< STUDIES AND EXERCISES IN FORMAL LOGIC IN FOEMAL LOGIC INCLUDING A GENERALIZATION OF LOGICAL PROCESSES IN THEIR APPLICATION TO COMPLEX INFERENCES BY JOHN NEVILLE KEYNES, M.A., Sc.D. UNIVERSITY LECTURER IS MORAL SCIENCE AND FORMERLY FELLOW OF PEMBROKE COLLEGE IN THE UNIVERSITY OF CAMBRIDGE THIRD EDITION RE-WRITTEN AND ENLARGED Uonfcon MACMILLAN AND CO. AND NEW YORK 1894 [The Riyht of Translation and Reproduction is reserved] First Edition (Crown 8vo.) printed 1884. Second Edition (Crown 8uo.) 1887. Third Edition (Demy 8vo.) 1894. Cambridge : PRINTED BY C. J. CLAY, M.A., AND SONS, AT THE UNIVERSITY PRESS. 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 modifica- tions 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. J. N. KEYNES. 6, HARVEY KOAD, CAMBRIDGE, 25 July 1894. 2015203 TABLE OF CONTENTS. INTRODUCTION. PAGE 1. Definition of Formal Logic 1 2. Logic and Language 2 3. Logic and Psychology 4 PART I. TERMS. CHAPTER I. GENERAL AND SINGULAR NAMES. 4. The Logic of Terms 6 5. Categorematic and Syncategorematic Words .... 8 6. General and Singular Names 9 7. Proper Names 10 8. Collective Names 11 CHAPTER II. CONCRETE AND ABSTRACT NAMES. 9. Distinction between Concrete and Abstract Names . . 14 10. Can Abstract Names be subdivided into General and Singular ? 17 11. The Logical Characteristics of Adjectives . . . .18 Vlll TABLE OF CONTENTS. CHAPTER III. CONNOTATION AND DENOTATION. SECTION 12. The Extension and Intension of Names .... 13. Connotation, Subjective Intension, and Comprehension . 14. Connotative Names 15. Are proper names connotative 1 16. Are any abstract names connotative ? . . . . 17. Extension and Denotation ....'.. 18. Dependence of Extension and Intension upon one another 19. Inverse Variation of Extension and Intension . 20. Formal and Material treatment of Connotation 21 to 23. Exercises CHAPTER IV. REAL, VERBAL, AND FORMAL PROPOSITIONS. 24. Real, Verbal, and Formal Propositions 42 25. Nature of the Analysis involved in Analytic Propositions . 45 26. 27. Exercises . 48 CHAPTER V. FURTHER DIVISIONS OF NAMES. 28. Contradictory Terms 49 29. Contrary Terms 50 30. Positive and Negative Names 51 31. Infinite or Indefinite Names 53 32. Relative Names 54 33. Simple Terms and Complex Terms 56 34 to 36. Exercises 57 PART II. PROPOSITIONS. CHAPTER I. PROPOSITIONS AND THEIR PRINCIPAL SUBDIVISIONS. 37. Kinds of Propositions 58 38. Categorical, Hypothetical, and Disjunctive Propositions . . 59 39. An analysis of the Categorical Proposition .... 60 40. The Quantity and Quality of Propositions .... 61 TABLE OF CONTENTS. IX SECTION PAGE 41. Indefinite Propositions 63 42. Singular Propositions 63 43. Multiple Quantification . . 65 44. Signs of Quantity 66 45. The Distribution of Terms in a Proposition .... 69 46. Distinction between Subject and Predicate .... 70 47. Infinite or Limitative Propositions .- 72 48. Complex Propositions and Compound Propositions ... 73 49. The Modality of Propositions 76 50 to 52. Exercises 79 CHAPTER II. THE OPPOSITION OF CATEGORICAL PROPOSITIONS. 53. The Square of Opposition 80 54. Contradictory Opposition 82 55. Contrary Opposition 87 56. The Opposition of Singular Propositions 88 57. Possible Relations of Propositions into which the same Terms or their Contradictories enter 89 58 to 61. Exercises .91 CHAPTER III. IMMEDIATE INFERENCES. 62. The Conversion of Categorical Propositions .... 93 63. Simple Conversion and Conversion per accidens ... 95 64. Inconvertibility of Particular Negative Propositions . . 96 65. Legitimacy of Conversion 97 66. Table of Propositions connecting any two terms ... 99 67. The Obversion of Categorical Propositions .... 100 68. The Contraposition of Categorical Propositions . . . 101 69. The Inversion of Categorical Propositions .... 103 70. The Validity of Inversion 106 71. Summary of Results 107 72. Table of Propositions connecting any two terms and their contradictories . . . . . . . . .110 73. Mutual Relations of the non-equivalent Propositions con- necting any two terms and their contradictories . .111 74. The Elimination of Negative Terms 113 75. Other Immediate Inferences 116 76. Reduction of immediate inferences to the mediate form . . 120 77 to 87. Exercises . . 122 X TABLE OF CONTENTS. CHAPTER IV. THE DIAGRAMMATIC REPRESENTATION OF PROPOSITIONS. SECTION PAGE 88. The use of Diagrams in Logic . . . . . .125 89. Baler's Diagrams 127 90. Lambert's Diagrams 133 91. Dr Venn's Diagrams 136 92. Expression of the possible relations between any two classes by means of the prepositional forms A, E, I, . . 137 93. Euler's diagrams and the class-relations between S, not-S, P, not-P 140 94. Lambert's diagrams and the class-relations between S, not-S, P, not-P 144 95, 96. Exercises 146 CHAPTER V. PROPOSITIONS IN EXTENSION AND IN INTENSION. 97. Fourfold Implication of Propositions in Connotation and Denotation 147 (i) Subject in denotation, predicate in connotation . . 149 (ii) Subject in denotation, predicate in denotation . . 151 (iii) Subject in connotation, predicate in connotation . . 154 (iv) Subject in connotation, predicate in denotation . . 156 98. The Reading of Propositions in Comprehension . . .158 CHAPTER VI. LOGICAL EQUATIONS AND THE QUANTIFICATION OF THE PREDICATE. 99. The employment of the symbol of Equality in Logic . .160 100. Types of Logical Equations 162 101. The expression of Propositions as Equations .... 165 102. The eight prepositional forms resulting from the explicit Quantification of the Predicate 166 103. Sir William Hamilton's fundamental Postulate of Logic . 167 104. Advantages claimed for the Quantification of the Predicate . 168 105. Objections urged against the Quantification of the Predicate . 169 106. The meaning to be attached to the word some in the eight prepositional forms recognised by Sir William Hamilton . 171 107. The use of some in the sense of some only . . . .174 108. The interpretation of the eight Hamiltonian forms of propo- sition, some being used in its ordinary logical sense . .175 TABLE OF CONTENTS. XI SECTION PAGE 109. The propositions U and Y 176 110. The proposition TJ 177 111. The proposition w . . 178 112. Sixfold schedule of propositions obtained by recognising Y and TJ, in addition to A, E, I, 179 113. 114. Exercises ; .180 CHAPTER VII. THE EXISTENTIAL IMPORT OF CATEGORICAL PROPOSITIONS. 115. Existence and the Universe of Discourse . . . .181 116. Formal Logic and the Existential Import of Propositions . 183 117. Various Suppositions concerning the Existential Import of Categorical Propositions . . . . . . .186 118. Immediate Inferences and the Existential Import of Proposi- tions 188 119. The Doctrine of Opposition and the Existential Import of Propositions 192 120. The relation between the propositions All S is P and All not-SisP 196 121. Jevons's Criterion of Consistency 197 122. The Existential Import of General Categorical Propositions . 199 123. The Existential Import of Singular Propositions . . . 209 124 to 126. Exercises 210 CHAPTER VIII. CONDITIONAL AND HYPOTHETICAL PROPOSITIONS. 127. The distinction between Conditional Propositions and Hypo- thetical Propositions 211 128. The Import of Conditional Propositions 214 129. The Opposition of Conditional Propositions .... 218 130. Immediate Inferences from Conditional Propositions . . 219 131. The Import of Hypothetical Propositions .... 220 132. The Opposition of Hypothetical Propositions .... 224 133. Immediate Inferences from Hypothetical Propositions . . 227 134 to 137. Exercises 228 CHAPTER IX. DISJUNCTIVE (OR ALTERNATIVE) PROPOSITIONS. 138. The terms Disjunctive and Alternative as applied to Pro- positions 230 139. Two types of Alternative Propositions 230 Xll TABLE OF CONTENTS. SECTION PAGE 140. The Import of Alternative Propositions . . . . . 232 141. The Reduction of Alternative Propositions to the form of Conditionals or Hypotheticals 235 142. The Opposition of Alternative Propositions .... 236 143. Immediate Inferences from Alternative Propositions . . 236 144. 145. Exercises 238 PART III. SYLLOGISMS. CHAPTER I. THE RULES OF THE SYLLOGISM. 146. The Terms of the Syllogism 239 147. The Propositions of the Syllogism 241 148. The Rules of the Syllogism 241 149. Corollaries from the Rules of the Syllogism .... 243 150. Restatement of the Rules of the Syllogism .... 245 151. Dependence of the Rules of the Syllogism upon one another . 246 152. Statement of the independent Rules of the Syllogism . . 248 153. Two negative premisses may yield a valid conclusion ; but not syllogistically 249 154. Other apparent exceptions to the Rules of the Syllogism . 252 155. Syllogisms with two singular premisses . . . . . 253 156. Charge of incompleteness brought against the ordinary syllo- gistic conclusion 254 157. The connexion between the dictum de omni et nullo and the ordinary rules of the syllogism 255 158 to 188. Exercises . .257 CHAPTER II. THE FIGURES AND MOODS OF THE SYLLOGISM. 189. Figure and Mood 264 190. The Special Rules of the Figures ; and the Determination of the Legitimate Moods in each Figure .... 264 191. Weakened Conclusions and Subaltern Moods . . . . 268 192. Strengthened Syllogisms 269 193. The peculiarities and uses of each of the four figures of the syllogism 270 194 to 200. Exercises 272 TABLE OF CONTEXTS. XIII CHAPTER III. THE REDUCTION OF SYLLOGISMS. SECTION PAGE 201. The Problem of Reduction 274 202. Indirect Reduction 274 203. The mnemonic lines Barbara, Celarent, &c 276 204. The direct reduction of Baroco and Bocardo .... 280 205. Indirect reduction of moods usually reduced ostensively . 281 206. Extension of the doctrine of Reduction 282 207. Dicta for Figures 2, 3, and 4, corresponding to the Dictum for Figure 1 283 208. Is Reduction an essential part of the doctrine of the Syllo- gism ? . . 285 209. The Fourth Figure . . . ...... .288 210. Indirect Moods . . . ... . . 290 211 to 221. Exercises 292 CHAPTER IV. THE DIAGRAMMATIC REPRESENTATION OF SYLLOGISMS. 222. Euler's diagrams and syllogistic reasonings .... 294 223. Lambert's diagrams and syllogistic reasonings . . . 297 224. Dr Venn's diagrams and syllogistic reasonings . . . 298 225 to 229. Exercises 299 CHAPTER V. CONDITIONAL AND HYPOTHETICAL SYLLOGISMS. 230. The Conditional Syllogism, the Hypothetical Syllogism, and the Hypothetico-Categorical Syllogism . . . . 300 231. Distinctions of Mood and Figure in the case of Conditional and Hypothetical Syllogisms 301 232. Fallacies in Hypothetical Syllogisms 302 233. The Reduction of Conditional and Hypothetical Syllogisms . 303 234. The Moods of the Hypothetico-Categorical Syllogism . . 303 235. Fallacies in Hypothetico-Categorical Syllogisms . . . 305 236. The Reduction of Hypothetico-Categorical Syllogisms . . 306 237. Is the reasoning contained in the hypothetico-categorical syllogism mediate or immediate ? 306 238 to 243. Exercises 310 XIV TABLE OF CONTENTS. CHAPTER VI. DISJUNCTIVE SYLLOGISMS. SECTION PAGE 244. The Disjunctive Syllogism 312 245. The modus ponendo tollens 314 246. The Dilemma 316 247 to 249. Exercises 321 CHAPTER VII. IRREGULAR AND COMPOUND SYLLOGISMS. 250. The Enthymeme 322 251. The Polysyllogism 324 252. The Epicheirema 325 253. The Sorites 325 254. The special rules of the Sorites 328 255. The possibility of a Sorites in a Figure other than the First . 329 x 256. Ultra-total Distribution of the Middle Term .... 332 257. The Quantification of the Predicate and the Syllogism . . 334 258. Table of valid moods resulting from the recognition of Y and r\ in addition to A, E, I, 337 259. Formal Inferences not reducible to ordinary Syllogisms . 341 260 to 266. Exercises 345 CHAPTER VIII. EXAMPLES OF ARGUMENTS AND FALLACIES. 267 to 289. Exercises 346 CHAPTER IX. PROBLEMS ON THE SYLLOGISM. 290. Bearing of the existential import of propositions upon the validity of syllogistic reasonings ..... 355 291. Connexion between the truth and falsity of premisses and con- clusion in a valid syllogism ...... 359 292. 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 . 361 293. Numerical Moods of the Syllogism 365 294 to 316. Exercises 368 TABLE OF CONTENTS. XV PART IV. A GENERALIZATION OF LOGICAL PROCESSES IN THEIR APPLICATION TO COMPLEX PROPOSITIONS. CHAPTER I. THE COMBINATION OF TERMS. SECTION PAGE 317. Complex Terms . 378 318. Order of Combination in Complex Terms .... 380 319. The Opposition of Complex Terms 381 320. Duality of Formal Equivalences in the case of Complex Terms 383 321. Laws of Distribution 384 322. Laws of Tautology 385 323. Laws of Development and Reduction 386 324. Laws of Absorption 387 325. Laws of Exclusion and Inclusion 387 326. Summary of Formal Equivalences of Complex Terms . . 388 327. The Conjunctive Combination of Alternative Terms . . 388 328 to 332. Exercises . . . . ; . . . .389 CHAPTER II. COMPLEX PROPOSITIONS AND COMPOUND PROPOSITIONS. 333. Complex Propositions 391 334. The Opposition of Complex Propositions .... 391 335. Compound Propositions ........ 392 336. The Opposition of Compound Propositions .... 393 337. Formal Equivalences of Compound Propositions . . . 394 338. The Simplification of Complex Propositions .... 395 339. The Resolution of Universal Complex Propositions into Equivalent Compound Propositions 397 340. The Resolution of Particular Complex Propositions into Equivalent Compound Propositions 398 341. The Omission of Terms from Complex Propositions . . 399 342. The Introduction of Terms into Complex Propositions . . 400 343. Interpretation of Anomalous Forms 401 344 to 346. Exercises 402 XVI TABLE OF CONTENTS. CHAPTER III. IMMEDIATE INFERENCES FROM COMPLEX PROPOSITIONS. SECTION PAGE 347. The Obversion of Complex Propositions 403 348. The Conversion of Complex Propositions .... 404 349. The Contraposition of Complex Propositions . . . . 405 350. Summary of the results obtainable by Obversion, Conversion, and Contraposition 408 351 to 366. Exercises 410 CHAPTER IV. THE COMBINATION OF COMPLEX PROPOSITIONS. 367. The Problem of combining Complex Propositions . . .414 368. The Conjunctive Combination of Universal Affirmatives . 414 369. The Conjunctive Combination of Universal Negatives . . 415 370. The Conjunctive Combination of Universals with Particulars of the same Quality 417 371. The Conjunctive Combination of Affirmatives with Negatives 418 372. The Conjunctive Combination of Particulars with Particulars 418 373. The Alternative Combination of Universal Propositions . 418 374. The Alternative Combination of Particular Propositions . 418 375. The Alternative Combination of Particulars with Universals 419 376 to 379. Exercises . . 419 CHAPTER V. INFERENCES FROM COMBINATIONS OF COMPLEX PROPOSITIONS. 380. Conditions under which a universal proposition affords in- formation in regard to any given term . . . .421 381. Information jointly afforded by a series of universal propo- sitions with regard to any given term .... 423 382. The Problem of Elimination 426 383. Elimination from Universal Affirmatives .... 426 384. Elimination from Universal Negatives . . . . .427 385. Elimination from Particular Affirmatives .... 428 386. Elimination from Particular Negatives 429 387. Order of procedure in the process of elimination . . . 429 388 to 428. Exercises . 430 TABLE OF CONTENTS. XV11 CHAPTER VI. THE INVERSE PROBLEM. SECTION PAGE 429. Nature of the Inverse Problem 446 430. A General Solution of the Inverse Problem .... 448 431. Another Method of Solution of the Inverse Problem . . 452 432. A Third Method of Solution of the Inverse Problem . . 453 433. Mr Johnson's Notation for the Solution of Logical Problems . 455 434. The Inverse Problem and Schroder's Law of Reciprocal Equivalences 456 435 to 445. Exercises 457 APPENDIX. ON THE DOCTRINE OF DIVISION. > 446. Logical Division . 461 447. Physical Division, Metaphysical Division, and Verbal Division 462 448. Rules of Logical Division 463 449. Division by Dichotomy 464 450. The place of the Doctrine of Division in Logic . . . 466 * INDEX 469 xvm REFERENCE LIST OF INITIAL LETTERS SHEWING THE AUTHOR- SHIP OR SOURCE OF QUESTIONS AND PROBLEMS. B = Professor J. I. Beare, Trinity College, Dublin ; C = University of Cambridge ; J =W. E. Johnson, King's College, Cambridge ; K=J. N. Keynes, Pembroke College, Cambridge; L = University of London ; M = University of Melbourne ; N= Professor J. S. Nicholson, University of Edinburgh; = University of Oxford ; 0'S=C. A. O'Sullivau, Trinity College, Dublin ; R =the late Professor G. Groom Robertson ; T =F. A. Tarleton, Trinity College, Dublin ; V=J. 