternatives. On this ground, quite apart from considerations of the ordinary use of language, I should support the non-exclusive interpretation of alternatives. The adoption of the exclusive interpretation would certainly render the manipulation of complex propositions much more complicated.

^{299} Obviously a disjunctive
proposition is a more condensed form of expression on the exclusive
than on the non-exclusive interpretation. Compare Mansel’s
*Aldrich*, p. 242, and *Prolegomena Logica*, p. 288.
“Let us grant for a moment the opposite view, and allow that the
proposition *All C is either A or B* implies as a condition of
its truth *No C can be both*. Thus viewed, it is in reality a
complex proposition, containing two distinct assertions, each of which
may be the ground of two distinct processes of reasoning, governed by
two opposite laws. Surely it is essential to all clear thinking that
the two should be separated from each other, and not confounded under
one form by assuming the Law of Excluded Middle to be, what it is not,
a complex of those of Identity and Contradiction”
(*Aldrich*, p. 242). It may be added that one paradoxical result
of the exclusive interpretation of alternatives is that *not either
P or Q* is not equivalent to *neither P nor Q*.

A further paradoxical result is pointed out by Mr G. R. T. Ross in
an article on the Disjunctive Judgment in *Mind* (1903, p. 492),
namely, that on the exclusive interpretation the disjunctives *A is
either B or C* and *A is either not B or not C* are identical
in their import; for in each case the real alternants are *B but not
C* and *C but not B*. Thus, to take an illustration borrowed
from Mr Ross, the two following propositions are (on the
interpretation in question) identical in their
import,—“Anyone who affirms that he has seen his own ghost
is either not sane or not telling what he believes to be the
truth,” “Anyone who affirms that he has seen his own ghost
is either sane or truthful.”

Mr Bosanquet and other writers who advocate the exclusive interpretation of disjunctives appear to have chiefly in view the expression in disjunctive form of a logical division or scientific classification. I should of course agree that such a division or classification is imperfect if the members of which it consists are not mutually exclusive as well as collectively exhaustive. This condition must also be satisfied when we make use of the disjunctive judgment in connexion with the doctrine of probability.300 It will, however, hardly be proposed to confine the disjunctive judgment to these uses. We frequently have occasion to state alternatives independently of any scientific classification or any calculation of probability; and we must not regard the bare form of the disjunctive judgment as expressing anything that we are not prepared to recognise as universally involved in its use.

^{300} In this connexion the
further condition of the “equality” in a certain sense of
the alternants has in addition to be satisfied.

It is of course always possible to express an alternative 281 statement in such a way
that the alternants are *formally* incompatible or exclusive.
Thus, not wishing to exclude the case of *A* being both *B*
and *C* we may write *A is B or bC* ;301 or, wishing to
exclude that case, *A is Bc or bC*. But in neither of these
instances can we say that the incompatibility of the alternants is
really given by the alternative proposition. It is a merely formal
proposition that *No A is both B and bC* or that *No A is both
Bc and bC*. The proposition *Every A is Bc or bC* does,
however, tell us that no *A* is both *B* and *C* ; and
when from our knowledge of the subject-matter it is obvious that we
are dealing with alternants that are mutually exclusive (and no doubt
this is a very frequent case), we have in the above form a means of
correctly and unambiguously expressing the fact. Where it is
inconvenient to use this form, it is open to us to make a separate
statement to the effect that *No A is both B and C*. All that is
here contended for is that the bare symbolic form *A is either B or
C* should not be interpreted as being equivalent to *A is either
Bc or bC*.

^{301} Where *b* =
*not-B*, and *c* = *not-C*. What is contained in this
paragraph is to some extent a repetition of what is given on page
278.

(2) We may pass on to consider the second main question that arises in connexion with the import of disjunctive (alternative) propositions, namely, whether such propositions are to be interpreted as modal or as merely assertoric.

In chapter 9 it was urged that the modal interpretation of the
typical hypothetical proposition *If A then C* must be regarded
as the more natural one, on the ground that we should not ordinarily
think it necessary to affirm the truth of *A* in order to
contradict the proposition, as would be necessary if it were
interpreted assertorically.302 Similarly the
enquiry as to how we should naturally contradict the typical
alternative propositions *Every S is either P or Q*, *Either X
or Y is true*, may help us in deciding upon the interpretation of
these propositions.

On the assertoric interpretation, the contradictories of the
propositions in question are *Some S is neither P nor Q*,
*Neither X nor Y is true* ; on the modal interpretation, they are
*An S need not be either P or Q*, *Possibly neither X nor Y is
true*. 282 There
can be no doubt that this last pair of propositions would not as a
rule be regarded as adequate to contradict the pair of alternatives;
and on this ground we may regard the assertoric interpretation of
alternatives as most in accordance with ordinary usage. There is also
some advantage in differentiating between hypotheticals and
alternatives by interpreting the former modally and the latter
assertorically, except in so far as a clear indication is given to the
contrary. It is not of course meant that modal alternatives are never
as a matter of fact to be met with or that they cannot receive formal
recognition; they can always be expressed in the distinctive forms
*Every S must be either P or Q*, *Either X or Y is necessarily
true*.

192. *Scheme of Assertoric
and Modal Propositions*.—By differentiating between forms of
propositions in the manner indicated in preceding sections we have a
scheme by which distinctive expression can be given to assertoric and
modal propositions respectively, whether they are simple or compound.

Thus the *categorical* form of proposition might be restricted to the
expression of *simple assertoric* judgments; the
*conditional* form to that of *simple modal* judgments; the
*disjunctive* (*alternative*)303 form to that of
*compound assertoric* judgments; and the *hypothetical* form
to that of *compound modal* judgments.

^{303} We are of course referring
here to disjunctive (alternative) propositions of the second type
only, alternative propositions of the first type being treated as
categoricals with alternative predicates. See section 190.

I have not in the present treatise attempted to adopt this scheme to the exclusion of other interpretations of the different propositional forms; but I have had it in view throughout, and I put it forward as a scheme the adoption of which might afford an escape from some ambiguities and misunderstandings.

193. *The Relation of
Disjunctive* (*Alternative*) *Propositions to Conditionals
and Hypotheticals*.—It may be convenient if we briefly
consider this question independently of the distinctions indicated in
the preceding section, the assumption being made that these different
types of propositions are interpreted either all assertorically or all
modally. On this assumption, alternative propositions are reducible to
the conditional or the true hypothetical form according to the type to
which they belong. Thus, 283 the proposition, “Every blood vessel
is either a vein or an artery,” yields the conditional,
“If any blood vessel is not a vein then it is an artery”;
the true compound alternative proposition, “Either there is a
future life or many cruelties go unpunished,” yields the true
hypothetical, “If there is no future life then many cruelties go
unpunished.”

It may be asked whether an alternative proposition does not require
a conjunction of *two* conditionals or hypotheticals in order fully to
express its import. This is not the case, however, on the view that
the alternants are not to be interpreted as necessarily exclusive. It
is true that even on this view an alternative proposition, such as
*Either X or Y*, is primarily reducible to two hypotheticals,
namely, *If not X then Y* and *If not Y then X*. But these
are contrapositives the one of the other, and therefore mutually
inferable. Hence the full meaning of the alternative proposition is
expressed by means of either of them.

On the exclusive interpretation, the alternative proposition
*Either X or Y* yields primarily four hypotheticals, namely,
*If X then not Y* and *If Y then not X* in addition to the
two given above. But these again are contrapositives the one of the
other. Hence the full import of the alternative proposition will now
be expressed by a conjunction of the two hypotheticals, *If X then
not Y* and *If not X then Y*.

This is denied by Mr Bosanquet, who holds that the disjunctive proposition yields a positive assertion not contained in either of the hypotheticals. “‘This signal light shews either red or green.’ Here we have the categorical element, ‘This signal light shews some colour,’ and on the top of this the two hypothetical judgments, ‘If it shews red it does not shew green,’ ‘If it does not shew red it does shew green.’ You cannot make it up out of the two hypothetical judgments alone; they do not give you the assertion that ‘it shews some colour.’”304 But surely the second of the two hypotheticals contains this implication quite as clearly and definitely as the disjunctive does.305

^{304} *Essentials of Logic*,
p. 124.

^{305} Mr Bosanquet’s
opinion that “the disjunction seems to complete the system of
judgments,” and that in some way it rises superior to other
forms of judgment, is apparently based on the view that it is by the
aid of the disjunctive judgment that we set forth the exposition of a
system with its various subdivisions. Apart, however, from the fact
that a disjunctive judgment does not necessarily contain such an
exposition, Mr Bosanquet’s doctrine appears to regard a
classification of some kind as representing the ideal of knowledge;
and this can hardly be allowed. We cannot, for example, regard the
classifications of such a science as botany as of equal importance
with the expressions of laws of nature, such as the law of universal
gravitation. And the ultimate laws on which all the sciences are based
are not expressed in the form of disjunctive propositions.

284 Returning to
the distinctions indicated in the preceding section, it is hardly
necessary to add that if the hypothetical *If not X then Y* is
interpreted modally, while the alternative *Either X or Y* is
interpreted assertorically, then the alternative can be inferred from
the hypothetical, but not *vice versâ*.

194. Shew how an alternative
proposition in which the alternants are not known to be mutually
exclusive (*e,g.*, *Either X or Y or Z is true*) may be
reduced to a form in which they necessarily are so. Write the new
proposition in as simple a form as possible. [K.]

195. Shew why the following
propositions are not contradictories: *Wherever A is present, B is
present and either C or B is also present* ; *In some cases where
A is present, either B or C or B is absent*. How must each of these
propositions in turn be amended in order that it may become the true
contradictory of the other? [K.]

196. *No P is both Q and
R*. Reduce this proposition (*a*) to the form of a conditional
proposition, (*b*) to the form of an alternative proposition.
Give the contradictory of the original proposition, of its conditional
equivalent, and of its alternative equivalent; and test your results
by enquiring whether the three contradictories thus obtained are
equivalent to one another. [K.]

197. *The Terms of the
Syllogism*.—A reasoning which consists of three propositions
of the traditional categorical form, and which contains three and only
three terms, is called a *categorical syllogism*.

Of the three terms contained in a categorical syllogism, two appear
in the conclusion and also in one or other of the premisses, while the
third appears in the premisses only. That which appears as the
predicate of the conclusion, and in one of the premisses, is called
the *major term* ; that which appears as the subject of the
conclusion, and in one of the premisses, is called the *minor
term* ;306 and that which appears in both the
premisses, but not in the conclusion (being that term by their
relations to which the mutual relation of the two other terms is
determined), is called the *middle term*.

^{306} The major and minor terms
are also sometimes called the *extremes* of the syllogism.

Thus, in the syllogism,—

All M is P, | |

All S is M, | |

therefore, | All S is P ; |

*S* is the minor term, *M* the
middle term, and *P* the major term.

286 These respective designations of the terms of a syllogism resulted from such a syllogism as that just given being regarded as typical. With the exception of the somewhat rare case in which the terms of a proposition are coextensive, the above syllogism may be represented by the following diagram. Here

clearly the major term is the largest in extent, and the minor the smallest, while the middle occupies an intermediate position.

But we have no guarantee that the same relation between the terms
of a syllogism will hold, when one of the premisses is negative or
particular. Thus, the syllogism—*No M is P*, *All S is
M*, therefore, *No S is P*—yields as one case

where the major term may be the smallest in extent, and the middle
the largest. Again, the syllogism—*No M is P*, *Some S is
M*, therefore, *Some S is not P*—yields as one case

where the major term may be the smallest in extent and the minor the largest.

Whilst, however, the middle term is not always a middle term in extent, it is always a middle term in the sense that by its means the two other terms are connected, and their mutual relation determined.

287 198. *The
Propositions of the Syllogism*.—Every categorical syllogism
consists of three propositions. Of these one is the *conclusion*.
The premisses are called the *major premiss* and the *minor
premiss* according as they contain the major term or the minor term
respectively.

Thus, | All M is P | (major premiss), |

All S is M | (minor premiss), | |

therefore, | All S is P | (conclusion). |

It is usual (as in the above syllogism) to state the major premiss
first and the conclusion last. This is, however, nothing more than a
convention. The order of the premisses in no way affects the validity
of a syllogism, and has indeed no logical significance, though in
certain cases it may be of some rhetorical importance. Jevons
(*Principles of Science*, 6, § 14) argues that the cogency of a
syllogism is more clearly recognisable when the minor premiss is
stated first. But it is doubtful whether any general rule of this kind
can be laid down. In favour of the traditional order, it is to be said
that in what is usually regarded as the typical syllogism (*All M is
P*, *All S is M*, therefore, *All S is P*) there is a
philosophical ground for stating the major premiss first, since that
premiss gives the general rule, of which the minor premiss enables us
to make a particular application.

199. *The Rules of the
Syllogism*.—The rules of the categorical syllogism as usually
stated are as follows:—

(1) *Every syllogism contains three and only three terms*.

(2) *Every syllogism consists of three and only three
propositions*.

These two so-called rules are not properly speaking rules for the
validity of an argument. They simply serve to *define* the
syllogism as a particular *form* of argument. A reasoning which does not
fulfil these conditions may be formally valid, but we do not call it a
syllogism.307 The four rules that follow 288 are really rules in the
sense that if, when we have got the reasoning into the form of a
syllogism, they are not fulfilled, then the reasoning is invalid.308

^{307} For example, *B is
greater than C*, *A is greater than B*, therefore, *A is
greater than C*.

Here is a valid reasoning which consists of three propositions. But
it contains more than three terms; for the predicate of the second
premiss is “greater than *B*,” while the subject of
the first premiss is “*B*.” It is, therefore, as it
stands, not a syllogism. Whether reasonings of this kind admit of
being reduced to syllogistic form is a problem which will be discussed
subsequently.

^{308} Apparent exceptions to
these rules will be shewn in sections 205 and 206 to result from the
attempt to apply them to reasonings which have not first been reduced
to syllogistic form.

(3) *No one of the three terms of a syllogism may be used
ambiguously; and the middle term must be distributed once at least in
the premisses*.

This rule is frequently given in the form: “The middle term
must be distributed once at least, and must not be ambiguous.”
But it is obvious that we have to guard against ambiguous major and
ambiguous minor as well as against ambiguous middle. The fallacy
resulting from the ambiguity of one of the terms of a syllogism is a
case of *quaternio terminorum*, that is, a fallacy of four terms.

The necessity of distributing the middle term may be illustrated by
the aid of the Eulerian diagrams. Given, for instance. *All P is
M* and *All S is M*, we may have any one of the five following
cases:—

Here all the five relations that are *à priori* possible
between *S* and *P* are still possible. We have, therefore,
no conclusion.

If in a syllogism the middle term is distributed in neither
premiss, we are said to have a fallacy of *undistributed middle*.

289 (4) *No term
may be distributed in the conclusion which was not distributed
in one of the premisses*.

The breach of this rule is called *illicit process of the
major*, or *illicit process of the minor*, as the case may be;
or, more briefly, *illicit major* or *illicit minor*.

(5) *From two negative premisses nothing can be inferred*.

This rule may, like rule 3, be very well illustrated by means of the Eulerian diagrams.

