The Project Gutenberg EBook of Studies and Exercises in Formal Logic, by John Neville Keynes This eBook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org. If you are not located in the United States, you'll have to check the laws of the country where you are located before using this ebook. Title: Studies and Exercises in Formal Logic Author: John Neville Keynes Release Date: May 24, 2019 [EBook #59590] Language: English Character set encoding: UTF-8 *** START OF THIS PROJECT GUTENBERG EBOOK STUDIES, EXERCISES IN FORMAL LOGIC *** Produced by Ed Brandon from material at the Internet Archive STUDIES AND EXERCISES IN F O R M AL L O G I C 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 ������ MACMILLAN AND CO., LIMITED NEW YORK: THE MACMILLAN COMPANY 1906 [The Right of Translation and Reproduction is reserved] First Edition (Crown 8vo.) printed 1884. Second Edition (Crown 8vo.) 1887. Third Edition (Demy 8vo.) 1894. Fourth Edition (Demy 8vo.) 1906. PREFACE TO THE FOURTH EDITION. 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. vi 1 PREFACE TO THE FIRST EDITION. 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. viii 1 PREFACE TO THE SECOND EDITION. 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. ix PREFACE TO THE THIRD EDITION. 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. TABLE OF CONTENTS. 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