'Venn, Gonville and Caius College, Cambridge; W=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 EXEECISES IN FOKMAL LOGIC. INTKODUCTION. V t 1. Definition of Formal Logic. Formal logic may be defined as the science which investigates those regulative principles of thought that have universal validity whatever may be the particular objects about which we are thinking. It is a science which is concerned with the form as distinguished from the matter of thought. In a proposition some kind of relation is affirmed between certain objects of thought. By the matter of the proposition we mean the particular things thus related ; by its form we mean the mode of their relation. For example, in the pro- positions, " All men are mortal," " All crystals are solid," the form is the same while the matter is different ; in the proposi- tions, " All crystals are solid," " Some rich men are not happy," the form as well as the matter is different. If we express the matter of propositions by symbols to which any signification we please may be assigned, then attention is concentrated on their form; e.g., All S is P, Some S is not P. The employment of non-significant symbols of this kind is accordingly advisable in dealing with most of the problems which fall within the scope of formal logic. When we speak of a reasoning as being formally valid, we mean that its validity is determined solely by its form and is in no way dependent upon the particular subject-matter to which K. L. 1 2 INTRODUCTION. it relates. The cogency of a formally valid argument will therefore be unaffected, if for the particular terms involved others are substituted. The following is an example : All whales are mammals ; Some water animals are whales ; there- fore, Some water animals are mammals. In formal inference the conclusion is always implicitly contained in the premisses ; and mere consistency compels assent to the conclusion if the premisses are once admitted. Formal logic is accordingly sometimes spoken of as the logic of mere consistency ; and it follows that the observance of the laws which formal logic investigates will not do more than secure freedom from self-contradiction and inconsistency. Absolute truth cannot be guaranteed by formal logic. At the same time, to draw out correctly all that is essentially involved in any given statement or set of statements is a function of fundamental importance ; and the performance of this function alone may in many cases lead to knowledge that is to all intents and purposes new. Whether formal logic properly constitutes the whole of logic is a disputed question that we need not here attempt to decide. Accepting the above definition of formal logic, it is at any rate open to us to recognise also another branch of the science, in which we take account of the matter of thought and are con- cerned with all the methods of reasoning and research by the aid of which it is possible to advance in the attainment of truth. 2. Logic and Language. Some logicians, in their treat- ment of the problems of formal logic, endeavour to abstract not merely from the matter of thought but also from the language which is the instrument of thought ; they seek to deal ex- clusively with the thought-products as they exist in the mind, not with these same products as expressed in language. This method of treatment is not adopted in the following pages. In adopting 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 com- plexity are as a matter of fact carried on by the aid of language. That language is in this sense the universal instrument of thought LOGIC AND LANGUAGE. 3 will not be denied by any one ; and it seems a fair corollary that the principles by which valid thought is regulated, and more especially the application of these principles, cannot be adequately discussed, unless some 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 in logic until a precise interpretation has been placed upon certain forms of words as representing thought. It frequently happens that in everyday discourse the same prepositional form may admit of different interpretations, accord- ing to the context of to the subject-matter of the statement. But of context and subject-matter formal logic has no cogni- zance. It is, therefore, necessary to determine definitely which of these interpretations is in our further investigations to be adopted. 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, in other words, mean net-all as well as not-none, or its full meaning may be not-none; the logician must determine at the outset in which of these senses he will employ the word. Again, a disjunctive statement in ordinary speech may be understood to imply that the different alterna- tives are mutually exclusive, or it may not; the logician must fix his meaning. Now if thought were considered exclusively in itself, such questions as these could not arise ; they have to do with the expression of thought in language. The fact that they do arise and cannot help arising shews that actually to eliminate all consideration of language from logic is an im- possibility. Moreover, all the thoughts with which logic is concerned are expressed in language, reasonings being combinations of propositions, and propositions combinations of terms. To analyse the import of terms and propositions ought, therefore, to be recognised as included amongst the functions of the science. To investigate the form of conclusions obtainable from pre- misses of a given form is indeed the main object of formal logic. For this alone a discussion of the import of propositions is indispensable. It may be added that in the case of some writers who 12 4 INTRODUCTION. attempt to rise above mere considerations of language, the only result 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 conceptual/at. 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 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 be a mental science. 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, therefore, is still concerned primarily with thought, and only secondarily with language as the instrument of thought. 3. Logic and Psychology. Since the laws regulating the processes of formal reasoning are laws which depend upon the constitution of our minds, they fall within the cognizance of psychology as well as of logic. But they are regarded from different points of view by these two sciences. Psychology deals with them as laws in the sense of uniformities, that is, as laws in accordance with which men are found by experience normally to think and reason ; psychology investigates also their genesis and origin. Logic, on the other hand, deals with them purely 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 psychology and logic from LOGIC AND PSYCHOLOGY. 5 a slightly different standpoint, it may be said that the former is concerned with the actual, the latter 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 reasonings only as regards their cogency, and with the dependence of one judg- ment upon another only in so far as it is a dependence in respect of proof. Logic has thus a unique character of its own, and is not a mere branch of psychology. Psychological and logical dis- cussions 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 psycho- logical side of logic is comparatively but little touched upon. The metaphysical questions also to which logic tends to give rise are as far as possible avoided. PAET I. TERMS. CHAPTER I. GENERAL AND SINGULAR NAMES. 4. The Logic of Terms. A name is defined by Hobbes as " a word taken at pleasure to serve for a mark which may raise in our minds a thought like to some thought we had before, and which, being disposed in speech and pronounced to others, may be to them a sign of what thought the speaker had or had not before in his mind." In this definition the words "taken at pleasure" have rightly been criticised on the ground that they suggest an arbitrary and capricious selection, which is not in accordance with any generally accepted theory of the origin and growth of language. It is true that when an astronomer discovers a new planet and names it, or when a florist raises a seedling dahlia and gives it a distinctive title, or when an inventor names a new invention, the choice made is purely voluntary. But there are not many names in common use whose origin can thus be accounted for. It is agreed that the formation of names has for the most part been a natural and spontaneous process, involving nothing of the nature of deliberate invention and selection. Hobbes's definition has been further criticised on the ground that it is not wide enough to cover a many-worded name. Not CHAP. I.] LOGIC OF TERMS. 7 all names consist of a single word, e.g., Prime Minister, Lord Chief Justice of England. Accepting these criticisms, we may substitute for Hobbes's definition the following : A name is a word or set of words serving as a mark to raise in our minds a given idea, and also to indicate to others what idea is before the mind of the speaker. A term is a name regarded as the logical subject or predicate of a proposition. Since every name is the mark of something of which an affirmation or negation can be made, it follows that any given name is capable of being used as a term in some proposition or other. It is in their character as terms that names are of im- portance to the logician, and 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 term. 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 appropri- ately to psychology, metaphysics, or grammar ; and in the case of some of the questions which arise it is impossible 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 attri- butes is the function of metaphysics rather than of logic. The distinction to which by far the greatest importance attaches from the logical standpoint is that between connotation and denotation. A concept is defined by Sir William Hamilton as "the cognition or idea of the general character or characters, point or points, in which a plurality of objects coincide." In other words, a concept is the mental equivalent of a general name. With those logicians who seek to exclude from their science all consideration of language a Logic of Concepts takes the place of a Logic of Terms ; and in their treatment of this part 8 TERMS. [PART i. of the subject they discuss problems of a markedly psycho- logical character, as, for example, the mode of formation of concepts and the controversy between conceptualism and nomi- nalism. Apart, however, from the fact that the so-called conceptualist school do not draw so clear a line of distinction between logic and psychology, the difference between the two schools is to a large extent merely one of phraseology. Practi- cally the same points, for example, are raised whether we dis- cuss the extension and intension of concepts or the denotation and connotation of names. 5. Categorematic and Syncategorematic Words. A catego- rematic word is one which can by itself be used as a name ; a syncategorematic word is one which cannot by itself be used as a name, but only in combination with one or more other words. Any noun substantive in the nominative case, or any other part of speech employed as equivalent to a noun substantive, may be used categorematically. It is a disputed question whether adjectives can be regarded as categorematic per se, or whether they can be used categore- matically only by a grammatical ellipsis. No doubt adjectives sometimes stand alone as the predicates or (less frequently) as the subjects of propositions. But, on the other hand, adjectives per se and apart from any context are not complete names, since they have an essentially dependent character and must always be understood to qualify some substantive, expressed or unexpressed. For example, in treating round and equiangular as names we must regard them as equivalent to some such ex- pressions as round object and equiangular figure respectively. Thus, in such a proposition as " the rich are not to be envied," " the rich " is equivalent to " rich persons " or " those who are rich," and the real subject of the proposition must be considered to be a substantive, although by a grammatical ellipsis it is expressed by an adjective. Again, when an adjective stands alone as the predicate of a proposition, we cannot discuss its characteristics as a name except by considering it to qualify either the substantive which appears as the subject or else the name of some wider class which includes the subject 1 . 1 Compare De Morgan, Formal Logic, p. 42. CHAP. I.] GENERAL AND SINGULAR NAMES. 9 Any part of speech, or the inflected cases of nouns sub- stantive, may be used categorematically by what has been termed a suppositio materialis, that is, by making a statement about the mere word itself regarded as a group of certain letters or as representing a certain sound ; e.g., " John's is a possessive case," "Rich is an adjective," " With is an English word." 6. 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 things ; a singular or individual name is a name which is understood in the particular circumstances under which it is employed to denote some one determinate thing only. The nature and logical importance of this distinction may be illustrated by considering names as the subjects of proposi- tions. 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 maybe 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 a particular portion of space, singular if we mean space in the aggregate. Water is general. Professor Bain takes a different view here; he says, "Names of Material earth, stone, 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 10 TERMS. [PAET I. 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. 1 A name is to be regarded as general if it may be potentially predicated of more than one object, although it accidentally happens that as a matter of fact it can be actually 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. 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 refer- ence, e.g., the first man, the present Lord Chancellor, the author of Paradise Lost. A special class of singular names which have sometimes been erroneously supposed to be the only true singular names will be discussed in the next section. 7. Proper Names. A proper name is a name assigned as a mark to distinguish an individual person or thing from others, but having no intrinsic significance beyond the fact of denoting the individual in question. Proper names form a sub-class of singular names, being distinguished from the singular names 1 Terms of the kind here under discussion are called by Jevons substantial terms. (See Principles of Science, chap. 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. CHAP. I.] PROPER NAMES. 11 discussed in the preceding section in that they denote individual objects without at the same time conveying any information as to the special attributes possessed by those objects. 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 under- stood 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 1 . The exception above referred to is 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 ellip- tical; written out more fully, they become all persons called Victoria, some individuals named John. 8. 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 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 . 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 29th regiment of foot, the English nation, the Bodleian Library. But if it is capable 1 Professor Bain brings out this distinction very clearly 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." 2 To collective name as above defined there is no distinctive antithetical term in ordinary use. Mr Johnson, however, suggests the word unitary. Thus a unitary name would be defined as one denoting an object which we regard as being itself the only unit with which we are immediately concerned. The anti- thesis between the collective and distributive use of names relates, as we shall see, to predication only. 12 TERMS. [PART i. 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 1 . 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 of the above alternatives must be regarded as misleading, if not actually erroneous; for the class of collective names overlaps, as we have just seen, each of the other classes. The correct and really important logical antithesis is be- tween the collective and 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*. 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 1 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). 2 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." CHAP. I.] COLLECTIVE USE OF NAMES. 13 taken together, while in the second it is true only of each of them taken separately ; in the first case, therefore, the tenn is used collectively, in the second distributively. Compare again the propositions, "The people filled the church," "The people all fell on their knees." 1 1 "When in an argument we pass from the collective to the distributive use of a term, or vice versa, we have what are technically called fallacies of division and of composition respectively. 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 fonner) 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. CHAPTER II. CONCRETE AND ABSTRACT NAMES. 