(6) *If one premiss is negative, the conclusion must be negative;
and to prove a negative conclusion, one of the premisses must be
negative*.309

^{309} This rule and the second
corollary given in the following section are sometimes combined into
the one rule, *Conclusio sequitur partem deteriorem* ;
*i.e.*, the conclusion follows the worse or weaker premiss both
in quality and in quantity, a negative being considered weaker than an
affirmative and a particular than a universal.

200. *Corollaries from the
Rules of the Syllogism.*—From the rules given in the
preceding section, three corollaries may be deduced:—310

^{310} The formulation of these
corollaries may in some cases help towards the more immediate
detection of unsound syllogisms.

(i) *From two particular premisses nothing can be inferred*.

Two particular premisses must be either

(*α*) both negative,

or (*β*) both affirmative,

or (*γ*) one negative and one affirmative.

But in case (*α*), no conclusion follows by rule 5.

In case (*β*), since no term can be distributed in two
particular affirmative propositions, the middle term cannot be
distributed, and therefore by rule 3 no conclusion follows.

In case (*γ*), if any valid conclusion is possible, it must
be negative (rule 6). The major term, therefore, will be distributed
in the conclusion; and hence we must have two terms distributed in the
premisses, namely, the middle and the major (rules 3, 4). But a
particular negative proposition and a particular affirmative
proposition between them distribute only one term. Therefore, no
conclusion can be obtained.

(ii) *If one premiss is particular, the conclusion must be
particular*.

290 We must have either

(*α*) two negative premisses, but this case is rejected
by rule 5;

or (*β*) two affirmative premisses;

or (*γ*) one affirmative and one negative.

In case (*β*) the premisses, being both affirmative and one
of them particular, can distribute but one term between them. This
must be the middle term by rule 3. The minor term is, therefore,
undistributed in the premisses, and the conclusion must be particular
by rule 4.

In case (*γ*) the premisses will between them distribute two
and only two terms. These must be the middle by rule 3, and the major
by rule 4 (since we have a negative premiss, necessitating by rule 6 a
negative conclusion, and therefore the distribution of the major term
in the conclusion). Again, therefore, the minor cannot be distributed
in the premisses, and the conclusion must be particular by rule 4.

De Morgan (*Formal Logic*, p. 14) gives the following proof of
this corollary:—“If two propositions *P* and *Q*
together prove a third *R*, it is plain that *P* and the
denial of *R* prove the denial of *Q*. For *P* and
*Q* cannot be true together without *R*. Now, if possible,
let *P* (a particular) and *Q* (a universal) prove *R*
(a universal). Then *P* (particular) and the denial of *R*
(particular) prove the denial of *Q*. But two particulars can
prove nothing.”311

^{311}Further attention will be
called in a later chapter to the general principle upon which this
proof is based. See section 264.

(iii) *From a particular major and a negative minor nothing can
be inferred*.

Since the minor premiss is negative, the major premiss must by rule
5 be affirmative. But it is also particular, and it therefore follows
that the major term cannot be distributed in it. Hence, by rule 4, it
must be undistributed in the conclusion, *i.e.*, the conclusion must be
*affirmative*. But also, by rule 6, since we have a negative
premiss, it must be *negative*. This contradiction establishes
the corollary that from the given premisses no conclusion can be
drawn.

The following mnemonic lines, attributed to Petrus Hispanus, 291 afterwards Pope John XXI., sum up the rules of the syllogism and the first two corollaries:

*Distribuas medium: nec quartus terminus adsit:
Utraque nec praemissa negans, nec particularis:
Sectetur partem conclusio deteriorem;
Et non distribuat, nisi cum praemissa, negetve*.

201. *Restatement of the
Rules of the Syllogism*.—It has been already pointed out that
the first two of the rules given in section 199 are to be regarded as a description of the syllogism rather than as rules for its validity.
Again, the part of rule 3 relating to ambiguity may be regarded as
contained in the proviso that there shall be only three terms; for, if
one of the terms is ambiguous, there are really four terms, and hence
no syllogism according to our definition of syllogism. The rules may,
therefore, be reduced to four; and they may be restated as
follows:—

A. *Two rules of distribution*:

(1) The middle term must be distributed once at least in the
premisses;

(2) No term may be distributed in the conclusion which was not
distributed in one of the premisses;

B. *Two rules of quality*:

(3) From two negative premisses no conclusion follows;

(4) If one premiss is negative, the conclusion must be negative;
and to prove a negative conclusion, one of the premisses must be
negative.312

^{312} The rules of quality might
also be stated as follows; To prove an affirmative conclusion, both
premisses must be affirmative; To prove a negative conclusion, one
premiss must be affirmative and the other negative.

202. *Dependence of the
Rules of the Syllogism upon one another*.—The four rules just
given are not ultimately independent of one another. It may be shewn
that a breach of the second, or of the third, or of the first part of
the fourth involves indirectly a breach of the first; or, again, that
a breach of the first, or of the third, or of the first part of the
fourth involves indirectly a breach of the second.

292 (i) *The
rule that two negative premisses yield no conclusion may be deduced
from the rule that the middle term must be distributed once at least
in the premisses*.

This is shewn by De Morgan (*Formal Logic*, p. 13). He takes
two universal negative premisses **E**, **E**. In whatever
figure they may be, they can be reduced by conversion to

*No P is M*,

*No S is M*.

Then by obversion they become (without losing any of their force),—

*All P is not-M*,

*All S is not-M* ;

and we have undistributed middle. Hence rule 3 is exhibited as a corollary from rule 1. For if any connexion between *S* and *P* can be
inferred from the first pair of premisses, it must also be inferable
from the second pair.

The case in which one of the premisses is particular is dealt with
by De Morgan as follows;—“Again, *No Y is X*, *Some Ys
are not Zs*, may be converted into

*Every X is* (*a thing
which is not Y*),*Some* (*things which are not Zs*)
*are Ys*,

in which there is no middle term.”

This is not satisfactory, since we may often exhibit a valid
syllogism in such a form that there appear to be four terms;
*e.g.*, *All M is P*, *All S is M*, may be reduced to
*All M is P*, *No S is not-M*, and there is now no middle term.

The case in question may, however, be disposed of by saying that if
we cannot infer anything from two negative premisses both of which are
universal, *à fortiori* we cannot from two negative premisses one
of which is particular.313

^{313} This argument holds good in
the special case under consideration even if we interpret particulars,
but not universals, as implying the existence of their subjects. For
the validity of the above proof that two universal negatives yield no
conclusion remains unaffected even if we allow to universals the
maximum of existential import.

(ii) *The rules that from two negative premisses nothing can be
inferred and that if one premiss is negative the conclusion must be
negative are mutually deducible from one another*.

The following proof that the second of these rules is deducible
from the first is suggested by De Morgan’s deduction of 293 the second corollary as
given in section 200. If two propositions *P* and *Q*
together prove a third *R*, it is plain that *P* and the
denial of *R* prove the denial of *Q*. For *P* and
*Q* cannot be true together without *R*. Now, if possible,
let *P* (a negative) and *Q* (an affirmative) prove *R*
(an affirmative). Then *P* (a negative) and the denial of
*R* (a negative) prove the denial of *Q*. But by hypothesis
two negatives prove nothing.

It may be shewn similarly that if we start by assuming the second of the rules then the first is deducible from it.

(iii) *Any syllogism involving directly an illicit process of
major or minor involves indirectly a fallacy of undistributed middle,
and vice versâ*.314

^{314} For this theorem and its
proof I am indebted to Mr Johnson.

Let *P* and *Q* be the premisses and *R* the
conclusion of a syllogism involving illicit major or minor, a term
*X* which is undistributed in *P* being distributed in
*R*. Then the contradictory of *R* combined with *P*
must prove the contradictory of *Q*. But any term distributed in
a proposition is undistributed in its contradictory. *X* is
therefore undistributed in the contradictory of *R*, and by
hypothesis it is undistributed in *P*. But *X* is the middle
term of the new syllogism, which is therefore guilty of the fallacy of
undistributed middle. It is thus shewn that any syllogism involving
directly a fallacy of illicit major or minor involves indirectly a
fallacy of undistributed middle.

Adopting a similar line of argument, we might also proceed in the opposite direction, and exhibit the rule relating to the distribution of the middle term as a corollary from the rule relating to the distribution of the major and minor terms.

203. *Statement of the
independent Rules of the Syllogism*.—The theorems
established in the preceding section shew that the first part of rule
4 (as given in section 201) is a corollary from rule 3, and that rule 3 is in its turn a corollary from rule 1; also that rules 1 and 2
mutually involve one another, so that either one of them may be
regarded as a corollary from the other. We are, therefore, left with
either rule 1 or rule 2 and also with the second part of rule 4; and
the independent rules of the syllogism may accordingly be stated as
follows: 294

(*α*) *Rule of Distribution*:—The middle term must be distributed
once at least in the premisses [**or**, as alternative with this,
No term may be distributed in the conclusion which was not distributed
in one of the premisses];

(*β*) *Rule of Quality*:—To prove a negative
conclusion one of the premisses must be negative.315

^{315} On examination it will be
found that the only syllogism rejected by this rule and not also
rejected directly or indirectly by the preceding rule is the
following:—*All P is M*, *All M is S*, therefore,
*Some S is not P*. In the technical language explained in the
following chapter, this is **AAO** in figure 4. So far, therefore,
as the first three figures are concerned, we are left with a single
rule, namely, a rule of distribution, which may be stated in either of
the alternative forms given above.

It should be clearly understood that it is not meant that every
invalid syllogism will offend *directly* against one of these two
rules. As a direct test for the detection of invalid syllogisms we
must still fall back upon the *four* rules given in section
201.316 All that we have succeeded in shewing is
that ultimately these four rules are not independent of one another.

^{316} If, for example, for our
rule of distribution we select the rule relating to the distribution
of the middle term, then the invalid syllogism,

All M is P, | |

No S is M, | |

therefore, | No S is P, |

does not directly involve a breach of either of our two independent rules. But if this syllogism is valid, then must also the following syllogism be valid:

All M is P (original major), | |

Some S is P (contradictory
of original conclusion), | |

therefore | Some S is M (contradictory
of original minor); |

and here we have undistributed middle. Hence the
rule relating to the distribution of the middle term establishes
*indirectly* the invalidity of the syllogism in question. The principle
involved is the same as that on which we shall find the process of
indirect reduction to be based.

Take, again, the syllogism: *PaM*, *SeM*, ∴
*SaP*. This does not directly offend against the rules given
above; but the reader will find that its validity involves the
validity of another syllogism in which a direct transgression of these
rules occurs.

204. *Proof of the Rule of
Quality.*—For the following very interesting and ingenious
proof of the Rule of Quality (as stated in the preceding section) I am
indebted to Mr R. A. P. Rogers, of Trinity College, Dublin. In this
proof the symbol *f _{n}*( ) is used to denote the form of
a proposition, the terms which the 295 proposition contains in any given case
being inserted within the brackets. Thus, if

Let *f*_{1}( ), *f*_{2}( ),
*f*_{3}( ) be propositions belonging to the traditional
schedule. Then “*f*_{1}(*P*, *M*),
*f*_{2}(*S*, *M*), ∴
*f*_{3}(*S*, *P*)” will be the expression
of a syllogism; and, since the syllogism is a process of formal
reasoning, if the above syllogism is valid in any case, it will hold
good if other terms are substituted for *S*, *M*, *P*
(or any of them). Thus, substituting *S* for *M*, and
*S* for *P*, if “*f*_{1}(*P*,
*M*), *f*_{2}(*S*, *M*), ∴
*f*_{3}(*S*, *P*)” is a valid syllogism,
then “*f*_{1}(*S*, *S*),
*f*_{2}(*S*, *S*), ∴
*f*_{3}(*S*, *S*)” will be a valid
syllogism.

It follows, by contraposition, that if
“*f*_{1}(*S*, *S*),
*f*_{2}(*S*, *S*), ∴
*f*_{3}(*S*, *S*)” is an invalid
syllogism, then “*f*_{1}(*P*, *M*),
*f*_{2}(*S*, *M*), ∴
*f*_{3}(*S*, *P*)” will be an invalid
syllogism.

If possible, let *f*_{1}( ) and *f*_{2}(
) be affirmative, while *f*_{3}( ) is negative. Then
*f*_{1}(*S*, *S*) and
*f*_{2}(*S*, *S*) will be formally true
propositions, while *f*_{3}(*S*, *S*) is
formally false. Hence *f*_{3}(*S*, *S*) cannot
be a valid inference from *f*_{1}(*S*, *S*) and
*f*_{2}(*S*, *S*); in other words,
“*f*_{1}(*S*, *S*),
*f*_{2}(*S*, *S*), ∴
*f*_{3}(*S*, *S*)” must be an invalid
syllogism. Consequently, “*f*_{1}(*P*.
*M*), *f*_{2}(*S*, *M*), ∴
*f*_{3}(*S*, *P*)” cannot be a valid
syllogism; that is, we cannot have a valid syllogism in which both
premisses are affirmative and the conclusion negative.

205. *Two negative
premisses may yield a valid conclusion; but not
syllogistically*.—Jevons remarks: “The old rules of
logic informed us that from two negative premisses no conclusion could
be drawn, but it is a fact that the rule in this bare form does not
hold universally true; and I am not aware that any precise explanation
has been given of the conditions under which it is or is not
imperative. Consider the following example,—*Whatever is not
metallic is not capable of powerful magnetic influence*, *Carbon
is not metallic*, therefore, *Carbon is not capable of powerful
magnetic influence*. Here we have two distinctly negative
premisses, and yet they yield a perfectly 296 valid negative conclusion. The syllogistic
rule is actually falsified in its bare and general statement”
(*Principles of Science*, 4, § 10).317

^{317} Lotze (*Logic*, § 89;
*Outlines of Logic*, §§ 40-42) holds that two negative premisses
invalidate a syllogism in figure 1 or figure 2, but not necessarily in
figure 3. The example upon which he relies is this,—*No M is
P*, *No M is S*, therefore, *Some not-S is not P*. The
argument in the text may be applied to this example as well as to the
one given by Jevons.

This apparent exception is, however, no real exception. The
reasoning (which may be expressed symbolically in the form, *No
not-M is P*, *No S is M*, therefore, *No S is P*) is
certainly valid; but if we regard the premisses as negative it has
four terms *S*, *P*, *M*, and *not-M*, and is
therefore no syllogism. Reducing it to syllogistic form, the minor
becomes by obversion *All S is not-M*, an affirmative
proposition.318 It is not the case, therefore, that we have
succeeded in finding a valid *syllogism* with two negative
premisses. In other words, while we must not say that from two
negative premisses nothing follows, it remains true that if a
syllogism regularly expressed has two negative premisses it is
invalid.319 It must not be considered that this is a
mere technicality, and that Jevons’s example shews that the rule
is at any rate of no practical value. It is not possible to formulate
specific rules at all except with reference to some defined form of
reasoning; and no given rule is vitiated either 297 theoretically or for
practical purposes because it does not apply outside the form to which
alone it professes to apply.320

^{318} It may be added that it is
in this form that the cogency of the argument is most easily to be
recognised. Of course every affirmation involves a denial and *vice
versâ* ; but it may fairly be said that in Jevons’s example
the primary force of the minor premiss, considered in connexion with
the major premiss, is to affirm that carbon belongs to the class of
non-metallic substances, rather than to deny that it belongs to the
class of metallic substances.