9. Nature of the distinction between Concrete and Abstract Barnes 1 . The distinction between concrete and abstract names 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. Our definitions may, there- fore, be made rather 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 ; an abstract name is the name of anything which is regarded as an attribute of some- thing else, i.e., as an attribute of subjects 2 . 1 The account of the distinction between concrete and abstract names given in this section differs from that given in previous editions ; but the substantial difference of view is not so great as might at first sight appear. The objection may perhaps be raised that the treatment of abstract and concrete names now adopted practically reduces the distinction between them to insignificance. No importance need, however, be attached to this objection, seeing that the dis- tinction cannot in any case be regarded as having much logical value. 2 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 altogether to avoid the term substance in this -connexion. CHAP. II.] CONCRETE AND ABSTRACT NAMES. 15 This distinction is in most cases easy of application; for example, 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 1 . 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 2 . Using the terms connote and denote in their technical sense, as defined in the following chapter, an abstract name denotes the quali- ties 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 associated. For when two names are given which are thus 1 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. At the same time, it is of course true that an abstract name involves a higher degree of abstraction than the corresponding concrete name. There is so much risk of confusion here that the terms abstract and concrete must be considered not well chosen for marking the distinction had in view. - Thus, in the case of every general concrete name there is or may be con- structed 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 Ctesariam and Bismarckism. These names, however, do not denote all the differentiating attributes of Ctesar 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 the last paragraph but one of section 15. 16 TERMS. [PART i. related, there will never be any difficulty in determining which is concrete and which is abstract in relation to the other. But whilst the distinction is absolute and unmistakeable when names are thus given in pairs, it is by no means always easy of application when we consider names in themselves and not in this definite relation to other names. We shall find the simplest solution of the various problems which arise by admitting at the outset that 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, which is primarily an abstract name can also be used in such a way that it is, ac- cording 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 land- scape. We arrive then 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 pos- sessing attributes. They are abstract names when viewed in one relation, concrete when viewed in another 1 . 1 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, CHAP. II.] ARE ANY ABSTRACT NAMES GENERAL ? 17 10. Can the distinction between Generals and Singulars be applied to Abstract Names? The question whether any ab- stract names can be considered general has given rise to much difference of opinion amongst logicians. On the one hand, it is argued that all 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 1 . 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"* when we talk of the Deity meaning thereby God, not the qualities of God. Jevons (Elementary Lessons in Logic, pp. 21, 22) gives other examples : " Rela- tion properly is the abstract name for the position of two people or things to each other, and those people are properly called relatives. But we constantly speak now of relations, meaning the persons themselves ; and when we want to indicate the abstract relation they have to each other we have to invent a new abstract name relationship. Nation has long been a concrete term, though from its form it was probably abstract at first ; but so far does the abuse of language now go, especially in newspaper writing, that we hear of a nationality, meaning a nation, although of course if nation is the concrete, nationality ought to be the abstract, meaning the quality of being a nation." 1 This represents the view taken by Jevons. "Abstract terms are strongly distinguished from general terms by possessing only one kind of meaning ; for as they denote qualities there is nothing which they can in addition imply. The adjective ' red' is the name of red objects, but it implies the possession by them of the quality redness ; but this latter term has one single meaning the quality alone. Thus it arises that abstract terms are incapable of number or plurality. Bed objects are numerically distinct each from each, and there are a multitude of such objects ; but redness is a single existence which runs through all those objects, and is the same in one as it is in another. It is true that we may speak of rednesses, meaning different kinds or tints of redness, just as we may speak of colours, meaning different kinds of colours. But in distinguishing kinds, degrees, or other differences, we render the terms so far concrete. In that they are merely red there is but one single nature in red objects, and so far as things are merely coloured, colour is a single indivisible quality. Redness, so far as it is redness merely, is one and the same everywhere, and possesses absolute oneness or unity" (Principles of Science, ch. 2, 3). - Thus Mill writes : " Some abstract names are certainly general. I mean those which are names not of one single and definite attribute, but of a class of K. L. 2 18 TERMS, [PART i. * A mode of reconciling these opposing views is to be found in the recognition that the same name may be regarded as either abstract or concrete according to the point of view taken. The name of an attribute can be described as general only in so far as the attribute is regarded as exhibiting character- istics which vary in different instances, only in so far, that is to say, as it is itself a subject of attributes. But from this point of view the name, according to the definitions given in the preceding section, is concrete. Thus in the proposition, " Some colours are painfully vivid," we are predicating an attribute of a subject ; similarly in the propositions, " All yellows are agree- able to me/' "Some courage is the result of ignorance"; we must, therefore, regard the subjects of these propositions as being here used in a concrete sense. We arrive then at the conclusions, first, that an abstract name considered strictly as such, that is to say, as the name of an attribute possessed by a certain class of objects, and without regard to any differences in the manner in which the attribute manifests itself, cannot properly be described as general ; but, secondly, that names which are primarily names of attributes can also be used in a concrete sense, and may then become at the same time general 1 . 11. The Logical Characteristics of Adjectives. The ques- tion whether adjectives can be regarded as names per se, or attributes. Such is the word colour, which is a name common to whiteness, redness, &c. Such is even the word whiteness, in respect of the various shades of whiteness to which it is applied in common ; the word magnitude, in respect of the various degrees of magnitude and the various dimensions of space ; the word weight, in respect of the various degrees of weight. Such also is the word attribute itself, the common name of all particular attributes. But when only one attribute, neither variable in degree nor in kind, is designated by the name ; as visibleness ; tangibleness ; equality ; squareness ; milk-whiteness ; then the name can hardly be considered general ; for though it denotes an attribute of many different objects, the attribute itself is always conceived as one, not many" (Logic, i. ch. 2, 4). 1 But like other general names they may be individualised by means of an individualising prefix, as in the propositions, "My health is not good," "Your happiness is my only object," " His honesty was not proof against such a temptation." CHAP. II.] ADJECTIVES. 19 only by a grammatical ellipsis, has been already considered. Whichever view may be taken on this point, it seems clear that if adjectives are classified as names at all, they are concrete and general. For an adjective is essentially a name which can be applied to whatever possesses a certain attribute ; great, for instance, is the name of whatever possesses the attribute of greatness. But a name which applies to an indeterminate number of objects is general, and the name of a subject of attributes is concrete. Hence the conclusion above indicated follows immediately from our definitions of concrete and general names. No qualification is necessary in order to meet the case of an adjective appearing as the predicate in such a proposition as Unpunctuality is irritating. Here unpanctuality, though an attribute, is regarded as itself the subject of a further attribute. It is, therefore, regarded from the point of view from which it is a concrete name, and the predicate of the proposition is ac- cordingly concrete also. An adjective may, however, form part of an abstract name, e.g., we may speak of great beauty or of great strength; it may also form part of a singular term, e.g., we may speak of Alex- ander the Great or of the great Goliath. In cases like these, therefore, combinations of names containing adjectives may be abstract or singular. But it does not follow that adjectives considered by themselves need be regarded as abstract or singular, any more than that such a term as man is itself singular because it forms part of the singular term the first man. 22 CHAPTER III. CONNOTATION AND DENOTATION. 12. The Extension and Intension of Names. 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 accord- ingly be employed to express in the most general way these two aspects of general names respectively 1 . Thus, by the extension of plane triangle we mean a certain class of geometrical figures, and by its intension certain pro- perties belonging to such figures. Similarly, by the extension of man is meant a certain class of material objects, and by its intension the properties of rationality, animality, &c., belonging to these objects. 13. 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 1 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 their meanings. CHAP. III.] INTENSION OF NAMES. 21 of view from which the properties of the corresponding class may be regarded; and it is to a want of discrimination between these points of view that we may largely attribute the con- troversies and misunderstandings to which the problem of the connotation of names has given rise in such abundance. (1) There are those properties which are essential to the class in the sense that the name implies them in its definition. Were any of this set of properties 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 properties 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 properties will include the marks Jby 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 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 properties actually possessed in common by every member of the class. These will in- clude all the qualities included under the two preceding 22 TERMS. [PART i. heads 1 , and usually many others in addition. The standpoint here taken is objective 2 . 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 have been selected by different logicians, sometimes without any clear recognition of divergence from the usage of other writers. It is of the greatest importance 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 will be 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 properties possessed in common by the class ; nor do we necessarily mean those particular properties which may be mentally associated with the name ; but we mean just those properties on account of the possession of which any individual is placed in the class, or 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 triangle, since it is not a property upon which the classification of triangles into equilateral and non-equilateral is based ; al- though 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 JSevv 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 - 1 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. 2 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. CHAP. III.] MEANINGS ASSIGNED TO CONNOTATION. 23 hoofed as part of the meaning of ruminant, and we may say with Mill that were an animal 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 here to be observed 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, therefore, is 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 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 24 TERMS. [PART i. common by all the objects so denoted" (Mind, 1881, p. 532). Professor Jevons also 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) 1 . While rejecting the use of the term connotation in any but the first of the above mentioned senses, it will be found desirable also to have terms which can be used with the other meanings which have been indicated. The three terms connotation, intension, and comprehension are commonly em- ployed 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 impli- cational 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, and we need not seek to invent a single term to be used as its equivalent 2 . Conventional intension or connotation will then include only those attributes which constitute the meaning of a name ; sub- jective intension will include those that are mentally associated with it, whether or not they are actually signified by it 3 ; ob- jective intension or comprehension will include all the attributes 1 Professor 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. 2 In the second edition what is here called subjective intension was called simply intension. It is, however, important to have a term which can be used in the more general sense, and for this purpose no other term seems as suitable as intension (with its correlative extension). 3 It is clear that subjective intension is likely to vary with each individual. How far connotation also may vary with each individual will be discussed in a subsequent section. CHAP. III.] CONNOTATIVE NAMES. 25 possessed in common by all members of the class denoted by the name. We might perhaps speak more strictly of the con- notation of the name itself, the subjective intension of the notion which is the mental equivalent of the name, and the compre- hension of the class which is denoted by the name 1 . 14. 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. ch. 2, 5). According to this definition, a connotative name must not only possess extension, but must also have a con- ventional intension assigned to it. 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 disputed one, 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 attri- butes" (Logic, I. p. 33). To this point also we shall return in a later section. 15. Are proper names connotative^ To this question ab- solutely contradictory answers are given by ordinarily clear thinkers as being obviously correct. To some extent, however, 1 The distinctions of meaning indicated in this section will be found absolutely essential for clearness of view in discussing certain questions to which we shall pass on immediately ; in particular, the questions whether proper names are connotative, and whether connotation and denotation necessarily vary inversely. 26 TERMS. [PART i. the divergence is merely verbal, the term connotation being used in different senses. Mill speaks decisively, "The only names of objects which connote nothing are proper names ; and these have, strictly speaking, no signification" (Logic, I. p. 36). The opposite view is taken by Jevons, Mr Bradley, and others. It is necessary at the outset to guard against a misconcep- tion which quite obscures the point really 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, 2nd ed. p. 27, with a reference to Mill in the foot-note). But Mill dis- tinctly says that some singular names are connotative, e.g., the sun 1 , the first emperor of Rome (Logic, I. pp. 34, 35). Again, Jevons says, "There would be an impossible breach of con- tinuity in supposing that after narrowing the extension of 'thing' successively down to animal, vertebrate, mammalian, man, Englishman, educated at Cambridge, mathematician, great logician, and so forth, thus increasing the intension all the time, the single remaining step of adding Augustus de Morgan, Professor in University College, London, could re- move all the connotation, instead of increasing it to the utmost point" (Studies in Deductive Logic, pp. 2, 3). But every one would allow that we may narrow down the extension of a term till it becomes individualised without destroying its connota- tion ; " 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. It must then be clearly understood that only one class of singular names, namely, proper names, are affirmed to be non- 1 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, vol. 2, 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 special attributes. CHAP. III.] ARE PROPER NAMES CONNOTATIVE ? 