^{319} By a syllogism regularly
expressed we mean a reasoning consisting of three propositions, which
not only contain between them three and only three terms, but which
are also expressed in the traditional categorical forms. Attention
must be called to this because, if we introduce additional
propositional forms of the kind indicated on page 146, we may have a valid reasoning with two negative premisses, which satisfies the
condition of containing only three terms; for example,

No M is P, | |

Some M is not S, | |

therefore, | There is something
besides S and P. |

It will be found that this reasoning is easily reducible to a valid
syllogism in *Ferison*.

^{320} A case similar to that
adduced by Jevons is dealt with in the *Port Royal Logic*
(Professor Baynes’s translation, p. 211) as
follows:—“There are many reasonings, of which all the
propositions appear negative, and which are, nevertheless, very good,
because there is in them one which is negative only in appearance, and
in reality affirmative, as we have already shewn, and as we may still
further see by this example: *That which has no parts cannot perish
by the dissolution of its parts; The soul has no parts; therefore, The
soul cannot perish by the dissolution of its parts*. There are
several who advance such syllogisms to shew that we have no right to
maintain unconditionally this axiom of logic, *Nothing can be
inferred from pure negatives* ; but they have not observed that, in
sense, the minor of this and such other syllogisms is affirmative,
since the middle, which is the subject of the major, is in it the
attribute. Now the subject of the major is not that which has parts,
but that which has not parts, and thus the sense of the minor is,
*The soul is a thing without parts*, which is a proposition
affirmative of a negative attribute.” Ueberweg also, who himself
gives a clear explanation of the case, shews that it was not
overlooked by the older logicians; and he thinks it not improbable
that the doctrine of qualitative aequipollence between two judgments
(*i.e.*, obversion) resulted from the consideration of this very
question (*System of Logic*, § 106). Compare, further,
Whately’s treatment of the syllogism, “No man is happy who
is not secure; no tyrant is secure; therefore, no tyrant is
happy” (*Logic*, II. 4, §
7).

The truth is that by the aid of the process of obversion the
premisses of *every* valid syllogism may be expressed as
negatives, though the reasoning will then no longer be technically in
the form of a syllogism; for example, the propositions which
constitute the premisses of a syllogism in *Barbara*—*All
M is P*, *All S is M*, therefore, *All S is P*—may
be written in a negative form, thus, *No M is not-P*, *No S is
not-M*, and the conclusion *All S is P* still follows.

206. *Other apparent
exceptions to the Rules of the Syllogism*.—It is curious that
the logicians who have laid so much stress on the case considered in
the preceding section do not appear to have observed that, as soon as
we admit more than three terms, other apparent breaches of the
syllogistic rules may occur in what are perfectly valid reasonings.
Thus, the premisses *All P is M* and *All S is M*, in which
*M* is not distributed, yield the conclusion *Some not-S is
not-P;*321 and 298 hence we might argue that undistributed
middle does not invalidate an argument. Again, from the premisses
*All M is P*, *All not-M is S*, we may infer *Some S is
not P*,322 although there is apparently an illicit
process of the major. It is unnecessary after what has been said in
the preceding section to give examples of valid reasonings in which we
have a negative premiss with an affirmative conclusion, or two
affirmative premisses with a negative conclusion, or a particular
major with a negative minor. Any valid syllogism which is affirmative
throughout will yield the first and, if it has a particular major,
also the last of these by the obversion of the minor premiss, and the
second by the obversion of the conclusion. The only syllogistic rules,
indeed, which still hold good when more than three terms are admitted
are the rule providing against illicit minor and the first two
corollaries.

^{321} By the contraposition of
both premisses this reasoning is reduced to the valid syllogistic
form, *All not-M is not-P*, *All not-M is not-S*, therefore,
*Some not-S is not-P*.

^{322} By the inversion of the
first premiss, this reasoning is reduced to the valid syllogistic
form, *Some not-M is not P*, *All not-M is S*, therefore,
*Some S is not P*. Compare section 104.

But of course none of the above examples really invalidate the syllogistic rules; for these rules have been formulated solely with reference to reasonings of a certain form, namely, those which contain three and only three terms. In every case the reasoning inevitably conforms to the rule which it appears to violate, as soon as, by the aid of immediate inferences, the superfluous number of terms has been eliminated.

207. *Syllogisms with two
singular premisses*.—Bain (*Logic*, *Deduction*, p.
159) argues that an apparent syllogism with two singular premisses
cannot be regarded as a genuine syllogistic or deductive inference;
and he illustrates his view by reference to the following syllogism:

Socrates fought at Delium, | |

Socrates was the master of Plato, | |

therefore, | The master of Plato fought at Delium. |

The argument is that “the proposition ‘Socrates was the
master of Plato and fought at Delium,’ compounded out of the two
premisses, is nothing more than a grammatical abbreviation,”
whilst the step hence to the conclusion is a mere omission of
something that had previously been said. “Now, we never 299 consider that we have
made a real inference, a step in advance, when we repeat *less* than we
are entitled to say, or drop from a complex statement some portion not
desired at the moment. Such an operation keeps strictly within the
domain of Equivalence or Immediate Inference. In no way, therefore,
can a syllogism with two singular premisses be viewed as a genuine
syllogistic or deductive inference.”

This argument leads up to some interesting considerations, but it proves too much. In the following syllogisms the premisses may be similarly compounded together:

All men are mortal, | ⎱ | All men are mortal and rational ; |

All men are rational, | ⎰ | |

therefore, Some rational beings are mortal. | ||

All men are mortal, | ⎱ | All men including kings are mortal ; |

All kings are men, | ⎰ | |

therefore, All kings are mortal.323 |

^{323} Compare with the above the
following syllogism which has two singular premisses:—The Lord
Chancellor receives a higher salary than the Prime Minister; Lord
Herschell is the Lord Chancellor; therefore, Lord Herschell receives a
higher salary than the Prime Minister. These premisses would
presumably be compounded by Bain into the single proposition,
“The Lord Chancellor, Lord Herschell, receives a higher salary
than the Prime Minister.”

Do not Bain’s criticisms apply to these syllogisms as much as to the syllogism with two singular premisses? The method of treatment adopted is indeed particularly applicable to syllogisms in which the middle term is subject in both premisses. But we may always combine the two premisses of a syllogism in a single statement, and it is always true that the conclusion of a syllogism contains a part of, and only a part of, the information contained in the two premisses taken together; hence we may always get Bain’s result.324 In other words, in the conclusion of every syllogism “we repeat less than we are entitled to say,” or, if we care to put it so, “drop from a complex statement some portion not desired at the moment.”

^{324} It may be pointed out that
the general method adopted by Boole in his *Laws of Thought* is
to sum up all his given propositions in a single proposition, and then
eliminate the terms that are not required. Compare also the methods
employed in Appendix C of the present work.

300 208. *Charge
of incompleteness brought against the ordinary syllogistic
conclusion*.—This charge (a consideration of which will
appropriately supplement the discussion contained in the preceding
section) is brought by Jevons (*Principles of Science*, 4, § 8)
against the ordinary syllogistic conclusion. The premisses
*Potassium floats on water*, *Potassium is a metal* yield,
according to him, the conclusion *Potassium metal is potassium
floating on water*. But “Aristotle would have inferred that
*some metals float on water*. Hence Aristotle’s conclusion
simply *leaves out some of the information afforded in the
premisses* ; it even leaves us open to interpret the *some
metals* in a wider sense than we are warranted in doing.”

In reply to this it may be remarked: first, that the Aristotelian
conclusion does not profess to sum up the whole of the information
contained in the premisses of the syllogism; secondly, that *some* must
here be interpreted to mean merely “not none,” “one
at least.” The conclusion of the above syllogism might perhaps
better be written “some metal floats on water,” or
“some metal or metals &c.” Lotze remarks in criticism
of Jevons: “His whole procedure is simply a repetition or at the
outside an addition of his two premisses; thus it merely adheres to
the given facts, and such a process has never been taken for a
*Syllogism*, which always means a movement of thought that uses
what is given for the purpose of advancing beyond it…… The meaning of
the Syllogism, as Aristotle framed it, would in this case be that the
occurrence of a floating metal Potassium proves that the property of
being so light is not incompatible with the character of metal in
general” (*Logic*, II. 3, note). This criticism is perhaps pushed a little too far. It is hardly a fair description of
Jevons’s conclusion to say that it is the mere sum of the
premisses; for it brings out a relation between two terms which was
not immediately apparent in the premisses as they originally stood.
Still there can be no doubt that the elimination of the middle term is
the very gist of syllogistic reasoning as ordinarily understood.

It may be added, as an *argumentum ad hominem* against Jevons,
that his own conclusion also leaves out some of the information
afforded in the premisses. For we cannot pass 301 back from the proposition *Potassium
metal is potassium floating on water* to either of the original
premisses.

209. *The connexion
between the Dictum de omni et nullo and the ordinary Rules of the
Syllogism*.—The *dictum de omni et nullo* was given by
Aristotle as the axiom on which all syllogistic inference is based. It
applies directly, however, to those syllogisms only in which the major
term is predicate in the major premiss, and the minor term subject in
the minor premiss (*i.e.*, to what are called syllogisms in figure 1).
The rules of the syllogism, on the other hand, apply independently of
the position of the terms in the premisses. Nevertheless, it is
interesting to trace the connexion between them. It will be found that
all the rules are involved in the *dictum*, but some of them in a
less general form, in consequence of the distinction just pointed out.

The *dictum* may be stated as follows:—“Whatever
is predicated, whether affirmatively or negatively, of a term
distributed may be predicated in like manner of everything contained
under it.”

(1) The *dictum* provides for three and only three terms;
namely, (i) a certain term which must be distributed, (ii) something
predicated of this term, (iii) something contained under it. These
terms are respectively the middle, major, and minor. We may consider
the rule relating to the ambiguity of terms to be also contained here,
since if any term is ambiguous we have practically more than three terms.

(2) The *dictum* provides for three and only three
propositions; namely, (i) a proposition predicating something of a
term distributed, (ii) a proposition declaring something to be
contained under this term, (iii) a proposition making the original
predication of the contained term. These propositions constitute
respectively the major premiss, the minor premiss, and the conclusion,
of the syllogism.

(3) The *dictum* prescribes not merely that the middle term
shall be distributed once at least in the premisses, but more
definitely that it shall be distributed in the *major*
premiss,—“Whatever is predicated of a term
*distributed*.”325

^{325} This is another form of
what will be found to be a special rule of figure 1, namely, that the
major premiss must be universal. Compare section 244.

302 (4) Illicit
process of the major is provided against indirectly. This fallacy can
be committed only when the conclusion is negative; but the words
“in like manner” declare that if there is a negative
conclusion, the major premiss must also be negative; and since in any
syllogism to which the *dictum* directly applies, the major term
is predicate of this premiss, it will be distributed in its premiss as
well as in the conclusion. Illicit process of the minor is provided
against inasmuch as the *dictum* warrants us in making our
predication in the conclusion only of what has been shewn in the minor
premiss to be contained under the middle term.

(5) The proposition declaring that something is contained under the
term distributed must necessarily be an affirmative proposition. The
*dictum* provides, therefore, that the premisses shall not both
be negative.326

^{326} It really provides that the
*minor* premiss shall be affirmative, which again is one of the
special rules of figure 1.

(6) The words “in like manner” clearly provide against
a breach of the rule that if one premiss is negative, the conclusion
must be negative, and *vice versâ*.

^{327} The following exercises may
be solved without any knowledge beyond what is contained in the
preceding chapter, the assumption however being made that if no rule
of the syllogism as given in section 199 or section 201 is broken, then the syllogism is valid.

210. If *P* is a mark
of the presence of *Q*, and *R* of that of *S*, and if
*P* and *R* are never found together, am I right in
inferring that *Q* and *S* sometimes exist separately? [V.]

The premisses may be stated as follows:

*All P is Q*,

*All R is S*,

*No P is R* ;

and in order to establish the desired conclusion we must be able to
infer at least one of the following,—*Some Q is not S*,
*Some S is not Q*.

But neither of these propositions can be inferred; for they
distribute respectively *S* and *Q*, and neither of these
terms is distributed in the given premisses. The question is,
therefore, to be answered in the negative.

303 211. If it be known concerning a syllogism in the Aristotelian system that the middle term is distributed in both premisses, what can we infer as to the conclusion? [C.]

If both premisses are affirmative, they can between them distribute only two terms, and by hypothesis the middle term is
distributed twice in the premisses; hence the minor term cannot be
distributed in the premisses, and it follows that the conclusion must
be particular.

If one of the premisses is negative, there may be three distributed
terms in the premisses; these must, however, be the middle term twice
(by hypothesis) and the major term (since the conclusion must now be
negative and will therefore distribute the major term); hence the
minor term cannot be distributed in the premisses, and it again
follows that the conclusion must be particular.

But either both premisses will be affirmative, or one affirmative
and the other negative; in any case, therefore, we can infer that the
conclusion will be particular.

212. Shew *directly* in
how many ways it is possible to prove the conclusions *SaP*,
*SeP* ; point out those that conform immediately to the *Dictum
de omni et nullo* ; and exhibit the equivalence between these and
the remainder. [W.]

(1) To prove *All S is P*.

Both premisses must be affirmative, and both must be universal.

*S* being distributed in the conclusion must be distributed in
the minor premiss, which must therefore be *All S is M*.

*M* not being distributed in the minor must be distributed in
the major, which must therefore be *All M is P*.

*SaP* can therefore be proved in only one way, namely,

All M is P, | |

All S is M, | |

therefore, | All S is P ; |

and this
syllogism conforms immediately to the *Dictum*.

(2) To prove *No S is P*.

Both premisses must be universal, and one must be negative while
the other is affirmative; *i.e.*, one premiss must be *E*
and the other *A*.

*First*, let the major be *E*, *i.e.*, either *No
M is P* or *No P is M*. In each case the minor must be
affirmative and must distribute *S* ; therefore, it will be *All
S is M*.

304 *Secondly*, let
the minor be *E*, *i.e.*, either *No S is M* or *No M
is S*. In each case the major must be affirmative and must
distribute *P* ; therefore, it will be *All P is M*.

We can then prove *SeP* in four ways, thus,—

(i) | MeP, |
(ii) | PeM, |
(iii) | PaM, |
(iv) | PaM, |

SaM, | SaM, | SeM, | MeS, | ||||

⎯⎯ | ⎯⎯ | ⎯⎯ | ⎯⎯ | ||||

SeP. | SeP. | SeP. | SeP. |

Of these, (i) only conforms immediately to the *dictum*, and
we have to shew the equivalence between it and the others.

The only difference between (i) and (ii) is that the major premiss
of the one is the simple converse of the major premiss of the other;
they are, therefore, equivalent. Similarly the only difference between
(iii) and (iv) is that the minor premiss of the one is the simple
converse of the minor premiss of the other; they are, therefore,
equivalent.

Finally, we may shew that (iv) is equivalent to (i) by transposing
the premisses and converting the conclusion.

213. Given that the major term is distributed in the premisses and undistributed in the conclusion of a valid syllogism, determine the syllogism. [C.]