27 connotative ; and in regard to these, as defined in section 7, it would seem to be hardly more than a verbal statement to say that they are so. Perhaps, however, those who regard proper names as connotative may criticise the definition referred to as question-begging ; and, in any case, there remains a source of ambiguity connected with the meaning of the term connotation. If we differentiate our terms and use them in the senses indicated in section 13, then we must say that while proper names have no connotation, they nevertheless have both sub- jective 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 1 . 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, the whole of the properties which belong to the individual denoted. If then by the connotation of a name we meant all the attributes possessed by the individuals denoted by it, or if we meant the attributes suggested by it, Jevons's view would be correct. One or other of these alternatives does appear to be what Jevons himself means, but it is distinctly not what Mill means; he means only those attributes which are in the strictest sense signified by the name. 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 1 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 Nyanzu. Some further remarks bearing on this point will be fouud later on in this section. 28 TERMS. [PART i. 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 John Duke Coleridge and the Lord Chief Justice of England, They denote the same individual, and I should recognise John Duke Coleridge and the Lord Chief Justice of England by the same attributes; but the names are not equivalent the one is given to a certain indi- vidual as a mere mark to distinguish him from others, and it 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. Nor is it true that such a name as " John Smith " connotes " Teuton, male, &c." John Smith might be a dahlia, or a race- horse, 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. Still, it may fairly be said that many, if not most, proper 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 cormotative when first imposed soon cease to CHAP. III.] ARE PROPER NAMES CONNOTATIVE ? 29 be connotative in the way in which other names are connota- tive. Their application is in no way dependent on the con- tinuance of the attribute with reference to which they were originally given. As Mill puts it, the name once given is inde- pendent 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 call him 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 1 . 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. Before leaving the subject of proper names, attention may be called to a class of singular names, such as Miss Smith, Captain Jones, President Cleveland, the Lake of Lucerne, the Falls of Niagara, which may be said to be partially but only partially connotative. Their peculiarity is that they contain as elements terms having a general and permanent signification, and that some change in the object denoted might conse- quently 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 their application. They occupy an intermediate position, therefore, between connotative singular names, such as the first man, and strictly proper names. 16. Are any abstract names connotative? A connotative 1 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). 30 TERMS. [PART i. name is one which denotes a subject and implies an attribute ; and, according to our definition of a concrete name, every name given to a class of things in virtue of some quality or set of qualities which they have in common is concrete. It follows immediately that all connotative names are concrete ; and hence, if we regard the names abstract and concrete as strictly contradictories, the above question may at once be answered in the negative. According, however, to the definitions adopted in the pre- ceding chapter, the terms abstract and concrete, as applied to names, are not mutually exclusive. In so far as attributes can be considered as themselves possessing attributes, attribute- names may be described either as abstract or as concrete, ac- cording to the point of view from which they are regarded, and it cannot, therefore, be said that no attribute-names are conno- tative. Thus virtue considered in relation to virtuous is to be described as abstract; it denotes the attribute which virtuous connotes, that is to say, the attribute on account of the posses- sion of which we describe a person or an action as virtuous. But considered in relation to the various virtues (courage, temperance, &c.) which constitute its sub-classes, virtue is to be -described as concrete ; it may then be said to denote these various virtues, and to connote the common property by reason of which they are classed together. Our conclusion then is that names of attributes names, therefore, which are primarily abstract may be regarded as connotative ; but only in so far as they take on at the same time a concrete character. 17. 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 usually an implied reference to some more or less limited universe of discourse. 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 ^CHAP. III.] EXTENSION AND DENOTATION. 31 fictitious persons or all beings possessing the essential charac- teristics of men whom we are able to conceive or imagine. 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 rather than the term extension when there is an explicit or im- plicit limitation to the objects actually to be found in some restricted universe. By the 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 extension, but its denotation in a universe which is in some way restricted by time, place, or circumstance may be zero 1 . In the sense here indicated, denotation is in certain respects the correlative of comprehension rather than of connotation. 1 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. Unicom, 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. Iii 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 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 must 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). 32 TERMS. [PART i. For in speaking of denotation we are, as in the case of compre- hension, taking an objective standpoint ; and there is, more- over, in the case of comprehension, as in that of denotation, a tacit reference to some defined universe of discourse. We shall, however, find that, in one very important respect, connotation and denotation are still correlatives. 18. Dependence of Extension and Intension upon one another 1 . Taking any class-name X, let us first suppose that there has been a conventional agreement to use it wherever a certain arbitrarily selected set of properties, P I} P 2 ,...P m , are present. This set of properties will constitute the connotation of X, and will, with reference to a given universe of discourse 2 , determine the denotation of the name, say, Q I} Q z ,...Q y ', that is, Q I} Qs>---Qy are a ^ tue individuals possessing in common the properties P t , P 2 ,...P m . These properties may not, and almost certainly will not, exhaust the properties common to Q I} Q. i} ...Q y . Let all the common properties be P,, P. 2 ,...P X ; they will include P I} P 2 , ...P m , and iu all probability more besides, and will constitute the comprehension of the class-name. Now it will always be possible in one or more ways to make out of Q t , Q 2 ,...Q y , a selection Q v , Q t ,...Q n , which shall be pre- cisely typical of the whole class 3 ; that is to say, Q lt Q 2 ,...Q n will possess in common those attributes and only those attributes (namely, P,, P i} ...P x ) which are possessed in common by Q I} Qv-Qy'- Qi> Qv'Qn may be called the exemplification or ex- 1 This section may be omitted on a first reading. 2 It will be assumed in the remainder of this section that we are throughout speaking with reference to a given universe of discourse. 3 It may chance to be necessary to make Q lt Q 2 , ... Q n coincide with Qi Qzi-'Qy But this must be regarded as the limiting case; usually a smaller number of individuals will be sufficient. 4 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 CHAP. III.] RELATIONS BETWEEN EXTENSION AND INTENSION. 33 tensive definition of X. The reason for selecting the name ex- tensive definition will appear in a moment. It will sometimes be convenient correspondingly to speak of the connotation of a name as its intensive definition. We have then, with reference to X, (1) Connotation: P^.-P; (2) Denotation: Q,...Q n ...Q y ; (3) Comprehension: P l ...P m ...P x ; (4) Exemplification: Q^.-Qn- 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 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 denned 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 lt P 2 , P 3 , P 4 , P 6 , P 6 ; and let A^ be an attribute possessed by Pj alone, A 2 an attribute possessed by P 2 alone, and so on. Then if we recog- nise A 1 or A 2 or A 3 as a distinct attribute, it is at once clear that Pj, P 2 , P 3 will no longer be typical of the whole class ; and the same result follows if not-A t 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 solicitor *. 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 pro- fessional 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. K. L. 3 34 TERMS. [PART i. we start from the connotation or from the exemplification everything else will come out the same 1 . 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 2 . The whole class thus constituted are, however, found to possess other properties in common be- sides 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 pre- cisely typical of the whole class 3 ; that is to say, his selected list would possess in common only such properties as are com- mon to the whole class. This selected class would constitute the exemplification of the name. 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 choice 4 (iv) connotation. 1 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. 2 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. 3 He would take metals as different from one another as possible, such as aluminium, antimony, copper, gold, iron, sodium, zinc. 4 It is ordinarily said that "of the denotation and connotation of a term one CHAP. III.] EXTENSIVE AND INTENSIVE DEFINITIONS. 35 It is important from a theoretical point of view to note the possibility of this second orcler of procedure ; and it may, more- over, 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 re- semblances long before they are able to give an intensive defi- nition 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 enume- rate 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 ivime of the class of which Q lt Q 2 ,...Q n are typical. This primitive form of definition may also be called definition by type 1 . Undoubtedly, however, it is far more usual, as well as really simpler, to start from an intensive definition, and this in nearly every case corresponds with the ultimate procedure of science. may, both cannot, be arbitrary," and this is broadly true. It is possible, how- ever, 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, con- notation 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 cquiaiujulur 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 possessing in common only those attributes which are also possessed by the whole class. 1 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. 32 36 TERMS. [PART i. For logical purposes, it is accordingly best to assume this order of procedure, unless a,n explicit statement is made to the contrary 1 . 19. Inverse Variation of Extension and Intension 9 . In general, as intension is increased or diminished, extension is diminished or increased accordingly, and vice versa. If, for ex- ample, rational is added to the connotation of animal, the deno- tation is diminished, since all irrational animals are now ex- cluded, whereas they were previously included. On the other hand, if the denotation of animal is to be extended so as to in- clude the vegetable kingdom, it can only be by omitting sensitive from the connotation. Hence the following law has been formu- lated : " In a series of common terms standing to one another in a relation of subordination 3 the extension and 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 sense. It is certainly not true that whenever the number of 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 conno- tation 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 in- tension of a name will necessarily be accompanied by some 1 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. 2 This section may be omitted on a first reading. 3 As in the Tree of Porphyry : Substance, Corporeal Substance (Body), Animate Body (Living Beiug), Sensitive Living Being (Animal), Rational Animal (Man). In this series of terms the intension is at each step increased, and the extension diminished. CHAP. III.] VARIATION OF DENOTATION WITH CONNOTATION. 37 diminution or increase of the extension as the case may be, and vice versa. We will discuss the alleged law in this form, con- sidering, first, connotation and denotation, exemplification and comprehension ; and, secondly, denotation and comprehension. 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, 6, c. It may happen that in fact wherever the attributes a and b are present, the attributes c and d are also present. 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 conno- tation 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 op- posite direction. Some individuals possessing the attributes a and b may lack the attributes c or d ; but no individuals pos- sessing 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 1 . 1 Since reference is here made to the actual denotation of a term in some more or less restricted universe of discourse, the above law may be said to turn partly on material, and not on purely formal, considerations. It should, there- fore, 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 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 JTs that 38 TERMS. [PART i. (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 exemplifi- cation (extensive definition) of a term is arbitrarily enlarged or restricted, the comprehension in an assigned universe of dis- course will either remain unaltered or will change in the op- posite direction. B. We may now consider the relation between the com- prehension and the denotation of a term. Let P 1} P 2 ,...P X be the totality of attributes possessed by the class X, and Q l} Q. 2> ... Q y the totality of objects included in the class X. Both these groups are objectively, not arbitrarily 1 , determined ; and the relation between them is reciprocal. P 1} P 2 ,...P X are the only attributes possessed in common by the objects Q l} Q z ,...Qy; and Qi, Qi,"'Q y are the only objects possessing all the attributes P P P *- \> * 2>"- r a;- 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 remits 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 F 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 F 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 versa. (a) Let X and F be determined by connotation or intensive are not d is necessarily excluded, while it was previously included, in the extension of the term X. Hence, if the connotation of a term is arbitrarily enlarged or restricted, the extension will be potentially restricted or enlarged accordingly. Cf. Jevons, Principles of Science, ch. xxx. 13. 1 What is arbitrary is the intensive definition (P lt P 2 , ... P m ) or the extensive definition (Q lt Q z , ... Q n ) which determines them both. CHAP. III.] VARIATION OF COMPREHENSION WITH DENOTATION. 