Since the major term is undistributed in the conclusion,
the conclusion—and, therefore, both premisses—must be
affirmative. Hence, in order to distribute *P*, the major premiss
must be *PaM* ; and in order to distribute *M* (which is not
distributed in the major premiss), the minor premiss must be
*MaS*. It follows that the syllogism must be

All P is M, | |

All M is S, | |

therefore, | Some S is P. |

214. Prove that if three
propositions involving three terms (each of which occurs in two of the
propositions) are together incompatible, then (*a*) each term is
distributed at least once, and (*b*) one and only one of the
propositions is negative.

Shew that these rules are equivalent to the rules of the syllogism.
[J.]

No two of the propositions can be formally incompatible
with one another, since they do not contain the same terms. But each
pair must be incompatible with the third, *i.e.*, the
contradictory of any one must be deducible from the other two. It
follows that 305 we
shall have three valid syllogisms, in which the given propositions
taken in pairs are the premisses, whilst the contradictory of the
third proposition is in each case the conclusion.328

Then (*a*) *each term must be distributed once at
least*. For if any one of the terms failed to be distributed at
least once, we should obviously have undistributed middle in one of
our syllogisms; and (since a term undistributed in a proposition is
distributed in its contradictory) illicit major or minor in the two
others. If, however, the above condition is fulfilled, it is clear
that we cannot have either undistributed middle, or illicit major or
minor. Hence rule (*a*) is equivalent to the syllogistic rules
relating to the distribution of terms.

Again, (*b*) *one of the propositions must be negative, but
not more than one of them can be negative*. For if all three were
affirmative, then (since the contradictory of an affirmative is
negative) we should in each of our syllogisms infer a negative from
two affirmatives; and if two were negative, we should have two
negative premisses in one of our syllogisms, and (since the
contradictory of a negative is affirmative) an affirmative conclusion
with a negative premiss in each of the others. If, however, the above
condition is fulfilled, it is clear that we cannot have either two
negative premisses, or two affirmative premisses with a negative
conclusion, or a negative premiss with an affirmative conclusion.
Hence rule (*b*) is equivalent to the syllogistic rules relating
to quality.

^{328} Every syllogism involves two others, in each of
which one of the original premisses combined with the contradictory of
the conclusion proves the contradictory of the other original premiss.
Hence the three syllogisms referred to in the text mutually involve
one another. Compare sections 264, 265.

215. Explain what is meant
by a *syllogism* ; and put the following argument into syllogistic
form:—"We have no right to treat heat as a substance, for it may
be transformed into something which is not heat, and is certainly not
a substance at all, namely, mechanical work.” [N.]

216. Put the following argument into syllogistic form:—How can anyone maintain that pain is always an evil, who admits that remorse involves pain, and yet may sometimes be a real good? [V.]

306 217. It has
been pointed out by Ohm that reasoning to the following effect occurs
in some works on mathematics:—“A magnitude required for
the solution of a problem must satisfy a particular equation, and as
the magnitude *x* satisfies this equation, it is therefore the
magnitude required.” Examine the logical validity of this
argument. [C.]

218. Obtain a conclusion
from the two negative premisses,—*No P is M*, *No S is
M*. [K.]

219. If it is false that the
attribute *B* is ever found coexisting with *A*, and not
less false that the attribute *C* is sometimes found absent from
*A*, can you assert anything about *B* in terms of *C*?
[C.]

220. Give examples (in
symbols—taking *S*, *M*, *P*, as minor, middle,
and major terms, respectively) in which, attempting to infer a
universal conclusion where we have a particular premiss, we commit
respectively one but one only of the following fallacies,—(*a*) undistributed middle, (*b*) illicit major, (*c*)
illicit minor. Give also an example in which, making the same attempt,
we commit none of the above fallacies. [K.]

221. Can an apparent syllogism break directly all the rules of the syllogism at once? [K.]

222. Can you give an instance of an invalid syllogism in which the major premiss is universal negative, the minor premiss affirmative, and the conclusion particular negative? If not, why not? [K.]

223. Shew that

(i) If both premisses of a syllogism are affirmative, and one but
only one of them universal, they will between them distribute only one
term;

(ii) If both premisses are affirmative and both universal, they
will between them distribute two terms;

(iii) If one but only one premiss is negative, and one but only one
premiss universal, they will between them distribute two terms;

(iv) If one but only one premiss is negative, and both premisses
are universal, they will between them distribute three terms. [K.]

224. Ascertain how many distributed terms there may be in the premisses of a syllogism more than in the conclusion. [L.]

225. If the minor premiss of a syllogism is negative, what do you know about the position of the terms in the major? [O’S.]

307 226. If the major term of a syllogism is the predicate of the major premiss, what do you know about the minor premiss? [L.]

227. How much can you tell
about a valid syllogism if you know (1) that only the middle term is
distributed;

(2) that only the middle and minor terms are distributed;

(3) that all three terms are distributed? [W.]

228. What can be determined respecting a valid syllogism under each of the following conditions: (1) that only one term is distributed, and that only once; (2) that only one term is distributed, and that twice; (3) that two terms only are distributed, each only once; (4) that two terms only are distributed, each twice? [L.]

229. Two propositions are
given having a term in common. If they are **I** and **A**, shew
that either no conclusion or two can be deduced; but if **I** and
**E**, always and only one. [T.]

230. Find out, from the rules of the syllogism, what are the valid forms of syllogism in which the major premiss is particular affirmative. [J.]

231. Given (*a*) that
the major premiss, (*b*) that the minor premiss, of a valid
syllogism is particular negative, determine in each case the
syllogism. [K.]

232. Given that the major premiss of a valid syllogism is affirmative, and that the major term is distributed both in premisses and conclusion, while the minor term is undistributed in both, determine the syllogism. [N.]

233. Shew *directly* in
how many ways it is possible to prove the conclusions *SiP*,
*SoP*. [W.]

234. Shew that if the rule that a negative conclusion requires a negative premiss be omitted from the general rules of the syllogism, the only invalid syllogism thereby admitted is such that, if its conclusion be false whilst its premisses are true, the three terms of the syllogism must be absolutely coextensive. [O’S.]

235. Find, by direct application of the fundamental rules of syllogism, what are the valid forms of syllogism in which neither of the premisses is a universal proposition having the same quality as the conclusion. [J.]

308 236. In what
cases will contradictory major premisses both yield conclusions when
combined with the same minor?

How are the conclusions related?

Shew that in no case will contradictory minor premisses both yield
conclusions when combined with the same major. [O’S.]

237. (*a*) All just
actions are praiseworthy; (*b*) No unjust actions are expedient;
(*c*) Some inexpedient actions are not praiseworthy; (*d*)
Not all praiseworthy actions are inexpedient.

Do (*c*) and (*d*) follow from (*a*) and (*b*)?
[K.]

238. Reduce the following
arguments to ordinary syllogistic form:

(i) *No M is S*, *Whatever is not M is P*, therefore,
*All S is P* ;

(ii) *It cannot be that no not-S is P*, for *some M is P* and *no M is S* ;

(iii) It is impossible for the three propositions, *All M is
P*, *Anything that is not M is not S*, *Some things that are
not P are S*, all to be true together;

(iv) *Everything is M or P*, *Nothing is both S and M*,
therefore, *All S is P*. [K.]

239. Shew that the following
syllogisms break directly or indirectly all the rules of the
syllogism:

(1) *All P is M*, *All S is M*, therefore, *Some S is
not P* ;

(2) *All M is P*, *All M is S*, therefore, *No S is
P*. [K.]

[The so-called rules that every syllogism contains three and only three terms, and that every syllogism consists of three and only three propositions, are not here included under the rules of the syllogism.]

240. In a circular argument
involving two valid syllogisms, *Q* and *U* are used as
premisses to prove *R*, while *R* and *V* are used as
premisses to prove *Q* ; shew that *U* and *V* must be a
pair of complementary propositions, *i.e.*, of the forms *All M
is N* and *All N is M* respectively. [J.]

241. Shew that if two valid syllogisms have a common premiss while the other premisses are contradictories, both the conclusions must be particular. [K.]

242. Given the premisses of
a valid syllogism, examine in what cases it is (*a*) possible,
(*b*) impossible, to determine which is the minor term and which
the major term. [J.]

243. *Figure and
Mood*.—By the *figure* of a syllogism is meant the position of
the terms in the premisses. Denoting the major, middle, and minor
terms by the letters *P*, *M*, *S* respectively, and
stating the major premiss first, we have four figures of the syllogism
as shewn in the following table:—

Fig. 1. | Fig. 2. | Fig. 3. | Fig. 4. |

M – P | P – M | M – P | P – M |

S – M | S – M | M – S | M – S |

⎯⎯ | ⎯⎯ | ⎯⎯ | ⎯⎯ |

S – P | S – P | S – P | S – P |

By the *mood* of a syllogism is meant the quantity and quality
of the premisses and conclusion. For example, *AAA* is a mood in
which both the premisses and also the conclusion are universal
affirmatives; *EIO* is a mood in which the major is a universal
negative, the minor a particular affirmative, and the conclusion a
particular negative. It is clear that if figure and mood are both
given, the syllogism is given.

244. *The Special Rules of
the Figures; and the Determination of the Legitimate Moods in each
Figure*.329—It may first of all be shewn that
certain combinations of premisses are incapable of yielding a valid
conclusion in any figure. *A priori*, there are possible the
following sixteen different combinations of premisses, the major
premiss being always stated first:—*AA*, *AI*,
*AE*, *AO*, *IA*, *II*, *IE*, *IO*,
*EA*, *EI*, *EE*, *EO*, *OA*, *OI*,
*OE*, *OO*. Referring back, however, to the syllogistic
rules and corollaries (as given in sections 199, 200), we find that
*EE*, 310
*EO*, *OE*, *OO* (being combinations of negative
premisses) yield no conclusion by rule 5; that *II*, *IO*,
*OI* (being combinations of particular premisses) are excluded by
corollary i.; and that *IE* is excluded by corollary iii., which
tells us that nothing follows from a particular major and a negative
minor.

^{329} The method of determination
here adopted is only one amongst several possible methods. Another is
suggested, for example, in sections 212, 233.

We are left then with the following eight possible
combinations:—*AA*, *AI*, *AE*, *AO*,
*IA*, *EA*, *EI*, *OA* ; and we may go on to
enquire in which figures these will yield conclusions. In pursuing
this enquiry, special rules of the various figures may be determined,
which, taken together with the three corollaries established in
section 200, replace the general rules of distribution. These special rules, supplemented by the general rules of quality and the
corollaries,330 will enable the validity of the different
moods to be tested by a mere inspection of the form of the
propositions of which they consist.

^{330} The general rules of
quality and the corollaries can be directly applied without reference
to the position of the terms in the premisses of a syllogism. This is
not the case with the general rules of distribution. The object of the
special rules is, in the case of each particular figure, to substitute
for the general rules of distribution special rules of quantity and
quality.

*The special rules*331 *and the
legitimate moods of Figure* 1.

^{331} As indicated in section
209, the special rules of figure 1 follow immediately from the
*dictum de omni et nullo*.

The position of the terms in figure 1 is shewn thus,—

*M – PS – M*

⎯⎯

*S – P*

and it can
be deduced from the general rules of the syllogism that in this
figure:—

(1) *The minor premiss must be affirmative*. For if it were
negative, the major premiss would have to be affirmative by rule 5,
and the conclusion negative by rule 6. The major term would therefore
be distributed in the conclusion, and undistributed in its premiss;
and the syllogism would be invalid by rule 4.

(2) *The major premiss must be universal*. For the middle
term, being undistributed in the affirmative minor premiss, must be
distributed in the major premiss.

311 Rule (1) shews
that *AE* and *AO* and rule (2) that *IA* and
*OA*, yield no conclusions in this figure. We are accordingly
left with only four combinations, namely, *AA*, *AI*,
*EA*, *EI* From the rules that a particular premiss cannot
yield a universal conclusion or a negative premiss an affirmative
conclusion, while conversely a negative conclusion requires a negative
premiss, it follows further that *AA* will justify either of the
conclusions *A* or *I*, *EA* either *E* or
*O*, *AI* only *I*, *EI* only *O*. There are
then six moods in figure 1 which do not offend against any of the
rules of the syllogism,332 namely, *AAA*, *AAI*, *AII*,
*EAE*, *EAO*, *EIO*.

^{332} Rule (2) provides against
undistributed middle, and rule (1) against illicit major. We cannot
have illicit minor, unless we have a universal conclusion with a
particular premiss, and this also has been provided against.

Mr Johnson points out that the following symmetrical rules may be
laid down for the correct distribution of terms in the different
figures; and that these rules (three in each figure) taken together
with the *rules of quality* are sufficient to ensure that
*no* syllogistic rule is broken.

(i) To avoid undistributed middle: in figure 1, If the minor is affirmative, the major must be universal; in figure 4, If the major is affirmative, the minor must be universal; in figure 2, One premiss must be negative; in figure 3, One premiss must be universal. (The last of these rules is of course superfluous if the corollaries contained in section 200 are supposed given.)

(ii) To avoid illicit major: in figures 1 and 3, If the conclusion is negative, the major must be negative and, therefore, the minor affirmative; in figures 2 and 4, If the conclusion is negative, the major must be universal.

(iii) To avoid illicit minor: in figures 1 and 2, If the minor is particular, the conclusion must be particular; in figures 3 and 4, If the minor is affirmative, the conclusion must be particular. (The first of these two rules is again superfluous as a special rule if the corollaries are supposed given.)

The above rules are substantially identical with those given in the text.

The actual validity of these moods may be established by shewing
that the axiom of the syllogism, the *dictum de omni et nullo*,
applies to them; or by taking them severally and shewing that in each
case the cogency of the reasoning is self-evident.

*The special rules and the legitimate moods of Figure 2*.

The position of the terms in figure 2 is shewn thus,—

*P – MS – M*

⎯⎯

*S – P* ;

312 and its special rules (which the reader is recommended to deduce from the general rules of
the syllogism for himself) are,—

(1) *One premiss must be negative* ;

(2) *The major premiss
must be universal*.

The application of these rules again leaves six moods, namely,
*AEE*, *AEO*, *AOO*, *EAE*, *EAO*, *EIO*.

Recourse cannot now he had directly to the *dictum de omni et
nullo* in order to shew positively that these moods are legitimate.
It may, however, be shewn in each case that the cogency of the
reasoning is self-evident. The older logicians did not adopt this
course; their method was to shew that, by the aid of immediate
inferences, each mood could be reduced to such a form that the
*dictum* did apply directly to it. The doctrine of reduction
resulting from the adoption of this method will be discussed in the
following chapter.

*The special rules and the legitimate moods of Figure 3*.

The position of the terms in this figure is shewn thus,—

*M – PM – S*

⎯⎯

*S – P* ;

and its special rules are,—

(1) *The minor must be affirmative* ;

(2) *The conclusion must be particular*.

Proceeding as before, we are left with six valid moods, namely,
*AAI*, *AII*, *EAO*, *EIO*, *IAI*, *OAO*.

*The special rules and the legitimate moods of Figure 4*.