3D definition. Thus, let X be determined by the set of properties PJ...P, and Fby the set P 1 ...P 7n+1 , which includes the addi- tional property P, n+1 . Then P m +i either does or does not always accompany P,...P. If the former, no object included in the denotation of X is excluded from that of F, so that the denotations of X and Y are the same ; and it follows that the comprehensions of X and Y are also the same. If the latter, then the denotation of F is less than that of X by all those objects that possess P l ...P m without also possessing P OT+1 . At the same time, the comprehension of F includes at least P m+ i in addition to the properties included in the com- prehension of X. (b) Let X and F be determined by exemplification or extensive definition. Thus, let X be determined by the set of examples Qi...Q n , and F by the set Qi...Q n +i, which includes the additional object Q n +\- Then Q n+1 either does or does not possess all the properties common to Qi-..Q n . If the former, no property included in the comprehension of X is excluded from that of F, so that the comprehensions of X and Fare .the same; and it follows that the denotations of X and F are also the same. If the latter, then the comprehension of F is less than that of X by all those properties that belong to Qi...Q n without also belonging to Q n+1 . At the same time, the denotation of F includes at least Q n+1 in addition to the objects included in the denotation of X. All cases have now been considered, and it has been shewn that the law above formulated holds good universally. This law and the laws given on pages 37 and 38 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 com- prehension or denotation, no reference is intended to changes in things or in our knowledge of them. The variation is always 40 TERMS. [PART i. supposed to have originated in some arbitrary alteration in the intensive or extensive definition of a given terra, 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 compre- hension 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 coming under the denotation of the name, and still its comprehension may remain the same. Or possibly new qualities may be developed by the whole of the class. In these cases, however, there is no arbi- trary alteration in the application or implication of the name, and hence no real exception to what has been above laid down. 20. Formal and Material treatment of Connotation. In speaking of the connotation of a name we may have in view either the signification that the name bears in common accep- tation, or some special meaning that a given individual may choose to assign to it. It has to be borne in mind 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 dif- ferent attributes in the connotation of the name ; and not un- frequently some of us may be unable to say precisely what is the meaning that we ourselves attach to the words we use. In formal logic, however, it is necessary to work on the assumption that every name has a fixed and definite connotation. In other words, we assume that every name employed is either used in its ordinary sense and that this is precisely determined, or else that, being used with a special meaning, this meaning is ad- hered to consistently and without equivocation. Formal logic is indifferent to what particular connotation is attached to any given term ; but it prescribes absolute consistency. Mill in his treatment of connotation goes beyond this. He discusses the principles in accordance with which the connota- tion of names should be determined 1 . This is the treatment of 1 Logic, Book i. chapter 8 ; Book iv. chapter 4. CHAP. III.] EXERCISES. 41 the subject proper to material or applied logic 1 . When we de- fine names already in use our object is to give them, that which formal logic assumes them to have, a fixed and definite connotation. It may be observed that in the case of an ideal language properly employed every name would have the same fixed and precise meaning for everyone. EXERCISES. 21. Enquire whether the following names are respectively con- notative or non-connotative : Ccesar, Czar, Lord Beaconsfteld, t/ie highest mountain in Europe, Mont Blanc, the Weisshorn, Greenland, the Claimant, the pole star, Homer, a Daniel come to judgment. [K.] 22. " A proper name at least connotes that the object called by the name is identical with that to which it has previously been as- signed." Examine this statement. [j.] 23. P means AB ; and it is found that all A's are B, but not all B's are A. Determine the relations between P and A in regard to connotation and denotation, and between the common attributes of A and of B. [j.] 1 If we are to advance in accurate knowledge, all sources of ambiguity must be cleared out of the way. One of the means, moreover, whereby we effect progress in science is by making our conceptions more clear, and our classi- fications more appropriate. For these ends we must aim at constructing precise definitions, which, however, we must be prepared to modify from time to time as the occasion arises. Applied logic, therefore, since it is concerned with all the means whereby we make progress in science, must discuss the principles in accordance with which scientific definitions should be constructed and employed. CHAPTER IV. REAL, VERBAL, AND FORMAL PROPOSITIONS. 24. 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 names ; as when a proposition predicates of a connotative subject some attribute not included in its connota- tion, or when a connotative term is predicated of a non-conno- tative subject. For example, all bodies have weight, the angles of any triangle are together equal to three 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 states only what is implied in the meaning of the terms involved, or which gives information only with regard to the application of names. 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 1 . 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 1 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 pro- positions under the general head of Terms. CHAP. IV.] VERBAL . PROPOSITIONS. 4S general they are trivial only in so far as their true nature is misunderstood; when, for example, people waste time in pre- tending to prove what has been already assumed in the mean- ing assigned to the terms employed 1 . 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., Tally is Cicero ; (b) where they are dictionary synonyms, e.g., wealth is riches, a story is a tale, chanty is love. In both these cases information is given only with regard to the application of names. Analytic propositions are also described as explicative and as essential. Very nearly the same distinction, therefore, as that between verbal and real propositions is expressed by the pairs of terms analytic and synthetic, explicative and amplia- tive, 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*. The distinction between real and verbal propositions as above given assumes that the use of terms is fixed by their con- notation and that this connotation is determinate 3 . Whether 1 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. 2 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). 3 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 44 TERMS. [PART i. any given proposition is as a matter of fact verbal or real will depend on the meaning which we attach to our terms; and since it is not the function of formal logic to decide this question, this science cannot attempt to determine under which category any given proposition should be placed. 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 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 41). 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." (3) There are propositions usually classed with verbal propositions which should more correctly be placed in a class by themselves, namely, those which are true whatever may be the meaning of the terms involved ; e.g., all A is A, No A is not- A, All A 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 G. These may be called formal propositions, since their validity is determined by their bare form 1 . Formal propositions are the only propositions whose truth is (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. 1 Propositions which are in appearance purely formal 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. CHAP. IV.] FORMAL PROPOSITIONS. 45 examined and guaranteed by formal logic itself irrespective of other sources of knowledge. In a sense they seem almost to coincide with the scope of formal logic ; for any formally valid reasoning can be expressed by a formal hypothetical proposition as in the last two of the examples given above. 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 appli- cation of the terms involved, by questions of fact concerning the things denoted by these terms 1 . 25. Nature of the Analysis involved in Analytic Proposi- tions*. Confusion is not unfrequently introduced into dis- cussions relating to analytic propositions by a want of clearness in regard to the nature of the analysis involved. As above identified with a division of the verbal proposition, an analytic proposition gives an analysis, partial or complete, of the conno- tation of the subject-term. Some writers, however, appear to have in view an analysis of the subjective intension of the sub- ject-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 in- tension 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 'syn- thetic' or 'analytic': its character varies with the knowledge possessed by various persons and at different times. If the 1 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 rerbal contradiction in terms and a formal contradiction in terms, the contra- diction 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. This section may be omitted on a first reading. 46 TERMS. [PART i. 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 con- tents 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." 1 If by intension is meant subjective intension, and by an analytic judgment one which analyses the intension of the .subject, the above statements are certainly unimpeachable. It is indeed so obviously true that in this sense synthetic judg- ments 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 say the least of it, to identify subjective intension with meaning*', and this is especi- ally the case in the present connexion, since it may with a cer- tain degree of plausibility be maintained 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 definitions are modified, and hence a 1 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 analyti- cally, 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). 2 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, vol. ii. p. 233). This criticism seems to lose its force if we clearly grasp the distinction between connotation and subjective intension. CHAP. IV.] ANALYTIC AND SYNTHETIC PROPOSITIONS. 47 form of words which is synthetic at one period may become analytic at another. 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 de- fensible either on the ground of logical expediency or of what actually occurs. As to logical expediency, it is a generally recognised principle of definition that we ought to aim at in- cluding in a definition the minimum number of properties necessary for identification rather than the maximum which it is possible to include 1 . 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. 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 is, there- fore, 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 reason- ably 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 im- portant. 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 1 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 propo- sition all S is P merely makes this verbal statement, and is uo assertion of any matter of fact at all. We are, therefore, involved in a kind of vicious circle. 48 TERMS. [PART i. rather to say that the same form of words now expresses a different proposition 1 . EXERCISES. 26. Enquire whether the following propositions are real or verbal : (a) Homer wrote the Iliad, (6) Milton wrote Paradise Lost. [a] 27. 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.] 1 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. CHAPTER V. FURTHER DIVISIONS OF NAMES. 28. Contradictory Terms. A pair of terms are called contradictories if they are 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 at the same time affirmed. The nature of this relation is expressed in the two laws of Contradiction and Excluded Middle : Nothing is at the same time both X and not-X ; Everything is X or not-X. 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 X and not-X 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." In the case of material contradictories, the contradictory relation usually holds good only in a fixed and very restricted universe of discourse. There are, however, some cases, as Dr Venn points out, where the range covered is very wide ; e.g., male and female, material and spiritual. In the case of formal contradictories, the universe of discourse must be determined K. L. 4 50 TERMS. [PART i. by the context, and may sometimes coincide with the whole universe of nam cable things. Properly speaking, formal logic is concerned with formal contradictories only ; at any rate the relation between terms which are formally contradictory is from the logical standpoint by far the most important question discussed in the present con- nexion. It should be added that in order formally to construct the contradictory of a given term it is best to use the negative prefix "not" or "non." Terms formed with other negative prefixes, e.g., unpleasant, impious, are often not the true con- tradictories of the corresponding positive terms. This will be further brought out in the following section. 29. Contrary Terms. The contrary 1 of a term is usually defined as the term denoting that which, in some particular universe, is furthest removed from that which is denoted by the original term; e.g., Hack and white, wise &ud foolish. 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. Such pairs of terms as pleasant and unpleasant, pious and impious, are notwithstanding the negative prefix con- traries, not contradictories ; for they admit of a mean. Thus, the word unpleasant, as Mill observes, does not connote the mere absence of pleasantness, but a less degree of what is signified by the word painful ; intermediate between pleasant and unpleasant may be that which is indifferent. By some writers, the term contrary is used in a wider- sense than the above, contrariety being identified with simple incompatibility; thus, blue and yellow, equally with black, would in this sense be called contraries of white*. 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 3 . 1 De Morgan uses the terms contrary and contradictory as equivalent, his definition of them corresponding to that given in the preceding section. 2 Compare the discussion of contrary propositions in section 55. 3 So long as we are confined to simple terms the relations of contrariety and CHAP. V.] POSITIVE AND NEGATIVE NAMES. 51 30. Positive and Negative Names. From the purely formal standpoint the distinction between positive and negative names may be expressed by saying that of two formal contradictories the one which is without the negative prefix is positive while that which has the negative prefix is negative; e.g., X and not-X are respectively positive and negative. The distinction, however, which is ordinarily drawn between positive and negative names is not purely formal. It may be expressed as follows: a positive name implies the presence of certain definite attributes, or, if non-connotative, denotes a particular person or thing e.g., man, Socrates ; a negative name implies the absence of one or other of certain definite attributes, or denotes everything with the exception of some particular person or thing e.g., not-man, not-Socrates. If it is asked which of two contradictory names is on this view positive and which negative, the answer is that the distinction lies in the manner in which the denotation of the name is determined. A strictly negative name has its denotation determined indirectly. It denotes an indefinite and unknown class outside a definite and limited class. In other words, we first mark off the class denoted by the positive name, and then the negative name denotes what is left. The fact that its denotation is thus determined is the distinctive characteristic of the negative name 1 . This distinction will usually coincide with the formal dis- tinction first drawn ; thus in either sense such terms as not- man, not-white will be negative. When, however, symbols are used, it is impossible to say which of two such terms as X and not-X really partakes of the indefinite character above ascribed to negative names, since, for example, there is nothing to repugnancy cannot be expressed formally or in mere symbols. But it is other- wise when we pass on to the consideration of complex terms. Thus, while A' And not-X are formal contradictories, XY and X not-Y may be said to be formal jepugnants, XY and not-X not-Y formal contraries. 1 It may be observed that in the case of material contradictories, it is often difficult to say which, if either, should be regarded as negative. For example, in the universe of property, personal and real may be considered contradictories ; but if we are to call one positive and the other negative it is not clear which should be which. 