The position of the terms in this figure is shewn thus,—

*P – MM – S*

⎯⎯

*S – P* ;

and the following may be given as its special rules,—

(1) *If the major is affirmative, the minor must be
universal* ;

(2) *If either premiss is negative, the major must be
universal* ; 313

(3) *If the minor is affirmative, the conclusion must be
particular*.333

^{333} The special rules of the
fourth figure are variously stated. They are given in the above form
in the *Port Royal Logic*, pp. 202, 203. See, also, section
255.

The result of the application of these rules is again six valid
moods, namely, *AAI*, *AEE*, *AEO*, *EAO*,
*EIO*, *IAI*.

Our final conclusion then is that there are 24 valid moods, namely, six in each figure.

In Figure 1, *AAA*, *AAI*, *EAE*, *EAO*,
*AII*, *EIO*.

In Figure 2, *EAE*, *EAO*,
*AEE*, *AEO*, *EIO*, *AOO*.

In Figure 3,
*AAI*, *IAI*, *AII*, *EAO*, *OAO*,
*EIO*.

In Figure 4, *AAI*, *AEE*, *AEO*,
*EAO*, *IAI*, *EIO*.

245. *Weakened Conclusions
and Subaltern Moods.*—When from premisses that would have
justified a universal conclusion we content ourselves with inferring a
particular (as, for example, in the syllogism *All M is P*,
*All S is M*, therefore, *Some S is P*), we are said to have
a *weakened conclusion*, and the syllogism is said to be a
*weakened syllogism* or to be in a *subaltern mood* (because
the conclusion might be obtained by subaltern inference334
from the conclusion of the corresponding unweakened mood).

^{334} In treating the syllogism
on the traditional lines it is assumed that *S*, *M*,
*P* all represent existing classes. Subaltern inference is,
therefore, a valid process.

In the preceding section it has been shewn that in each figure there are six moods which do not offend against any of the syllogistic rules: so that in all there are 24 distinct valid moods. Five of these, however, have weakened conclusions; and, since we are not likely to be satisfied with a particular conclusion when the corresponding universal can be obtained from the same premisses, these moods are of no practical importance. Accordingly when the moods of the various figures are enumerated (as in the mnemonic verses) they are usually omitted. Still, their recognition gives a completeness to the theory of the syllogism, which it cannot otherwise possess. There is also a symmetry in the result of 314 their recognition as yielding exactly six legitimate moods in each figure.335

^{335} It has been remarked that
19 being a prime number at once suggests incompleteness or
artificiality in the common enumeration.

The subaltern moods are,—

In Figure 1, *AAI*, *EAO* ;

In Figure 2, *EAO*, *AEO* ;

In Figure 4, *AEO*.

It is obvious that there can be no weakened conclusion in Figure 3, since in no case is it possible to infer more than a particular conclusion in this figure.

*AAI* in Figure 4 is sometimes spoken of as a subaltern mood.
But this is a mistake. With the premisses *All P is M*, *All M
is S*, the conclusion *Some S is P* is certainly in one sense
weaker than the premisses would warrant since the universal conclusion
*All P is S* might have been inferred. But *All P is S* is
not the universal corresponding to *Some S is P*. The subjects of
these two propositions are different; and we infer all that we
possibly can about *S* when we say that *some S is P*. In
other words, regarded as a mood of figure 4, this mood is not a
subaltern. *AAI* in figure 4 is thus differentiated from
*AAI* in figure 1, and its inclusion in the mnemonic verses
justified.

246. *Strengthened
Syllogisms*.—If in a syllogism the same conclusion can still
be obtained although for one of the premisses we substitute its
subaltern, the syllogism is said to be a *strengthened syllogism*. A
strengthened syllogism is thus a syllogism with an unnecessarily
strengthened premiss.336

^{336} Compare De Morgan,
*Formal Logic*, pp. 91, 130. De Morgan calls a syllogism
*fundamental*, when neither of its premisses is stronger than is
necessary to produce the conclusion (*Formal Logic*, p.
77).

For example, the conclusion of the syllogism—

All M is P, | |

All M is S, | |

therefore, | Some S is P, |

could equally
be obtained from the premisses *All M is P*, *Some M is S* ;
or from the premisses *Some M is P*, *All M is S*.

By trial we may find that *every syllogism in which there*
315 *are two
universal premisses with a particular conclusion is a strengthened
syllogism, with the single exception of AEO in the fourth
figure*.337

In a full enumeration there are two strengthened syllogisms in each figure:—

In Figure 1, *AAI*, *EAO* ;

In Figure 2, *EAO*, *AEO* ;

In Figure 3, *AAI*, *EAO* ;

In Figure 4, *AAI*, *EAO*.

It will be observed that in figures 1 and 2, a syllogism having a
strengthened premiss may also be regarded as a syllogism having a
weakened conclusion, and *vice versâ* ; but that in figures 3 and
4, the contrary holds in both cases. The only syllogism with a
weakened conclusion in either of these figures is *AEO* in figure
4; and in this mood no conclusion is obtainable if either of the
premisses is replaced by its subaltern.

If syllogisms containing either a strengthened premiss or a weakened conclusion are omitted, we are left with 15 valid moods, namely, 4 in each of the first three figures and 3 in figure 4.

247. *The peculiarities
and uses of each of the four figures of the syllogism*.338—*Figure* 1. In this figure it is
possible to prove conclusions of all the forms **A**, **E**,
**I**, **O**; and it is the *only* figure in which a
universal affirmative conclusion can be proved. This alone makes it by
far the most useful and important of the syllogistic figures. All
deductive science, the object of which is to establish universal
affirmatives, tends to work in *AAA* in this figure.

Another point to notice is that only in this figure is it the case that both the subject of the conclusion is subject in the premisses, and the predicate of the conclusion predicate in the premisses; in figure 2 the predicate of the conclusion is subject in the major premiss; in figure 3 the subject of the conclusion is predicate in the minor premiss; and in figure 4 there is a double inversion.339 This no doubt partly 316 accounts for the fact that a reasoning expressed in figure 1 so often seems more natural than the same reasoning expressed in any other figure.340

^{339} The double inversion in
figure 4 is one of the reasons given by Thomson for rejecting that
figure altogether. Compare section 262.

^{340} Compare Solly, *Syllabus
of Logic*, pp. 130 to 132.

*Figure* 2. In this figure, only negatives can be proved; and
therefore it is chiefly used for purposes of disproof. For example,
*Every real natural poem is naïve* ; *those poems of Ossian
which Macpherson pretended to discover are not naïve* (*but
sentimental*)*; hence they are not real natural poems*
(Ueberweg, *System of Logic*, § 113). It has been called the
*exclusive* figure; because by means of it we may go on excluding
various suppositions as to the nature of something under
investigation, whose real character we wish to ascertain (a process
called *abscissio infiniti*). For example, *Such and such an
order has such and such properties*, *This plant has not those
properties* ; therefore, *It does not belong to that order*. A
syllogism of this kind may be repeated with a number of different
orders till the enquiry is so narrowed down that the place of the
plant is easily determined. Whately (*Elements of Logic*, p. 92)
gives an example from the diagnosis of a disease.

*Figure* 3. In this figure, only particulars can be proved. It
is frequently useful when we wish to take objection to a universal
proposition laid down by an opponent by establishing an instance in
which such universal proposition does not hold good.

It is the natural figure when the middle term is a singular term,
especially if the other terms are general. It has been already shewn
that if one and only one term of an affirmative proposition is
singular, that term is almost necessarily the subject. For example,
such a reasoning as *Socrates is wise*, *Socrates is a
philosopher*, therefore, *Some philosophers are wise*, can
only with great awkwardness be expressed in any figure other than
figure 3.

*Figure* 4. This figure is seldom used, and some logicians
have altogether refused to recognise it. We shall return to a
discussion of it subsequently. See section 262.

Lambert in his *Neues Organon* expresses the uses of the
different syllogistic figures as follows: “The first figure is
suited to the discovery or proof of the properties of a thing; 317 the second to the
discovery or proof of the distinctions between things; the third to
the discovery or proof of instances and exceptions; the fourth to the
discovery or exclusion of the different species of a genus.”

248. Why is *IE* an
inadmissible, while *EI* is an admissible, mood in every figure
of the syllogism? [L.]

249. What moods are good in
the first figure and faulty in the second, and *vice versâ*? Why
are they excluded in one figure and not in the other? [O.]

250. (i) Shew that *O*
cannot stand as premiss in figure 1, as major in figure 2, as minor
in figure 3, as premiss in figure 4.

(ii) Shew that it is impossible to have the conclusion in *A*
in any figure but the first. What fallacies would be committed if
there were such a conclusion to a reasoning in any other figure? [C.]

251. Two valid syllogisms in the same figure have the same major, middle, and minor terms, and their major premisses are subcontraries; determine—without reference to the mnemonic verses—what the syllogisms must be. [K.]

252. Prove, by general reasoning, that any mood valid both in figure 2 and in figure 3 is valid also in figure 1 and in figure 4. [C.]

253. Shew, without
individual reference to the different figures, that *EAO* is a
strengthened syllogism in every figure, and that *AAI* is a
strengthened syllogism whenever it is valid. [K.]

254. Shew, by general reasoning, that every valid syllogism in which the middle term is twice distributed contains a strengthened premiss. Does it follow that it must have also a weakened conclusion? [K.]

255. Shew that the following
*two* rules would suffice as the special rules for the fourth figure:
(i) The conclusion and major cannot have the same form unless it be
particular affirmative; (ii) The conclusion and minor cannot have the
same form unless it be universal negative. [J.]

256. *The Problem of
Reduction*.—By *reduction* is meant a process whereby
the reasoning contained in a given syllogism is expressed in some
other mood or figure. Unless an explicit statement is made to the
contrary, reduction is supposed to be to figure 1.

The following syllogism in figure 3 may be taken as an example:

All M is P, | |

Some M is S, | |

therefore, | Some S is P. |

It will be seen that by simply converting the minor premiss, we have precisely the same reasoning in figure 1.

This is an example of *direct* or *ostensive* reduction.

257. *Indirect
Reduction*.—A proposition is established *indirectly*
when its contradictory is proved false; and this is effected if it can
be shewn that a consequence of the truth of its contradictory would be
self-contradiction.

The method of indirect proof is in several cases adopted by Euclid;
and it may be employed in the reduction of syllogisms from one mood to
another. Thus, *AOO* in figure 2 is usually reduced in this
manner. The argument may be stated as follows:—

From the premisses,—

All P is M, | |

Some S is not M, | |

it follows that | Some S is not P ; |

for if this conclusion is not true, then,
by the law of excluded 319 middle, its contradictory (namely, *All
S is P*) must be so; and, the premisses being given true, the three
following propositions must all be true, namely,

All P is M, |

Some S is not M, |

All S is P. |

But combining the first and the third of these we have a syllogism in figure 1, namely,

All P is M, | |

All S is P, | |

yielding the conclusion |
All S is M. |

*Some S is not M* and *All S is M* are, therefore, true
together; but, by the law of contradiction, this is absurd, since they
are contradictories.

Hence it has been shewn that the consequence of supposing *Some S
is not P* false is a self-contradiction; and we may accordingly
infer that it is true.

It will be observed that the only syllogism made use of in the above argument is in figure 1; and the process may, therefore, be regarded as a reduction of the reasoning to figure 1.

This method of reduction is called *Reductio ad impossibile*,
or *Reductio per impossibile*,341 or *Deductio ad
impossibile*, or *Deductio ad absurdum*. It is the only way of
reducing *AOO* in figure 2 or *OAO* in figure 3 to figure 1,
unless negative terms are used (as in obversion and contraposition);
and it was adopted by the old writers in consequence of their
objection to negative terms.

^{341} Compare Mansel’s *Aldrich*, pp. 88, 89.

It will be shewn later on in this chapter that by employing the method of indirect reduction systematically we can bring out with great clearness the relation between the different moods and figures of the syllogism.

258. *The mnemonic lines
Barbara, Celarent, &c.*—The mnemonic
hexameter verses (which are spoken of by De Morgan as “the magic
words by which the different moods have been denoted for many
centuries, words which I take to be more full of meaning than any that
ever were made”) are usually given as follows: 320

*Barbără*, *Cēlārent*, *Dărĭi*, *Fĕrĭō*que prioris:

*Cēsărĕ*, *Cāmēstres*, *Festīnŏ*, *Bărōcŏ*, secundae:

Tertia, *Dāraptī*, *Dĭsămis*, *Dātīsĭ*, *Fĕlapton*,

*Bōcardō*, *Fērīsŏn*, habet: Quarta insuper addit

*Brāmantip*, *Cămĕnes*, *Dĭmăris*, *Fēsāpŏ*,
*Frĕsīson*.

Each valid mood in every figure, unless it be a subaltern mood, is
here represented by a separate word; and in the case of a mood in any
of the so-called imperfect figures (*i.e.*, figures 2, 3, 4), the
mnemonic gives full information for its reduction to figure 1, the
so-called perfect figure.

The only meaningless letters are *b* (not initial), *d*
(not initial), *l*, *n*, *r*, *t* ; the
signification of the remainder is as follows:—

The *vowels* give the quality and quantity of the propositions
of which the syllogism is composed; and, therefore, really give the
syllogism itself, if the figure is also known. Thus, *Camenes* in
figure 4 represents the syllogism—

All P is M, | |

No M is S, | |

therefore, | No S is P. |

The *initial letters* in the case of figures 2, 3, 4 shew to
which of the moods of figure 1 the given mood is to be reduced,
namely, to that which has the same initial letter. The letters
*B*, *C*, *D*, *F* were chosen for the moods of
figure 1 as being the first four consonants in the alphabet.

Thus, *Camestres* is reduced to *Celarent*,—

All P is M, |
⟍ ⟋ | No M is S, | |

No S is M, |
⟋ ⟍ | All P is M, | |

therefore, | No S is P. |
therefore, | No P is S, |

therefore, | No S is P.
342 |

^{342} The *order* of
inference in this and in other reductions might be made clear by the
use of arrows, representing inference, as follows:

All P is M, | ⟍ ↗ | No M is S, |

No S is M, | ⟋ ↘ | All P is M, |

↓ | ||

No S is P. | ← | No P is S, |

*s* (in the middle of a word) indicates that in the process of
reduction the preceding proposition is to be simply converted. 321 Thus, in reducing
*Camestres* to *Celarent*, as shewn above, the minor premiss
is simply converted.

*s* (at the end of a word) shews that the conclusion of the
*new* syllogism has to be simply converted in order that the
given conclusion may be obtained. This again is illustrated in the
reduction of *Camestres*. The final *s* does not affect the
conclusion of *Camestres* itself, but the conclusion of
*Celarent* to which it is reduced.343

^{343} This peculiarity in the
signification of *s* and *p* when they are *final*
letters is sometimes overlooked. The point to be noted is that the
conclusion of the syllogism originally given is not, like the original
premisses, a datum from which we set out, but a result that we have to
reach. It follows that the conclusion to be manipulated, if any, must
be the conclusion of the syllogism obtained by reduction, not the
conclusion of the original syllogism. This is clearly shewn in the
case of *Camestres* by the method adopted in the last preceding
note to illustrate the reduction of *Camestres* to
*Celarent*. The reduction of *Disamis*, *Bramantip*,
*Camenes*, *Dimaris* to figure 1 might be illustrated
similarly.

*p* (in the middle of a word) signifies that the preceding
proposition is to be converted *per accidens* ; as, for example,
in the reduction of *Darapti* to *Darii*,—

All M is P, | All M is P, | ||

All M is S, | Some S is M, | ||

therefore, | Some S is P. |
therefore, | Some S is P. |

*p* (at the end of a word344) implies that the
conclusion *obtained by reduction* is to be converted *per
accidens*. Thus, in *Bramantip*, the *p* does not relate
to the **I** conclusion of the mood itself;345 it really relates
to the **A** conclusion of the syllogism in *Barbara* which is
given by reduction. Thus,—

All P is M, | ⟍ ⟋ | All M is S, | |

All M is S, | ⟋ ⟍ | All P is M, | |

therefore, | Some S is P. |
therefore, | All P is S, |

therefore, | Some S is P. |

^{345} Compare, however, Hamilton,
*Logic*, I. p. 264, and Spalding,
*Logic*, pp. 230, 1.

*m* indicates that in reduction the premisses have to be
transposed (*metathesis praemissarum*); as just shewn in the case
of *Bramantip*, and also in the case of *Camestres*.

*c* signifies that the mood is to be reduced *indirectly*
(*i.e.*, by 322
*reductio per impossibile* in the manner shewn in the preceding
section); and the position of the letter indicates that in this
process of indirect reduction the first step is to omit the premiss
preceding it, *i.e.*, the other premiss is to be combined with
the contradictory of the conclusion (*conversio syllogismi*, or
*ductio per contradictoriam propositionem sive per impossibile*),
The letter *c* is by some writers replaced by *k*,
*Baroko* and *Bokardo* being given as the mnemonics, instead
of *Baroco* and *Bocardo*.

The following lines are sometimes added to the verses given above, in order to meet the case of the subaltern moods:—

Quinque Subalterni, totidem Generalibus orti,

Nomen habent nullum, nec, si bene colligis, usum.346

^{346} The mnemonics have been
written in various forms. Those given above are from Aldrich, and they
are the ones that are in general use in England. Wallis in his
*Institutio Logicae* (1687) gives for the fourth figure,
*Balani, Cadere, Digami, Fegano, Fedibo*. P. van Musschenbroek in
his *Institutiones Logicae* (1748) gives *Barbari, Calentes,
Dibatis, Fespamo, Fresisom*. This variety of forms for the moods of
figure 4 is no doubt due to the fact that the recognition of this
figure at all was quite exceptional until comparatively recently.
Compare sections 262, 263.

According to Ueberweg (*Logic*, § 118) the mnemonics
run,—

*Barbara, Celarent* primae, *Darii Ferio*que.

*Cesare, Camestres, Festino, Baroco* secundae.

Tertia grande sonans recitat *Darapti, Felapton,
Disamis, Datisi, Bocardo, Ferison*. Quartae

Sunt

Ueberweg gives *Camestros* and *Calemos* for the weakened
moods of *Camestres* and *Calemes*. This is not, however,
quite accurate. The mnemonics should be *Camestrop* and
*Calemop*.

Professor Carveth Read (*Logic*, pp. 126, 7) suggests an
ingenious modification of the verses, so as to make each mnemonic
immediately suggest the figure to which the corresponding mood
belongs, at the same time abolishing all the unmeaning letters. He
takes *l* as the sign of the first figure, *n* of the
second, *r* of the third, and *t* of the fourth. The lines
(to be scanned, says Professor Read, discreetly) then run

*Ballala, Celallel, Dalii, Felio*que prioris.

*Cesane, Camesnes, Fesinon, Banoco* secundae.

Tertia *Darapri, Drisamis, Darisi, Ferapro,
Bocaro, Ferisor* habet. Quanta insuper addit

Professor Mackenzie suggests that, if this plan is adopted, it
would be better to take *r* for the first figure (*figura
recta*, the straightforward figure), *n* for the second figure
(*figura negativa*), *t* for the third figure (*figura
tertia* or *particularis*), and *l* for the fourth figure
(*figura laeva*, the left-handed figure). Compare also Mrs Ladd
Franklin, *Studies in Logic*, Johns Hopkins University, p. 40.

323 259. *The
direct reduction of Baroco and Bocardo*.—These moods may be
reduced directly to the first figure by the aid of obversion and
contraposition as follows.347

*Baroco*:—

All P is M, | |

Some S is not M, | |

therefore, | Some S is not P, |

is reducible to *Ferio* by the
contraposition of the major premiss and the obversion of the minor,
thus,—

No not-M is P, | |

Some S is not-M, | |

therefore, | Some S is not P. |

^{347} Another method is to reduce
*Baroco* and *Bocardo* by the process of ἔκθεσις
to other moods of figures 2 and 3, and thence to figure 1. Ueberweg
writes, “*Baroco* may also be referred to *Camestres*
when those (some) *S* of which the minor premiss is true are
placed under a special notion and denoted by *Sʹ*. Then the
conclusion must hold good universally of *Sʹ*, and
consequently particularly of *S*. Aristotle calls such a
procedure ἔκθεσις” (*Logic*, § 113). As regards
*Bocardo*, “Aristotle remarks that this mood may be proved
without apagogical procedure (*reductio ad impossibile*) by the
ἐκθέσθαι or λαμβάνειν of that part of the
middle notion which is true of the major premiss. If we denote this
part by *N*, then we get the premisses; *NeP* ; *NaS*:
from which follows (in *Felapton*) *SoP* ; which was to be
proved” (§ 115). The procedure is, however, rather more
complicated than appears in the above statements. In the case of
*Baroco* (*PaM*, *SoM*, ∴ *SoP*), let the
*S*’s which are not *M* (of which by hypothesis there
are some) be denoted by *X* ; then we have *PaM*, *XeM*,
∴ *XeP* (*Camestres*); but *XaS*, and hence we
have further *XeP*, *XaS*, ∴ *SoP*
(*Felapton*). In the case of *Bocardo* (*MoP*,
*MaS*, ∴ *SoP*), let the *M*’s which are
not *P* (of which by hypothesis there are some) be denoted by
*N* ; then we have *MaS*, *NaM*, ∴ *NaS*
(Barbara); and hence *NeP*, *NaS*, ∴ *SoP*
(*Felapton*). The argument in both cases suggests questions
connected with the existential import of propositions; but the
consideration of such questions must for the present be
deferred.

*Faksoko* has been suggested as a mnemonic for this method of
reduction, *k* denoting obversion, so that *ks* demotes
obversion followed by conversion (*i.e.*, contraposition).

Whately’s mnemonic *Fakoro* (*Elements of Logic*,
p. 97) does not indicate the obversion of the minor premiss (*r*
being with him an unmeaning letter).

Some M is not P, | |

All M is S, | |

therefore, | Some S is not P, |

is reducible to *Darii* by the
contraposition of the major premiss and the transposition of the
premisses, thus,—

All M is S, | |

Some not-P is M, | |

therefore, | Some not-P is S. |

*Some not-P is S* is not indeed our original conclusion, but
the latter can be obtained from it by conversion followed by
obversion. This method of reduction may be indicated by
*Doksamosk* (which again is obviously preferable to
*Dokamo*, suggested by Whately, since the latter would make it
appear as if we immediately obtained the original conclusion in
*Darii*.)

260. *Extension of the
Doctrine of Reduction*.—The doctrine of reduction may be
extended, and it can be shewn not merely that any syllogism may be
reduced to figure 1, but also that it may be reduced to any given mood
(not being a subaltern mood) of that figure.348 This position will
obviously be established if we can shew that *Barbara*,
*Celarent*, *Darii*, and *Ferio* are mutually reducible
to one another.

*Barbara* may be reduced to *Celarent* by the obversion
of the major premiss and also of the new conclusion thereby obtained.
Thus, using arrows, as in the note on page 320,

All M is P, | → | No M is not-P, |

All S is M, | → | All S is M, |

↓ | ||

All S is P. | ← | No S is not-P. |

Conversely, *Celarent* is reducible to *Barbara* ; and in
a similar manner, by obversion of major premiss and conclusion,
*Darii* and *Ferio* are reducible to one another.

It will now suffice if we can shew that *Barbara* and
*Darii* are mutually reducible to one another. Clearly the only
method possible here is the indirect method.

Take *Barbara*,

MaP, | |

SaM, | |

⎯⎯ | |

∴ | SaP ; |

325 for, if not, then we
have *SoP* ; and *MaP*, *SaM*, *SoP* must be true
together. From *SoP* by first obverting and then converting (and
denoting *not-P* by *Pʹ*) we get *PʹiS*, and
combining this with *SaM* we have the following syllogism in *Darii*,—

SaM, | |

PʹiS, | |

⎯⎯ | |

∴ | PʹiM. |

*PʹiM* by
conversion and obversion becomes *MoP* ; and therefore *MaP*
and *MoP* are true together; but this is impossible, since they
are contradictories. Therefore, *SoP* cannot be true,
*i.e.*, the truth of *SaP* is established.

Similarly, *Darii* may be indirectly reduced to
*Barbara*.349

MaP, | (i) | |

SiM, | (ii) | |

⎯⎯ | ||

∴ | SiP. |
(iii) |

The contradictory of (iii) is *SeP*, from which we obtain
*PaSʹ*. Combining with (i), we have—

PaSʹ, | ||

MaP, | ||

⎯⎯ | ||

∴ | MaSʹ | in Barbara. |

But from this conclusion we may obtain *SeM*,
which is the contradictory of (ii).

^{349} It has been maintained,
that this reduction is unnecessary, and that, to all intents and
purposes, *Darii* is *Barbara*, since the “some
*S*” in the minor is, and is known to be, the *same
some* as in the conclusion. Compare section 269.

261. *Is Reduction an
essential part of the Doctrine of the Syllogism?*—According
to the original theory of reduction, the object of the process is to
be sure that the conclusion is a valid inference from the premisses.
The validity of a syllogism in figure 1 may be directly tested by
reference to the *dictum de omni et nullo*: but this dictum has
no direct application to syllogisms in the remaining three figures.
Thus, Whately says, “As it is on the *dictum de omni et
nullo* that all reasoning *ultimately* depends, so all arguments may
be in one way or other brought into some one of the four moods in the
first figure: and a syllogism is, in that case, said to be
*reduced*” (*Elements of Logic*, p. 93). Professor
Fowler puts the same position somewhat more guardedly, “As we
have adopted no canon for the 2nd, 3rd, and 4th figures, we have as
yet 326 no positive
proof that the six moods remaining in each of those figures are valid:
we merely know that they do not offend against any of the syllogistic
rules. But if we can *reduce* them, *i.e.*, bring them back to the
first figure, by shewing that they are only different statements of
its moods, or in other words, that precisely the same conclusions can
be obtained from equivalent premisses in the first figure, their
validity will be proved beyond question” (*Deductive
Logic*, p. 97).

Reduction is, on the other hand, regarded by some logicians as both
*unnecessary* and *unnatural*. It is, in the first place,
said to be *unnecessary*, on the ground that the *dictum de
omni et nullo* has no claim to be regarded as the paramount law for
all valid inference.350 In sections 270 to 272 it will be shewn that dicta can be formulated for the other figures, which may be regarded
as making them independent of the first, and putting them on a level
with it. It may also be maintained that in any mood the validity of a
particular syllogism is as self-evident as that of the *dictum de
omni et nullo* itself; and that, therefore, although axioms of
syllogism are useful as generalisations of the syllogistic process,
they are needless in order to establish the validity of any given
syllogism. This view is indicated by Ueberweg.

^{350} Compare Thomson, *Laws of
Thought*, p. 172.

Reduction is, in the second place, said to be *unnatural*,
inasmuch as it often involves the substitution of an unnatural and
indirect for a natural and direct predication. Figures 2 and 3 at any
rate have their special uses, and certain reasonings fall naturally
into these figures rather than into the first figure.351

^{351} Compare a quotation from
Lambert (*Neues Organon*, §§ 230, 231) given by Sir W. Hamilton
(*Logic*, II. p. 438).

The following example is given by Thomson (*Laws of Thought*,
p. 174): “Thus, when it was desirable to shew by an example that
zeal and activity did not always proceed from selfish motives, the
natural course would be some such syllogism as the following. The
Apostles sought no earthly reward, the Apostles were zealous in their
work; therefore, 327
some zealous persons seek not earthly reward.” In reducing this
syllogism to figure 1, we have to convert our minor into “Some
zealous persons were Apostles,” which is awkward and unnatural.

Take again this syllogism, “Every reasonable man wishes the
Reform Bill to pass, I don’t, therefore, I am not a reasonable
man.” Reduced in the regular way to *Celarent*, the major
premiss becomes, “No person wishing the Reform Bill to pass is
I,” yielding the conclusion, “No reasonable man is I.”

Further illustrations of this point will be found if we reduce to figure 1, syllogisms with such premisses as the following:—All orchids have opposite leaves, This plant has not opposite leaves; Socrates is poor, Socrates is wise.

The above arguments justify the position that reduction is not a necessary part of the doctrine of the syllogism, so far as the establishment of the validity of the different moods is concerned.352

^{352} Hamilton (*Logic*,
I. p. 433) takes a curious position in
regard to the doctrine of reduction. “The last three
figures,” he says, “are virtually identical with the
first.” This has been recognised by logicians, and hence
“the tedious and disgusting rules of their reduction.” But
he himself goes further, and extinguishes these figures altogether, as
being merely “accidental modifications of the first,” and
“the mutilated expressions of a complex mental process.” A
somewhat similar position is taken by Kant in his essay *On the
Mistaken Subtilty of the Four Figures*. Kant’s argument is
virtually based on the two following propositions: (1) Reasonings in
figures 2, 3, 4 require to be implicitly, if not explicitly, reduced
to figure 1, in order that their validity may be apparent; for
example, in *Cesare* we must have covertly performed the
conversion of the major premiss in thought, since otherwise our
premisses would not be conclusive; (2) No reasonings ever fall
naturally into any of the moods of figures 2, 3, 4, which are,
therefore, a mere useless invention of logicians. On grounds already
indicated, both these propositions must be regarded as erroneous. A
further error seems to be involved in the following passage from the
same essay of Kant’s: “It cannot be denied that we can
draw conclusions legitimately in all these figures. But it is
incontestable that all except the first determine the conclusion only
by a roundabout way, and by interpolated inferences, and that *the
very same conclusion would follow from the same middle term in the
first figure by pure and unmixed reasoning*.” The latter part
of this statement cannot be justified in such a case as that of
*Baroco*.