42 52 TERMS. [PART i. prevent our having originally written X for not-white, in which case white becomes not-X, and X is the really negative term. It is clear, therefore, that our second distinction does not depend on purely formal considerations. We may perhaps avoid confusion between the two distinctions by speaking of terms as formally negative or materially negative as the case may be. Mill observes that "names which are positive in form are often negative in reality, and others are really positive though their form is negative." The fact that a positive term may be negative in form results from the circumstance that the negative prefix is sometimes given to the contrary of a term, although the contrary of a positive term is itself as a rule positive also. For instance, to repeat an example already quoted from Mill in a slightly different connexion, "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 be, with reference to a limited universe of discourse, negative in force, e.g., alien, foreign 1 . From the standpoint of formal logic, however, it is a matter of indifference whether any given term is materially positive or negative. What the formal logician is really concerned with is the relation between contradictory terms. Not-X is the contradictory of X, and X is the contradictory of not-X, which- ever of the terms may be more strictly the positive and the negative respectively 2 . 1 The term Turanian, as employed in the science of language, is another example. This term is used to denote groups lying outside the Aryan and Semitic groups, but not distinguished by any positive characteristics which they possess in common. 2 To the distinction between positive and negative names, Mill adds a class of names called privative. " A privative name is equivalent in its signification to a positive and a negative name taken together ; being the name of something which has once had a particular attribute, or for some other reason might have been expected to have it, but which has it not. Such is the word blind, which is not equivalent to not seeing, or to not capable of seeing, for it would not, except by a poetical or rhetorical figure, be applied to stocks and stones " (Logic, i. p. 44). Perhaps also idle, which Mill gives as a negative name, should CHAP. V.] INFINITE NAMES. 53 31. Infinite or Indefinite Names. Infinite and indefinite are designations applied to names having a thoroughgoing negative character ; to such a name, for example, as not-white, understood as denoting 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. Some writers hold that no significant term can be really infinite or indefinite in the sense above indicated. They say that if a term like not-white is to have any significance at all it must be understood as denoting, not all things whatsoever except white things, but only things that are black, red, green, yellow, &c., that is, all coloured things except such as are white. In other words, the universe of discourse which any pair of contradictory terms X and not-X between them exhaust is considered to be necessarily limited to the proximate genus of which X is a species ; as, for example, in the case of white and not-white, the universe of colour. This view must be regarded as erroneous. It is based on the argument that it is an utterly impossible feat to hold together a chaotic mass of the most different things in any one idea 1 . 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 2 . rather be regarded as privative. It does not mean merely "not- working," but "not-working where there is the capacity to work." Sometimes indeed by a figure of speech we refer to inanimate objects as "lying idle"; bat in the strict sense of the term we should hardly speak of a stone as "idle." It cannot be said that the separate recognition of a class of privative names as above defined is of logical importance ; and it may be added that by some logicians the term privative is used as simply equivalent to negative. 1 Compare Lotze, Logic, 40. 2 Mrs Ladd Franklin states very well the counter-argument. "The intent of the positive term and of the negative term are extremely different ; the one involves a combination of quality-elements, the other an alternation of absences of quality-elements. When, therefore, Lotze says that it remains a for ever 54 TERMS. [PART i. To say that not-X is unmeaning if it is interpreted as embracing everything in the universe except X, is to say what is in reality self-contradictory; for in this very statement a meaning is assigned to not-X in the sense under discussion. If, moreover, we are unable to denote by not-X all things whatso- ever except X, it is difficult to see in what way we shall be able to denote these things when we have occasion to refer to them 1 . From our present point of view it is important again to call attention to the distinction between such forms as unholy, inhuman, discourteous and such forms as non-holy, non-human, non-courteous. The latter may be used with reference to any universe of discourse which may be selected, however extensive it may be. But not so the former ; in their case there is undoubtedly a restriction to some particular universe of dis- course which is more or less limited in its range. We can, for example, speak of a table as non-human, although we cannot speak of it as inhuman. A want of clear recognition of this distinction may perhaps be partly responsible for the denial that any terms can properly be described as infinite or in- definite. 32. Relative Names. A name is relative, when, over and above the object which it denotes, it implies in its signification insoluble task to abstract the qualities of the not-man, he says what is true but unimportant. Not-man is not destitute of import, as Lotze says it is, but its intent consists in an alternation of deficiencies of some one, at least, of the elements of the intent of man " (Mind, January, 1892, pp. 130, 1). 1 Writers who take the view which we are here criticizing must in con- sistency deny the universal validity of the process of immediate inference called obversion (cf. section 67). 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 Virtu* is either blue or it is not blue ; but Lotze himself does not thus deny the universality of the law of excluded middle. CHAP. V.] RELATIVE NAMES. 55 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 rela- tions 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 under circumstances of change. We cannot think of any object except as distinguished from something else. Every term, therefore, implies its negative as an object 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 dis- tinction." 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 and the subjective and ob- jective 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 some- thing else which is called by a correlative name, e.g., husband, 56 TERMS. [PART i. 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 shepherd and sheep, the acts of tending and watching 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. 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. 33. Simple Terms and Complex Terms. A simple term may be defined as one which is represented by a single symbol ; e.g., 8, P, Q. The combination of simple terms yields a complex term. Complex terms will be discussed in detail in Part iv ; but it is desirable to call attention at once to the two fundamental ways in which simple terms may be combined so as to yield complex terms. In the first place, terms may be combined conjunctively. Thus, the complex term PQ is formed by the conjunctive combination of the simple terms P and Q, and it denotes whatever belongs both to the class P and to the class Q. In the second place, terms may be combined alternatively (or, as it is more usually expressed, disjunctively). Thus, the complex term P or Q is formed by the alternative combination of the simple terms P and Q, and it denotes whatever belongs either to the class P or to the class Q or to both these classes. The elements of a complex term formed by conjunctive combination may be spoken of as determinants; the elements of a complex term formed by alternative combination as alternants. Thus in the term AB, A and B are determinants ; in the term A or B, they are alternants. Complex terms may of course themselves be combined in CHAP. V.] EXERCISES. 57 the above ways ; and hence terms of the second or any higher degree of complexity may involve both conjunctive and altern- ative combination, e.g., PQ or QR. EXERCISES. 34. 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 justi- fying your example in each case. [K.] 35. 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 of course vary according to the sense in which it is used.] 36. It has been maintained that the doctrine of terms is extra- logical. Justify or controvert this position. [j.] PART II. PROPOSITIONS. CHAPTER I. PROPOSITIONS AND THEIR PRINCIPAL SUBDIVISIONS. 37. Kinds of Propositions. A proposition may be defined as a sentence indicative or declaratory (as distinguished, for ex- ample, from sentences imperative or interrogative); in other words, a proposition is a sentence making an affirmation or denial, as All 8 is P, No vicious man is happy. It is the verbal expression of a judgment. Kant classified judgments according to four different prin- ciples (Quantity, Quality, Relation, and Modality) each yielding three subdivisions, as follows : (1) Quantity, (i) Singular This 8 is P. (ii) Particular Some 8 is P. (iii) Universal All 8 is P. (2) Quality. (i) Affirmative All 8 is P. (ii) Negative No 8 is P. (iii) Infinite All 8 is not-P. (3) Relation, (i) Categorical S is P. (ii) Hypothetical... If 8 is P then Q is R. (iii) Disjunctive Either S is P or Q is R. CHAP. I.] KINDS OF PROPOSITIONS. 59 (4) Modality, (i) Problematic ...Smay be P. (ii) Assertoric S is P. (iii) Apodeictic S must be P. The above arrangement is open to certain criticisms, but from its symmetry it will serve as a useful starting point in our discussion of the various kinds of propositions. 38. Categorical, Hypothetical, and Disjunctive Proposi- tions. The usual division of propositions according to relation is into categorical, hypothetical, and disjunctive. A proposition is categorical if the affirmation or denial which it contains is absolute, as All S is P; Some rich men are not to be envied. It is hypothetical (or conditional) if the affirmation or denial is made under a condition, as If S is P, Q is R; Where ignorance is bliss, 'tis folly to be wise. It is disjunctive if the affirmation or denial is made with an alternative, as Either S is P or Q is R ; He is either a knave or a fool 1 . 1 In lieu of the above threefold division, some logicians commence with a twofold division, the second member of which is again subdivided, the term hypothetical being employed sometimes in a wider and sometimes in a narrower sense. To prevent confusion, it may be helpful to give the following table of the usage of one or two modern logicians with regard to this division. Whately, Mill, and Bain : 1. Categorical. 2. Hypothetical, ( or Compound W Conditional. or Complex. I & Disjunctive. Hamilton and Thomson : 1. Categorical. . ((1) Hypothetical. 2. Conditional. < ; ' _.. . ( (2) Disjunctive. Fowler (following Boethius) : 1. Categorical. 2. Conditional ( (1) Conjunctive. or Hypothetical. \ (2) Disjunctive. Mansel gives at once the threefold division : 1. Categorical. 2. Hypothetical or Conditional. 3. Disjunctive. A distinction between conditionals and hypothetical*, differing from all the above, will be suggested later on. The term alternative will also be used as synonymous with disjunctive. See chapters 8 and 9. 60 PROPOSITIONS. [PART n. For the present we shall concern ourselves entirely with categorical propositions, deferring the consideration of the im- port of hypothetical and disjunctives, and also the question whether the different forms are or are not mutually con- vertible. 39. An analysis of the Categorical Proposition. A cate- gorical proposition consists of two names (which are respec- tively 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 con- stitute its matter, while the remaining two the copula and the sign of quantity constitute its form 1 . The subject is that name about which affirmation or denial is made. The predicate is that name which is affirmed or denied of the subject. When propositions are brought into strictly logical form it is desirable that the subject should precede the predicate ; but in ordinary discourse this order is sometimes inverted for the sake of literary effect, e.g., in the proposition Sweet are the uses of adversity. The means of discriminating between subject and predicate in doubtful cases will be discussed subsequently. The sign of quantity attached to the subject indicates the extent to which the individuals denoted by the subject-term are referred to 2 . Thus, in the proposition All 8 is P the sign of 1 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. 2 Miss Jones (General Logic, p. 10) distinguishes between term and term- name. Thus, in the proposition " Some mistakes are irremediable," some mistakes is said to be a term, mistake a term-name. This usage may sometimes be convenient, but on the whole it seems better to adhere to the traditional use of the word term and not include in its signification the sign of quantity attached to the subject of a proposition. This of course necessitates our regarding quantity as a distinct element of a proposition, as in the text. We should accordingly hold that in the propositions All S is P, Some S is P, the terms are the same, while the quantity of the propositions differs. Miss Jones, on the other hand, would consider the term-names the same, but the terms themselves different. It may be observed that with this usage we can no longer say that every syllogism contains three and only three terms. CHAP. I.] ELEMENTS OF THE PROPOSITION. 61 quantity is all, and we accordingly understand the affirmation 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 logical elements of the proposition are by no means always separately expressed in the propositions of ordi- nary 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 logically. 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 All subject lovers of virtue are copula predicate lovers of angling ; and in this form the four different logical elements are made distinct. The older logicians distinguished between proposi- tions 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. 40. The Quantity and Quality of Propositions. Proposi- tions are 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 1 . Propositions are also divided 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' 1 . The combination of these two principles of division yields four fundamental forms of proposition as follows : 1 Other ways of dividing propositions according to their quantity, including Kant's threefold division, will be referred to subsequently. 2 Kant's threefold division according to quality will be considered in sec- tion 47. 62 PROPOSITIONS. [PART n. (1) the universal affirmative All S is P (or Every 8 is P, or Any S is P, or All S's are P's) usually denoted by the symbol A ; (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 also be found convenient sometimes to use the following SaP=AllSisP; SiP = Some S is P ; SeP = NoSisP; SoP = Some S is not P. The above 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 = AllMisP; PoQ = Some P is not Q. It will further be found convenient sometimes to denote not-S by S', not-P by P', and so on. Thus we shall have S'aP' = All not-S is not-P ; PiQ' = Some P is not-Q. The universal negative should not be written in the form All S is not P 1 ; for this form is ambiguous and would usually be interpreted as 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 1 Similar remarks apply to the form Every S is not P. CHAP. I.] INDEFINITE PROPOSITIONS. 63 proposition, and is equivalent to " Some things that glitter are not gold." 41. Indefinite Propositions. According to quantity, propo- sitions have sometimes been divided into (1) Universal, (2) Particular, (3) Singular, (4) Indefinite 1 . Singular propositions will be discussed in the following section. 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 or prein- designate would be a better term to employ. There can be no doubt that, as Mansel remarks, " the true indefinite proposition is in fact the particular, the statement some A is B being applicable to an uncertain number of instances, from the whole class down to any portion of it. For this reason particular propositions were called indefinite by Theophrastus" (Aldrioh, p. 49). 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 2 . 42. Singular Propositions. By a singular or individual proposition is meant a proposition in which the affirmation or 1 This is a further expansion of Kant's threefold division into universal, particular, and singular. 