At the same time, no treatment of the syllogism can be 328 regarded as scientific
or complete until the *equivalence* between the moods in the
different figures has been shewn; and for this purpose, as well as for
its utility as a logical exercise, a full treatment of the problem of
reduction should be retained.353

262. *The Fourth
Figure*.—Figure 4 was not as such recognised by Aristotle;
and its introduction having been attributed by Averroës to Galen, it
is frequently spoken of as the *Galenian Figure*. It does not
usually appear in works on Logic before the beginning of the
eighteenth century, and even by modern logicians its use is sometimes
condemned. Thus Bowen (*Logic*, p. 192) holds that “what is
called the fourth figure is only the first with a converted
conclusion; that is we do not actually reason in the fourth, but only
in the first, and then if occasion requires, convert the conclusion of
the first.” This account of figure 4 cannot, however, be
accepted, since it will not apply to *Fesapo* or *Fresison*.
For example, from the premisses of *Fesapo* (*No P is M* and
*All M is S*) no conclusion whatever is obtainable in figure 1.354

^{354} For the most part the
critics of the fourth figure seem to identify it altogether with
*Bramantip*. The following extract from Father Clarke’s
*Logic* (p. 337) will serve to illustrate the contumely to which
this poor figure is sometimes subjected: “Ought we to retain it?
If we do, it should be as a sort of syllogistic Helot, to shew how low
the syllogism can fall when it neglects the laws on which all true
reasoning is founded, and to exhibit it in the most degraded form
which it can assume without being positively vicious. Is it capable of
reformation? Not of reformation, but of extinction…… Where the same
premisses in the first figure would prove a universal affirmative,
this feeble caricature of it is content with a particular; where the
first figure draws its conclusion naturally and in accordance with the
forms into which human thought instinctively shapes itself, this
perverted abortion forces the mind to an awkward and clumsy process
which rightly deserves to be called ‘inordinate and
violent.’” Father Clarke’s own violence appears to
be attributable mainly to the fact that figure 4 was not, as such,
recognised by Aristotle.

Thomson’s ground of rejection is that in the fourth figure
the order of thought is wholly inverted, the subject of the conclusion
having been a predicate in the premisses, and the predicate a subject.
“Against this the mind rebels; and we can ascertain that the
conclusion is only the converse of the real one, by proposing to
ourselves similar sets of premisses, to 329 which we shall always find ourselves
supplying a conclusion so arranged that the syllogism is in the first
figure, with the second premiss first” (*Laws of Thought*,
p. 178). As regards the first part of this argument, Thomson himself
points out that the same objection applies partially to figures 2 and
3. It no doubt helps to explain why as a matter of fact reasonings in
figure 4 are not often met with;355 but it affords no
sufficient ground for altogether refusing to recognise this figure.
The second part of Thomson’s argument is, for a reason already
stated, unsound. The conclusion, for example, of *Fresison*
cannot be “the converse of the real conclusion,” since
(being an **O** proposition) it is not the converse of any other
proposition whatsoever.

^{355} The reasons why figure 4,
“with its premisses looking one way, and its conclusion
another,” is seldom used, are elaborated by Karslake, *Aids to
the Study of Logic*, I. pp. 74, 5.

It is indeed impossible to treat the syllogism scientifically and
completely without admitting in some form or other the moods of figure
4. In an *à priori* separation of figures according to the
position of the major and minor terms in the premisses, this figure
necessarily appears, and it yields conclusions which are not directly
obtainable from the same premisses in any other figure. It is not
actually in frequent use, but reasonings may sometimes not unnaturally
fall into it; for example, None of the Apostles were Greeks, Some
Greeks are worthy of all honour, therefore, Some worthy of all honour
are not Apostles.

263. *Indirect
Moods*.—The earliest form in which the mnemonic verses
appeared was as follows:—

*Barbara*, *Celarent*, *Darii*, *Ferio*,
*Baralipton*,

*Celantes*, *Dabitis*, *Fapesmo*, *Frisesomorum,*

*Cesare*, *Camestres*, *Festino*, *Baroco*, *Darapti*,

*Felapton*, *Disamis*, *Datisi*, *Bocardo*,
*Ferison*.356

^{356} First published in the
*Summulae Logicales* of Petrus Hispanus, afterwards Pope John
XXI., who died in 1277. The mnemonics occur in an earlier unpublished
work of William Shyreswood, who died as Chancellor of Lincoln in 1249.

Aristotle recognised only three figures: the first figure, which he
considered the type of all syllogisms and which he 330 called the perfect
figure, the *dictum de omni et nullo* being directly applicable
to it alone; and the second and third figures, which he called
imperfect figures, since it was necessary to reduce them to the first
figure, in order to obtain a test of their validity.

Before the fourth figure, however, was commonly recognised as such,
its moods were recognised in another form, namely, as *indirect* moods
of the first figure; and the above mnemonics—*Baralipton*,
*Celantes*, *Dabitis*, *Fapesmo*,
*Frisesomorum*—represent these moods so regarded.357

^{357} From the 14th to the 17th
century the mnemonics found in works on Logic usually give the moods
of the fourth figure in this form, or else omit them altogether.
Wallis (1687) recognises them in both forms, giving two sets of
mnemonics.

The conception of indirect moods may be best explained by starting
from a definition of figure, which contains no reference to the
distinction between major and minor terms, and which accordingly
yields only three figures instead of four, namely: Figure 1, in which
the middle term is subject in one of the premisses and predicate in
the other; Figure 2, in which the middle term is predicate in both
premisses; Figure 3, in which the middle term is subject in both
premisses. The moods of figure 1 may then be distinguished as direct
or indirect according as the position of the terms in the conclusion
is the same as their position in the premisses or the reverse.358
Thus, with 331 the
premisses *MaP*, *SaM*, we have a direct conclusion
*SaP*, and an indirect conclusion *PiS*. These are
respectively *Barbara* and *Baralipton*. Similarly,
*Celantes* corresponds to *Celarent*, and *Dabitis* to
*Darii*. With the premisses *MeP*, *SiM*, we obtain the
direct conclusion *SoP*, but nothing can be inferred of *P*
in terms of *S*. There is, therefore, no indirect mood
corresponding to *Ferio*. On the other hand, *Fapesmo* and
*Frisesomorum* (the *Fesapo* and *Fresison* of the
fourth figure) have no corresponding direct moods.

^{358} It follows that if we
compare the conclusion of an indirect mood with the conclusion of the
corresponding direct mood (where such correspondence exists), we shall
find that the terms have changed places. Mansel’s definition of
an indirect mood as “one in which we do not infer the immediate
conclusion, but its converse” (*Aldrich*, p. 78) must, however, be
rejected for the reason that it cannot be applied to *Fapesmo*
and *Frisesomorum*, which are indirect moods having no
corresponding valid direct moods at all. In these we cannot be said to
infer “the converse of the immediate conclusion,” for
there is no immediate conclusion. Mansel deals with these two moods
very awkwardly. “*Fapesmo* and *Frisesomorum*,”
he remarks, “have negative minor premisses, and thus offend
against a special rule of the first figure; but this is checked by a
counterbalancing transgression. For by simply converting **O**, we
alter the distribution of the terms, so as to avoid an illicit
process.” But the notion that we can counterbalance one
violation of law by committing a second cannot be allowed. The truth
of course is that, in the first place, the special rules of the first
figure as ordinarily given do not apply to the indirect moods; and in
the second place, the conclusion **O** is not obtained by
conversion at all.

Clearly it is no more than a formal difference whether the five moods in question are recognised in the manner just indicated, or as constituting a distinct figure; but, on the whole, the latter alternative seems less likely to give rise to confusion.

The distinction between direct and indirect moods as above
expressed is for obvious reasons confined to the first figure. It will
be observed, however, that in the traditional names of the indirect
moods of the first figure the minor premiss precedes the major, and if
we seek to apply a distinction between direct and indirect moods in
the case of the second and third figures, it can only be with
reference to the conventional order of the premisses. Thus, in the
second figure, taking the premisses *PeM*, *SaM*, we may
infer either *SeP* or *PeS*, and if we call a syllogism
direct or indirect according as the major premiss precedes the minor,
or *vice versâ*, then *PeM*, *SaM*, *SeP* will be
a direct mood, and *PeM*, *SaM*, *PeS* an indirect
mood. The former of these syllogisms is *Cesare*, and the latter
is *Camestres* with the premisses transposed.359
Hence the latter will immediately become a direct mood by merely
changing the order of the premisses; and the artificiality of the
distinction is at once apparent. The result will be found to be
similar in other cases, and the distinction may, therefore, be
rejected so far as figures 2 and 3 are concerned.

^{359} Take, again, the premisses
*MaP*, *MoS*. Here there is no direct conclusion, but only
an indirect conclusion *PoS*. This, however, is merely
*Bocardo* with the premisses transposed.

264. *Further discussion
of the process of Indirect Reduction*.—The discussion of the
problem of reduction in the preceding pages has in the main followed
the traditional lines. It 332 is, however, desirable to treat the
process of indirect reduction in a rather more independent and
systematic manner. By doing so, we shall find that the process enables
us to exhibit very clearly and symmetrically the relations between the
first three figures, and also the distinctive functions of these
figures.

The argument on which indirect reduction is based is one of which
we have several times made use (*e.g.*, in the proof of the
second corollary adopted from De Morgan in section 200, and in certain of the proofs contained in section 202), namely, that if *X* and *Y* together prove *Z*, then *X* and the denial of
*Z* must prove the denial of *Y*, and *vice versâ*.

The process may conveniently be exhibited as the contraposition of
a hypothetical. Thus, from the proposition *X being given, if Y then
Z* we may infer by contraposition *X being given, if not Z then
not Y* ; and we can equally pass back from the contrapositive to the
original proposition.

Since the contradictory of the conclusion of a syllogism may be
combined with *either* of the original premisses, it follows that every
valid syllogism carries with it the validity of *two* other syllogisms.
Hence all valid syllogisms must be capable of being arranged in sets
of three which are mutually equivalent.

The three equivalent syllogisms may be symmetrically expressed as
follows (where *P* and *Pʹ*, *Q* and *Qʹ*,
*R* and *Rʹ* are respectively contradictories):

(i) premisses, *P* and *Q* ; conclusion *Rʹ* ;

(ii) premisses, *Q* and *R* ; conclusion *Pʹ* ;

(iii) premisses, *R* and *P* ; conclusion *Qʹ*.

It must be understood that the order of the premisses in these syllogisms is not intended to indicate which is major and which minor.

265. *The
Antilogism*.—Each of the three equivalent syllogisms just
given involves further the formal incompatibility of the three
propositions *P*, *Q*, *R* (compare section 214). Three
propositions, containing three and only three terms, which are thus
formally incompatible with one another, constitute what has been
called by Mrs Ladd Franklin an *antilogism*.360 Thus, 333 the syllogism,
“*MaP*, *SaM*, therefore, *SaP*,” has for
its equivalent antilogism, “*MaP*, *SaM*, *SoP*
are three propositions that are formally incompatible with one another.”

^{360} See Baldwin’s
*Dictionary of Philosophy*, art. *Symbolic Logic*. It is
shewn in this article that the whole of syllogistic reasoning may be
summed up in the following antilogism, the symbolism of section 138
being made use of,—

[(*AB* = 0)(*bC* = 0)(*AC* > 0)] = 0.

The fifteen moods containing neither a strengthened premiss nor a weakened conclusion may, by the aid of conversions and obversions, be obtained from this antilogism according as the contradictory of one or other of the three incompatibles is taken as the conclusion.

266. *Equivalence of the
Moods of the first three Figures shewn by the Method of Indirect
Reduction*.—If one of our three equivalent syllogisms is in
one of the first three figures, then it can be shewn that the two
others will be in the remaining two of these figures.

Thus, let *P*, *Q*, ∴ *Rʹ* be in figure
1, the minor premiss being stated first. It may then be written

S ⎯ M, M ⎯ P,
∴ (S ⎯ P)ʹ. | (1) |

The second syllogism becomes

M ⎯ P,
S ⎯ P, ∴
(S ⎯ M)ʹ; | (2) |

and the third is

S ⎯ P, S ⎯ M,
∴ (M ⎯ P)ʹ. | (3) |

It will be seen that (2) is in figure 2, and (3) in figure 3.

Next, let *P*, *Q*, ∴ *Rʹ* be in figure
2, the major premiss being stated first. We then have for our three
syllogisms,—

P ⎯ M,
S ⎯ M, ∴
(S ⎯ P)ʹ; | (1) |

S ⎯ M, S ⎯ P,
∴ (P ⎯ M)ʹ; | (2) |

S ⎯ P, P ⎯ M,
∴ (S ⎯ M)ʹ. | (3) |

Here (2) is in figure 3, (3) in figure 1.

Finally, let *P*, *Q*, ∴ *Rʹ* be in
figure 3, the major premiss being stated first. We have

M ⎯ P, M ⎯ S,
∴ (S ⎯ P)ʹ; | (1) |

M ⎯ S, S ⎯ P,
∴ (M ⎯ P)ʹ; | (2) |

S ⎯ P, M ⎯ P,
∴ (M ⎯ S)ʹ. | (3) |

Here (2) is in figure 1, (3) in figure 2.

Hence we see that, starting with a syllogism in any one of the first three figures (the minor premiss preceding the major in figure 1, but following it in figures 2 and 3), and taking the 334 propositions in the above cyclic order, then the figures will always recur in the cyclic order 1, 2, 3.361

^{361} If we were to start with a
syllogism in figure 1, the major premiss being stated first, then the
cyclic order of figures would be 1, 3, 2, and in figures 2 and 3 the
minor premiss would precede the major.

It follows that (as we already know to be the case) there must be an equal number of valid syllogisms in each of the first three figures, and that they may be arranged in sets of equivalent trios. These equivalent trios will be found to be as follows (sets containing strengthened premisses or weakened conclusions being enclosed in square brackets);

Barbara, Baroco, Bocardo;

[**AAI**, **AEO**, Felapton;]

Celarent, Festino, Disamis;

[**EAO**, **EAO**, Darapti;]

Darii, Camestres, Ferison;

Ferio, Cesare, Datisi.

The corresponding antilogisms are **AAO**, [**AAE**,]
**EAI**, [**EAA**,] **AIE**, **EIA**.362

^{362} The position of the terms
in these antilogisms corresponds to that of figure 1, the major
premiss being stated first.

267. *The Moods of Figure
4 in their relation to one another*.—We have seen that in
the equivalent trios of syllogisms yielded by the process of indirect
reduction we never have in any one trio more than one syllogism in
figure 1, or in figure 2, or in figure 3. Figure 4 is, however,
self-contained in the sense that if we start with a syllogism in this
figure, both the other syllogisms will be in the same figure.
Proceeding as in the last section, we may shew this as follows, the
major premiss being stated first:363

P ⎯ M, M ⎯ S,
∴ (S ⎯ P)ʹ; | (1) |

M ⎯ S, S ⎯ P,
∴ (P ⎯ M)ʹ; | (2) |

S ⎯ P, P ⎯ M,
∴ (M ⎯ S)ʹ. | (3) |

^{363} It will be found that it
comes to just the same thing if the minor premiss is stated
first.

It follows that in figure 4 the number of valid syllogisms must be some multiple of three. The number is, as we know, six. There are, therefore, two equivalent trios; and they will be found to be as follows: 335

[Bramantip, **AEO**, Fesapo;]

Camenes, Fresison, Dimaris.