2 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; AH 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 (see sec- tion 44). 64 PROPOSITIONS. [PART n. 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 usually 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 1 . Such propositions have, however, certain peculiarities of their own, as will be pointed out subsequently ; e.g., they have not like other universal propositions a contrary distinct from their contradictory. Hamilton distinguishes between universal and singular pro- positions, the predication being in the former case of a whole undivided, and in the latter case of a unit indivisible. The 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 propo- sitions (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 1 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 for, as we shall find, the particular is not inconsistent with the universal but its indefinite character. 2 Lotze (Logic, 68) distinguishes between general and universal judgments. In the former the predication is of the whole of an indefinite class, including both examined and unexamined cases. In the latter we have merely a summation of what is found to be true in every individual instance of the subject. "The universal judgment is only a collection of many singular judg- CHAP. I.] SINGULAR PROPOSITIONS. 65 whereby to call attention to the distinction wherever 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 8 is P"; singular definite, "This S is P"; singular indefinite, "A certain 8 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. 43. 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 sometimes unhappy, In some countries all foreigners are un- popular. This differentiation may be carried further so as to ments, 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 every one who partakes in it." In applied logic the distinction here indicated may be of importance ; a somewhat similar distinction is indicated by Mill in his treatment of "inductions improperly so-called." But it cannot be regarded as a formal distinction ; it depends not so much on the propositions themselves as on the manner in which they are obtained. There is no sufficient justification for Lotze's implication that propositions of the form all S is P are always in his sense universal, while those of the form S is P are always in his sense general. K. L. 5 66 PROPOSITIONS. [PART n. yield triple or any higher order of quantification. Thus, we have triple quantification in the proposition, In all countries all foreigners are sometimes unpopular. 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*. 44. Signs of Quantity. A brief discussion is necessary in regard to the precise signification to be attached to the various signs of quantity which may be recognised by the logician. 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, but 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 therefore as covering every case (including all) which is in- consistent with none ; secondly, as any quantity intermediate between none and all, carrying with it, therefore, the implication not all as well as not none. In ordinary speech the latter of these two meanings is the more usual 2 . This may, however, be 1 For a further development of the notion of multiple quantification see Mr Johnson's articles on the Logical Calculus in Mind, 1892. 2 We might indeed go further and say that in ordinary speech some usually means comiderably 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). It may be added that in regarding some as implying no more than at least one, we are in another way departing from the ordinary usage of language, which would generally regard it as implying more than one, that is, at least two. On this point, compare Mansel's Aid-rich, p. 59, and Venn's Empirical Logic, pp. 222, 3. It should perhaps be added that on rare occasions some may in ordinary speech carry with it a still more definite implication. For example, the proposition "Some truth is better kept to oneself" may be so emphasized as to make it perfectly clear to what particular kind of truth reference is made. This is, however, extra-logical. Logically the proposition must be treated as particular, or it must be written in another form, "All truth of a certain specified kind is better kept to oneself." Thus, Spalding remarks, "The logical 'some' is totally indeterminate in its reference to the constitutive CHAP. I.] SIGNS OF QUANTITY. 67 regarded as one of the implications or suggestions of ordinary speech that logic cannot recognise ; and accordingly with most modern logicians the logical interpretation of some is limited to one at least. Using the word in this sense, if we want to express Some, but not all, 8 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, as already suggested, 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 1 . We shall henceforth interpret some in this indefinite sense unless an explicit indication is given to the contrary. All is ambiguous, inasmuch as it 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 dis- tributively, 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 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 our sign of quantity. In general, all is to be interpreted distributively, unless by the context or in some other way an indication is given to the contrary. Any, as the sign of quantity of a categorical proposition, e.g., Any S is P, introduces a universal statement; for P is affirmed indiscriminately of S, whatever particular S we may happen to have selected, and it is therefore practically affirmed of the whole of the subject. Hence in such a proposition any is, as a rule 2 , equivalent to all in its distributive sense. When not the objects. It is always aliqui, never quidam ; it designates some objects or other of the class, not some certain objects definitely pointed out " (Logic, p. 63). 1 It will hardly do to define some as an indefinite quantity or number without further explanation, since this would include none as a limiting case. - This qualification is introduced because, as Miss Jones points out, a proposition commencing with any may sometimes be in effect a singular pro- position, and, when this is so, any and all are no longer interchangeable. ""Any may occur as subject indicator in a proposition in which, by the signi- 52 68 PROPOSITIONS. [PART n. subject of a categorical proposition, however, any may have a different signification. For example, in the hypothetical pro- position If any A is B, C is D, it has the same indefinite character which we logically ascribe to some 1 ; since the ante- cedent condition is satisfied if a single A is B. The proposition might indeed be written If one or more A is B, C is D, Propositions of the forms Most S's are P's, Few S's are P's, are called plurative propositions. Most may be logically in- terpreted as equivalent to at least one more than half 2 . Few has a negative force ; and Few S's are P's may be regarded as equivalent to Most S's are not P's 3 . Formal logicians (excepting fication of S or P, the application of the subject is restricted to one individual ; e.g., 'Any one who wins this race will have a silver cup'" (General Logic, p. 70). An affirmation is still made of the whole of the subject, but the subject consists of a single individual only. Miss Jones adds as another illustration the proposition "Any one may have my ticket"; this proposition, however, seems really general, so far as the potentiality of having the ticket is concerned. 1 It appears to have this meaning (a) in the principal clause of an inter- rogative sentence, e.g., Are any subscribers dissatisfied because some non- subscribers were admitted? (b) in the subordinate clause of a negative sentence, e.g., Some people do not think that any men are perfect ; (c) in the antecedent clause of a pure hypothetical, e.g., If any men are perfect, some men are mistaken. This does not, however, apply to the antecedent of a conditional, in the sense in which conditionals are distinguished from hypotheticals in chapter 8; e.g., the proposition "If any flower is scarlet, it is scentless" is equivalent to the proposition "All scarlet flowers are scentless." 2 We are here somewhat departing from popular usage just as when we interpret some as equivalent to at least one more than none. No doubt a larger excess over half and none, as the case may be, is usually contemplated when the words most and some are used in ordinary discourse. 3 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, on the ground that it contains both an affirmation and a negation, though one of them in a concealed way. The proposition Few S's are P's may also be interpreted in a slightly different way, as meaning that an absolutely small number of S's are P's. It can then be written in the form, The number of S's which are P's is small. If, however, we recognise few as a logical' sign of quantity, it is necessary to give it a fixed interpretation, and on the whole that indicated in the text seems the most satisfactory that can be adopted. It should be added that a feiv has not the same signification &sfew, but must be regarded as affirmative, and, generally, as simply equivalent to some; e.g., A few S's are P's=Sotiie 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 CHAP. I.] NUMERICALLY DEFINITE PROPOSITIONS. 69 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 are P's, and Few S's are P's to Some S's are not P's. Some- times, 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. Numerically definite propositions are those in which a predication is made of some definite proportion of a class; e.g., Two-thirds o/S are P. A certain ambiguity may lurk in numeri- cally 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 1 . But some such words as at least can of course be used when it is not professed to state more than the minimum proportion of S's that are P's. 45. The Distribution of Terms in a Proposition. A term is said to be distributed when reference is made to all the indi- viduals denoted by it ; it is said to be undistributed when they are only referred to partially, i.e., when information is given 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 iu this way is singular, not general, it is singular indefinite, not singular definite ; for what small band is alluded to is left indeterminate. 1 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. 70 PROPOSITIONS. [PART n. 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 sub- ject is distributed in a universal, and undistributed in a par- ticular 1 , 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 8, 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 8 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 predi- cate is distributed, although the subject is not 2 . 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. 46. The Distinction between the Subject and the Predicate of a Proposition. 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 1 Some being used in the sense of some, it may be all. If by some we under- stand 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. 2 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 6. CHAP. I.] SUBJECT AND PREDICATE. 71 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 1 . It follows that the subject must be given first in idea, since we cannot assert anything, 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 neces- sarily 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 predicate are pure synonyms 2 or if the proposition is practically reduced to an equation, as in the doctrine of the quantification of the predi- cate, 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, however, " (Some) great is (all) Diana of the Ephesians." But leaving this particular doctrine on one side, it cannot be said that the order of terms is always a 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 informa- 1 Hence the subject is generally speaking that term which is comparatively unemphatic, whilst the predicate is comparatively emphatic, the point at issue being its applicability to the subject. There may, however, be exceptions to this rule, and sometimes the only emphatic word in a proposition is the sign of quantity. 2 For illustrations of this point, and on the general question raised in this section, compare Venn, Empirical Logic, pp. 208 to 214. 72 PROPOSITIONS. [PART n. tion 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 1 . If none of these considerations are 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 distri- bution 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 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." 47. 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 nega- tive, S is not-P is spoken of as infinite or limitative*. Logically, however, the last judgment (which is equivalent to the second in meaning) must be regarded as simply affirmative. As shewn in section 30, it is impossible to say which of the terms P or not-P is really infinite ; and it is, therefore, also impossible to 1 The subject is often a substantive and the predicate an adjective. Compare section 97. 2 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 infinitte those whose subject, as well as those whose predicate, is negative. Thus Father Clarke defines propositiones infanta 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). CHAP. I.] COMPLEX AND COMPOUND PROPOSITIONS. 73 say which of the propositions S is P or S is not-P is really infinite or limitative. Hence they must be regarded as belong- ing to the same type of proposition, and we have to fall back upon the twofold division into affirmative and negative. 48. Complex Propositions and Compound Propositions. A complex proposition may be defined as a proposition which has a complex term either as subject or predicate : for example, AB is C, S is P or Q, XY is Z or W. But just as terms may be combined conjunctively or alternatively, so may propositions themselves ; and a compound proposition may be defined as one which consists of the conjunctive or alternative combination of other propositions : for example, P and Q are both true, P is true or Q is true. The distinction between complex propositions and compound propositions is commonly overlooked. Thus, it is usual to class simply as disjunctives, (a) propositions which may be regarded as categorical with a disjunctive predicate, e.g., Every S is P or Q, Every blood vessel is either a vein or an artery ; and (6) proposi- tions which consist in the disjunctive (alternative) combination of two distinct propositions, e.g., All S is P or some X is not Y, Either free will is a fact or the sense of obligation an illusion. It will presently be shewn that the distinction is really of fundamental importance in the case both of disjunctives and hypotheticals. Leaving this point, however, for the present, we may here notice certain classes of complex and compound propositions which have in a somewhat unsystematic way been commonly recognised in logical text-books. The distinctions indicated are of some historical interest, but for the most part they are of little scientific importance. Exponible Propositions. Some propositions, which are not compound in form, can nevertheless be resolved into a conjunc- tion of two or more simpler propositions which are independent of one another. Propositions which are in this way susceptible of analysis are termed exponible. One of the chief difficulties arising in the logical interpretation of a good many propositions relates to the question whether they are or are not to be re- garded as exponible. For example, is the plurative proposition 74- PROPOSITIONS. [PART n. Most S's are P's to be interpreted as implying not only that the majority of S's are P's but also that some .S's are not P's ? If the latter view is taken, then the proposition is exponible, but not otherwise. Other examples will be given below. Copulative Propositions. Complex propositions which can be analysed into a conjunction of two or more affirmative propositions having the same subject are termed copulative. For example, the proposition All P is QR is in form complex, not compound, but it is obviously resolvable into the conjunc- tion All P is Q and all P is R\ Copulative propositions fall, therefore, within the class of exponibles. Remotive Propositions. Complex propositions which can be analysed into a conjunction of two or more negative propositions having the same subject are termed remotive; e.g., No P is Q or R, which is resolvable into the conjunction No P is Q and no P is R. Remotives, as well as copulatives, fall within the class of exponibles. Exceptive Propositions. Propositions in which the subject is limited by some such word as unless or except are termed exceptive; for example, All P is Q, unless it happens to be R. If we interpret a proposition of this type as implying not merely that every P which is not R is Q, but also that every P which is R is not Q, then exceptives must be regarded as forming another class of exponibles. The above proposition may on the same interpretation also be resolved into All P is either Q or R, but no P is both of these. Exclusive Propositions. Propositions which contain some such word as only or alone whereby the predicate is limited to the subject, for example, Only S is P, S alone is P, are termed exclusive. Propositions of this kind may be written in the form Some S is all P ; but this is not one of the forms recognised in the traditional scheme as given in section 40 2 . In order to deal with exclusives under the traditional scheme it is necessary to replace them by one of the equivalent forms 1 The resolution of complex propositions into compound propositions will be considered more systematically in Part iv. 2 For a further discussion of propositions of the form Some S is all P, see chapter 6. CHAP. I.] EXCLUSIVE PROPOSITIONS. 