The equivalent antilogisms are [**AAE**,] **AEI**. Comparing
this result with that obtained in the preceding section, we see that
the only valid antilogistic combinations are **AAO** and
**AEI**, with the addition of **AAE** (in which one of the three
propositions is unnecessarily strengthened).364

268. *Equivalence of the
Special Rules of the First Three Figures*.—Let the following
be a valid syllogism in figure 1,—

(minor) | S ⎯ M, | (1) | |

(major) | M ⎯ P, | (2) | |

(conclusion) | ∴ |
(S ⎯ P)ʹ. | (3) |

Then the corresponding valid syllogism in figure 2 will be

(major) |
M ⎯ P, | (2) | |

(minor) |
S ⎯ P, | contradictory of (3) | |

(conclusion) | ∴ | (S ⎯ M)ʹ; | contradictory of (1) |

and the corresponding valid syllogism in figure 3 will be

(major) | S ⎯ P, | contradictory of (3) | |

(minor) |
S ⎯ M, | (1) | |

(conclusion) | ∴ |
(M ⎯ P)ʹ. | contradictory of (2) |

The special rules of figure 1 are

minor | affirmative, |

major | universal, |

that is, (1) must be affirmative, (2) must be universal.

In figure 2, (2) is the major, and the contradictory of (1) is the conclusion. Therefore, in figure 2 we must have the rules,—

major | universal, |

conclusion | negative [and hence one premiss
negative]. |

In figure 3, (1) is the minor, and the contradictory of (2) is the conclusion. Therefore, in figure 3 we must have the rules,—

minor | affirmative, |

conclusion | particular. |

Thus the special rules of figures 2 and 3 are shewn to be deducible from the special rules of figure 1. We might equally 336 well start from the special rules of figure 2 or of figure 3 and deduce the rules of the two other figures.365

^{365} The complete rules for the
antilogisms of the first three figures, as given at the end of section
266, are (*a*) first proposition universal, (*b*) second
proposition affirmative, (*c*) third proposition opposite in
quality to the first, and (unless it is strengthened) opposite in
quantity to the second. These rules replace all general
rules.

269. *Scheme of the Valid
Moods of Figure* l.—So far as the nature of the reasoning
involved is concerned, there is practically no distinction between
*Barbara* and *Darii*, or between *Celarent* and
*Ferio*. For in each case, if *S* is the minor term, the
*S*’s referred to in the conclusion are precisely *the
same S*’s as those referred to in the minor premiss.

Again, the only difference between *Barbara* and
*Celarent*, or between *Darii* and *Ferio*, is that the
universal rule which the minor premiss enables us to apply to a
particular case is in *Barbara* and *Darii* a universal
affirmation, while in *Celarent* and *Ferio* it is a
universal denial.

We may, therefore, sum up all four moods in the following scheme:366

All B is C (or is not C), | (Rule) | |

All (or some) A is B, | (Case) | |

therefore, | All (or some) A is C (or is not C). | (Result) |

^{366} Compare C. S. Peirce in the
*Johns Hopkins Studies in Logic*, p. 148, and Sigwart,
*Logic*, i. p. 354. Sigwart gives the following formula:

If anything is M it is P (or is not P), | |

Certain subjects S are M, | |

therefore, | They are P (or are not
P). |

This way of setting out the valid moods of figure 1 shews clearly
how they are all included under the *dictum de omni et nullo*.

270. *Scheme of the Valid
Moods of Figure* 2.—Applying the principle of indirect
reduction, we may immediately obtain from the scheme given in the last
preceding section the following scheme, summing up the valid moods of
figure 2:367 337

All B is C (or is not C), | (Rule) | |

Some (or all) A is not C (or is C), | (Denial of Result) | |

therefore, | Some (or all) A is not
B. | (Denial of Case) |

^{367} Sigwart’s way of
putting it (*Logic*, i. p. 354) is that in figure 2, instead of
inferring from ground to consequence, we infer from invalidity of
consequence to invalidity of ground; and he gives the following
scheme:

If anything is P it is M (or is not M), | |

Certain subjects S are not M (or are M), | |

therefore, | They are not P. |

This scheme may be expressed in the following
dictum,—“If a certain attribute can be predicated,
affirmatively or negatively, of every member of a class, any subject
of which it cannot be so predicated does not belong to the
class.”368 This dictum may, like the *dictum de omni
et nullo*, claim to be axiomatic, and it is related to the valid
syllogisms of figure 2 just as the *dictum de omni et nullo* is
related to the valid syllogisms of figure 1.369

^{368} The dictum for figure 2,
sometimes called the *dictum de diverso*, is expressed in the
above form by Mansel (*Aldrich*, p. 86). It was given by Lambert
in the form, “If one term is contained in, and another excluded
from, a third term, they are mutually excluded.” This is at
least expressed loosely, since it would appear to warrant a universal
conclusion, if any conclusion at all, in *Festino* and
*Baroco*. Bailey (*Theory of Reasoning*, p. 71) gives the
following pair of maxims for figure 2,—“When the whole of
a class possess a certain attribute, whatever does not possess the
attribute does not belong to the class. When the whole of a class is
excluded from the possession of an attribute, whatever possesses the
attribute does not belong to the class.”

^{369} Lambert is usually regarded
as the originator of the idea of framing dicta that shall be directly
applicable to figures other than the first. Thomson, however, points
out that it is an error to suppose that Lambert was the first to
invent such dicta. “More than a century earlier, Keckermann saw
that each figure had its own law and its own peculiar use, and stated
them as accurately, if less concisely, than Lambert” (*Laws of
Thought*, p. 173, note). Distinct principles for the second and
third figures are laid down also in the *Port Royal Logic*, which
was published in 1662.

271. *Scheme of the Valid
Moods of Figure* 3.—Dealing with figure 3 in the same way as
we have done with figure 2, we get the following scheme, summing up
the valid moods of that figure:

Some (or all) A is
not C (or is C), | (Denial of Result) | |

All (or
some) A is B, | (Case) | |

therefore, | Some B is not C
(or is C). | (Denial of Rule) |

It is not easy to express this scheme in a single self-evident maxim.370 Separate dicta of an axiomatic character may, 338 however, be formulated for the affirmative and negative moods respectively of figure 3, namely, “If two attributes can both be affirmed of a class, and one at least of them universally so, then these two attributes sometimes accompany each other,” “If one attribute can be affirmed while another is denied of a class, either the affirmation or the denial being universal, then the former attribute is not always accompanied by the latter.”371

^{370} Lambert gave the following
*dictum de exemplo* for figure 3:—“Two terms which
contain a common part partly agree, or if one contains a part which
the other does not, they partly differ.” This maxim is open to
exception. The proposition “If one term contains a part which
another does not, they partly differ” applied to *MeP*,
*MaS*, would appear to justify *PoS* just as much as
*SoP*, or else to yield an alternative between these two. Mr
Johnson gives a single formula for figure 3, namely, “A
statement may be applied to part of a class, if it applies
wholly [or at least partly] to a set of objects that are at least
partly [or wholly] included in that class.” This is correct, but
perhaps not very easy to grasp.

^{371} These dicta (or dicta
corresponding to them) are sometimes called respectively the *dictum
de exemplo* and the *dictum de excepto*.

272. *Dictum for
Figure* 4.—The following *dictum*, called the *dictum
de reciproco*, was formulated by Lambert for figure 4:—“If no *M* is *B*, no *B* is this or that
*M* ; if *C* is (or is not) this or that *B*, there are
*B*’s which are (or are not) *C*.” The first part
of this *dictum* is intended to apply to *Camenes*, and the
second part to the remaining moods of the fourth figure; but the
application can hardly in either case be regarded as self-evident.
Several other axioms have been constructed for figure 4; but they are,
as a rule, little more than a bare enumeration of the valid moods of
that figure, whilst at the same time they are less self-evident than
these moods considered individually. The following axiom, however,
suggested by Mr Johnson, is not open to these criticisms: “Three
classes cannot be so related, that the first is wholly included in the
second, the second wholly excluded from the third, and the third
partly or wholly included in the first.” This *dictum*
affirms the validity of two antilogisms; in other words, it declares
the mutual incompatibility of each of the following trios of
propositions: *XaY*, *YeZ*, *ZiX* ; *XaY*,
*YeZ*, *ZaX* ; and it will be found that these incompatibles
yield the six valid moods of the fourth figure.372

273. Reduce *Barbara*
to *Bocardo*, *Bocardo* to *Baroco*, *Baroco* to
*Barbara*. [K.]

274. Reduce *Ferio* to figure 2, *Festino* to figure 3,
*Felapton* to figure 4. [K.]

275. Reduce *Camestres*
to *Datisi*. Why cannot *Camestres* be reduced either
directly or indirectly to *Felapton*? Can *Felapton* be
reduced to *Camestres*? [K.]

276. Assuming that in the
first figure the major must be universal and the minor affirmative,
shew by *reductio ad absurdum* that the conclusion in the second
figure must be negative and in the third particular. [J.]

277. State the following argument in a syllogism of the third figure, and reduce it, both directly and indirectly, to the first:—Some things worthy of being known are not directly useful, for every truth is worthy of being known, while not every truth is directly useful. [M.]

278. State the figure and
mood of the following syllogism; reduce it to the first figure; and
examine whether there is anything unnatural in the argument as it
stands:—

None who dishonour the king can be true patriots; for a
true patriot must respect the law, and none who respect the law would
dishonour the king. [J.]

279. “Rejecting the
fourth figure and the subaltern moods, we may say with Aristotle:
**A** is proved only in one figure and one mood, **E** in two
figures and three moods, **I** in two figures and four moods,
**O** in three figures and six moods. For this reason, **A** is
declared by Aristotle to be the most difficult proposition to
establish, and the easiest to overthrow; **O**, the reverse.”
Discuss the fitness of these data to establish the conclusion. [K.]

280. Prove, from the general
rules of the syllogism, that the number of possible moods,
irrespective of difference of figure, is 11.

In the 19 moods of the mnemonic verses, only 10 out of the possible
11 moods are represented. Find the missing mood, and account for its
absence from the verses. [L.]

281. Given

(1) the conclusion of a syllogism in the first figure,

(2) the minor premiss
of a syllogism in the second figure,

(3) the major premiss of a
syllogism in the third figure,

340 examine in each case how far the quality
and quantity of the two remaining propositions of the syllogism can be
determined (it being given that the syllogism does not contain a
strengthened premiss or a weakened conclusion).

Express the result, as far as possible, in general terms in each figure. [J.]

282. Find out in which of the valid syllogistic moods the combination of one premiss with the subcontrary of the conclusion would establish the subcontrary of the other premiss. [L.]

283. Construct a syllogism
in accordance with each of the following two dicta:—

(1) Any object that is found to lack a property known to belong to
all members of a class must be excluded from that class;

(2) If any objects that have been included in a class are found to
lack a certain property, then that property cannot be predicated of
all members of the class.

Assign the mood and figure of each argument, and shew the relations
between the above dicta and the *dictum de omni et nullo*. [L.]

284. Shew that any given mood may be directly reduced to any other mood, provided (1) that the latter contains neither a strengthened premiss nor a weakened conclusion, and (2) that if the conclusion of the former is universal, the conclusion of the latter is also universal [K.]

285. Shew that any given mood may be directly or indirectly reduced to any other mood, provided that the latter has not either a strengthened premiss or a weakened conclusion, unless the same is true of the former also. [K.]

286. Examine the following statement of De Morgan’s:—“There are but six distinct syllogisms. All others are made from them by strengthening one of the premisses, or converting one or both of the premisses, where such conversion is allowable; or else by first making the conversion, and then strengthening one of the premisses.” [K.]

287. Shew, by the aid of the process of indirect reduction, that the special rules for Figure 4 given in section 244 are mutually deducible from one another. [RR.]

288. *The application of
the Eulerian diagrams to syllogistic reasonings*.—In shewing
the application of the Eulerian diagrams to syllogistic reasonings we
may begin with a syllogism in *Barbara*:

All M is P, | |

All S is M, | |

therefore, | All S is P. |

The premisses must first be represented separately by means of the diagrams. Each yields two cases; thus,—

To obtain the conclusion, each of the cases yielded by the major
premiss must now be combined with each of those yielded by the minor.
This gives four combinations,373 and whatever is
true of *S* in terms of *P* in all of them is the conclusion
required.

^{373} These combinations afford a
complete solution of the problem as to what class-relations between
*S*, *M*, and *P* are compatible with the premisses;
and similarly in other cases. The syllogistic conclusion is obtained
by the elimination of *M*.

In each case *S* either coincides with *P* or is included
within *P* ; hence *all S is P* may be inferred from the
given premisses.

Next, take a syllogism in *Bocardo*. The application of the
diagrams is now more complicated. The premisses are

Some M is not P, |

All M is S. |

The major premiss yields three cases, namely,

and the minor premiss two cases, namely,

343 Taking them together we have six combinations, some of which themselves yield more than one case:—

344 So far as
*S* and *P* are concerned (*M* being left out of
account) these nine cases are reducible to the following three:

The conclusion, therefore, is *Some S is not P*.

It must be admitted that this is very complex, and that it would be a serious matter if in the first instance we had to work through all the different moods in this manner.374 Still, for purposes of illustration, this very complexity has a certain advantage. It shews how many relations between three terms in respect of extension are left to us, even with two premisses given.

^{374} Ueberweg, however, takes
the trouble to establish in this way the validity of the valid moods
in the various figures. Thomson (*Laws of Thought*, pp. 189, 190)
introduces comparative simplicity by the use of dotted lines. His
diagrams are, however, incorrect.

289. *The application of
Lambert’s diagrammatic scheme to syllogistic
reasonings*.—As applied to syllogisms, Lambert’s lines
are much less cumbrous than Euler’s circles. The main point to
notice is that it is in general necessary that the line standing for
the middle term should not be dotted over any part of its extent.375
This condition can be satisfied by selecting the appropriate
alternative form in the case of **A**, **I**, and **O**
propositions, as given in section 127. As examples we may represent
*Barbara*, *Baroco*, *Datisi*, and *Fresison* by
Lambert’s method.

^{375} The following
representation of *Barbara*,

illustrates the kind of error that is likely to result if the above
precaution is neglected. If this representation were correct we should
be justified in inferring *Some P is not S* as well as *All S
is P*.

290. *The application of
Dr Venn’s diagrammatic scheme to syllogistic
reasonings*.—Syllogisms in *Barbara*, *Camestres*,
*Datisi*, and *Bocardo* may be taken in order to shew how Dr
Venn’s diagrams can be used to illustrate syllogistic reasonings.

The premisses of *Barbara*,

*All M is P*,

*All S is M*,

exclude certain compartments as shewn in the following diagram:

This yields at once the conclusion *All S is P*.

346 Similarly for
*Camestres* we have the following:

For *Datisi* we have

*Bocardo* yields

It will be remembered that this scheme is based upon a particular
interpretation of propositions as regards their existential import.
The student will find it useful to attempt to represent by Dr
Venn’s diagrams a mood containing a strengthened premiss, for
example, *Darapti*.

291. Represent
*Celarent* by the aid of Euler’s diagrams. Will the same
set of diagrams serve for any other of the syllogistic moods? [K.]

292. Represent by means of
the Eulerian diagrams the moods *Festino*, *Datisi*, and
*Bramantip*. [K.]

293. Determine (i) by the
aid of Euler’s diagrams, (ii) by ordinary syllogistic methods,
what is all that can be inferred about *S* and *P* in terms
of one another from the following premisses, *Some M is P*,
*Some M is not P*, *Some P is not M*, *Some S is not
M*, *All M is S*. [K.]

294. Represent in Lambert’s scheme the mood