75 All P is S, No not-S is P. In making this transformation, however, we have not kept the original subject and predicate ; we have in fact performed upon the given proposition a process of immediate inference. It is convenient to discuss exclusive propositions in the present section, but the view that all exclusives are exponible must be regarded as erroneous 1 . When this view is taken, it is not quite clear whether it is intended to resolve such a proposition as S alone is P into (a) All S is P and no not-S is P or into (6) Some S is P and no not-S is P. The first of these alternatives, however, must be rejected on the ground that S alone is P does not necessarily * imply that all S is P ; for example, Graduates alone are eligible does not imply that all graduates are so, since other qualifications may also be neces- sary for eligibility. The second alternative appears to overlook the fact that Some S is P is itself an immediate inference from No not-S is P, and that hence there is no resolution into two distinct propositions at all. If a proposition were considered exponible simply because it could be resolved into two pro- positions, whether or not these propositions were independent of one another, then every proposition would be exponible. Discretive Propositions. Compound propositions in which we express different judgments, denoting that difference by the particles but, nevertheless, or the like, expressed or understood, are called discretive; e.g., "Fortune may take away wealth, but it cannot take away virtue" (Port Royal Logic, p. 136). Inceptive Propositions and Desitive Propositions. "When we say that a thing has commenced or ceased to be such, we form two judgments one what the thing was before the time of which we speak, the other what it is after; and thus these propositions, of which the one class is called inceptives, the other desitives, are compound in sense : e.g., The Jews commenced, after the return from the captivity of Babylon, to disuse their ancient characters, which are those which are now called the 1 Exclusives and exceptives are sometimes spoken of as interchangeable forms. But in order that this may he the case we must of course interpret both as exponible, or neither. J No doubt it may in ordinary discourse be often so understood. 76 PROPOSITIONS. [PART n. Samaritan ; The Latin language has for five hundred years ceased to be common in Italy" (Port Royal Logic, p. 143). It is clear that, as here interpreted, inceptives and desitives belong to the class of exponibles. 49. The Modality of Propositions. Different accounts of the modality of propositions are given by different writers. Whately (Logic, Book ii. chapter 2, 1) divides categorical propositions into (a) pure and (6) modal, according as they assert simply that the subject does or does not agree with the predicate, or express in what mode or manner it agrees : Brutus killed Cossar, An intemperate man will be sickly, are given as examples of pure propositions ; Brutus killed Ccesar justly, An intemperate man will probably be sickly, as examples of modals. A clear distinction ought, however, to be drawn between these examples. In passing from the statement that Brutus killed Caesar to the statement that Brutus killed Csesar justly, the addition is obviously a qualification or modification of the predicate, and there is no reason why it should not logically be included in the predicate. Compare the propositions, Browning is a poet, Browning is a great poet. But the distinction be- tween affirming something as a fact and affirming its probability is a distinction of quite a different kind, and one that does not admit of so summary a treatment. Whately's account of modal propositions may, therefore, be at once rejected, on the ground that it classes together propositions which fundamentally differ from one another in their character. 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 unchange- able, 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 considera- tions, and do not relate to the varying degrees of belief with which different propositions are accepted. The resulting CHAP. I.] MODALITY OF PROPOSITIONS. 77 classification, so far as it can be clearly interpreted in ac- cordance with modern ideas, corresponds broadly with the ordinary fourfold classification of propositions into universal affirmative, particular affirmative, particular negative, and universal negative ; and independently of this it seems to have little or no value. On this ground, taken in connexion with its inherent vagueness and obscurity, the scholastic doctrine of modals may now be regarded as obsolete and as having only an historical interest 1 . 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. In criticizing this division from the point of view of the logician, Dr Venn urges that whereas the distinction between the problematic judgment and the assertoric concerns the quantity of belief with which the judgments are entertained, that between the apodeictic judgment and the assertoric con- cerns quality rather than quantity of belief, and this is a distinction of which the logician cannot properly take account. " The belief with which an assertory judgment is entertained is full belief, else it would not differ from the problematic ; and 1 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 cannot be formally expressed, so that propositions of the same form may have a different modality, then the argument that the doctrine of modals is extra-logical is perfectly sound. 78 PROPOSITIONS. [PART n. therefore in regard to the quantity of belief, as distinguished from the quality or character of it, there is no difference be- tween it and the apodeictic " (Logic of Chance, p. 313). It is further urged that problematic judgments form a class admitting of all degrees and that they cannot be satisfactorily treated except in connexion with the general theory of probability 1 . On the whole, if the doctrine of modals is to be retained at all in logic it must be by regarding it as having relation to the grounds upon which judgments are formed. This view and its consequences are excellently stated by Mr Johnson. "Modality refers to the grounds on which the thinker forms his judgment. It, therefore, expresses a relation between the thinker on the one hand and a certain proposition on the other hand. The real terms, then, of the modal proposition are the thinker and his relation to some judgment which is propounded to him. Thus the proposition S must be P asserts (say) that Any rational being is bound by his rationality (or, it may be, by his spatial or moral intuitions) to judge that 8 is P. Now the contradictory of a modal proposition such as S must be P is always another modal proposition such as S may be not-P, which would mean on the above shewing A rational being is not bound by his rationality (or by his spatial or moral intui- tions, as the case may be) to judge that S is P. The modal proposition is, therefore, simply an assertoric on a different plane concerned with the relations between different sorts of terms. It follows, then, that whereas a modal must always be contradicted by a modal, an assertoric must always be con- tradicted by an assertoric" (Mind, 1892, pp. 18, 19) 2 . 1 On the whole subject of modality compare Venn's Logic of Chance, chapter 13. 2 An account of modality, which somewhat resembles the above, but which can be only partially accepted, is given by Mr Bradley, who identifies the assertoric proposition with the categorical, and regards the apodeictic and the problematic as different phases of the hypothetical or conditional, what must be and what may be not being declared as actual facts, but both being inferred on the strength of a condition and subject to a condition. "It is easy to give the general sense in which we use the term necessity. A thing is necessary if it is taken not simply in and by itself, but by virtue of something else and because of something else. Necessity carries with it the idea of mediation, of dependency, of inadequacy to maintain an isolated position and to stand and CHAP. I.] EXERCISES. 79 EXERCISES. 50. Determine the quality of each of the following propositions, arid 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.] 51. Everything is eitJier 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, /, E, categorical propositions. [c.] 52. Express each of the following statements in one or more of the strict categorical forms admitted in logic : (i) No one can be rich and happy unless he is aluo 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.] act alone and self -supported. A thing is not necessary when it simply is ; it is necessary when it is, or is said to be, because of something else" (Principles of Logic, p. 183) . In a conditional or hypothetical proposition, however, the statement that the consequent follows from the antecedent may be itself merely assertoric. If, for example, we compare the propositions All spoilt children are troublesome and If a child is spoilt then he will be troublesome, it is clear that the latter is no more apodeictic than the former. Each affirms a matter of fact and the same matter of fact. This point will arise again when we are considering the import of the conditional proposition as contrasted with the categorical. But it seems desirable to make it clear at this stage that there is no justification for considering that there is always a modal distinction between these two kinds of propositions. Possibility is treated by Mr Bradley as a form of hypothetical necessity. When we say that anything is possible we mean that it would exist as fact, if something else were fact. What separates the species possible from the genus necessary is the implication that part of the antecedent exists but that we are in ignorance as regards the remaining part. "Take a judgment such as this, Given abed then E must follow. Add to it the judgment, or the supposition, that ab exists, while cd is not known to exist, and we get the possible. E is now a possibility" (p. 187). But are we not here assuming that cd is a possibility? The argument does not seem to be pursued far enough back. A far more consistent and satisfactory account of the problematic judgment is given when it is regarded as the form which the contradictory of an apodeictic judgment will take, as explained in the text. CHAPTER II. THE OPPOSITION OF CATEGORICAL PROPOSITIONS 1 . 53. The Square of Opposition. 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 2 . 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. (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 3 . 1 Complications arising in connexion with the implication or non-implication of existence in propositions will for the present be postponed. For a further discussion of the doctrine of opposition, see section 119. 2 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 con- venient 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. 3 Mr Stock writes, "When we say that of two subcontrary propositions, if one be false, the other is true, we are not taking the propositions I and in CHAP. II.] THE SQUARE OF OPPOSITION. 81 (4) From a given universal proposition, the truth of the particular having the same quality follows, but not vice versd 1 . This is subaltern opposition, the universal being called the subalternant, and the particular the subalternate or subaltern. SaP and SiP. SeP and SoP. All the above relations are indicated in the ancient square of opposition. . Contraries . cc c -S 1 Subcontraries their now accepted logical meaning as indefinite, but rather in their popular sense as 'strictly particular' propositions. For if I and were taken as indefinite propositions, meaning ' some, if not all,' the truth of I would not exclude the possibility of the truth of A, and similarly the truth of would not exclude the possibility of the truth of E. Now A and E may both be false. Therefore, I and 0, being possibly equivalent to them, may both be false also. In this case the doctrine of contradiction breaks down as well. For I and may, on this shewing, be false, without their contradictories E and A being thereby rendered true" (Deductive Logic, pp. 139, 140). This criticism of the received doctrine of opposition is based on a fallacy. If some is interpreted in its strict logical sense, then I may be true along with either A or 0, but both these coincidences cannot occur together. In other words, whilst I and are always formally consistent, we can never infer the one from the other, inasmuch as additional information may at any time shew that they are not as a matter of fact both true. We have this case whenever I is an understatement in the sense that A might have been affirmed in its place. Mr Stock does not appear to recognise that to say that either of two things may occur is not the same thing as to say that they may both occur together. We may add that it is in any case infelicitous to speak of I and as being possibly equivalent to A and E. 1 It is perhaps desirable to warn the student at once that this result and some of our other results may need to be modified when we take into account K. L. 6 82 PROPOSITIONS. [PART n. Propositions must of course be brought to such a form that they have the same subject and the same predicate before the terms of opposition can be directly applied to them ; for example, All S is P and Some P is not S are not contradictories. We may distinguish between two different points of view from which the doctrine of opposition may be regarded, namely, first, 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 : 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 ; 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 ; 1 being given false, A is false, E true, O true; O being given false, A is true, E, false, I tru-e. The legitimacy of the above inferences may be considered to depend exclusively on the three fundamental laws of thought : namely, the Law of Identity A is A ; the Law of Contradiction A is not not- A ; the Law of Excluded Middle A is either B or not-B. Thus, from the truth of All S is P we may infer the truth of Some 8 is P by the Law of Identity 1 , and the falsity of Some S is not P by the Law of Contradiction ; from the falsity of All S is P we may infer the truth of Some S is not P by the Law of Excluded Middle; and similarly in other cases. 54. Contradictory Opposition. The doctrine of opposition given in the preceding section is primarily applicable only to the fourfold schedule of propositions ordinarily recognised ; but it is of essential importance to understand clearly the nature of later on the existential import 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. 1 This of course is on the assumption that the existential import of universals and particulars is the same. CHAP. II.] CONTRADICTORY OPPOSITION. 83 contradictory opposition whatever may be the schedule of propositions with which we are dealing. To deny the truth of a proposition is equivalent to affirming the truth of its contradictory ; and vice versd. 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 to- gether, 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. The nature of contradictory opposition may be illustrated by asking what are the contradictories of the following proposi- tions Few S are P ; Two-thirds of the army are abroad ; None but the brave deserve the fair. Few S are P is equivalent to Most S are not P, and we might hastily be inclined to give as the contradictory Most S are P. Both these propositions would, however, be false in the case in which exactly one half S was P. The true contradictory, therefore, is At least one half S is P. This example shews, that if we once travel outside the limits set by the old logic, and recognise the signs of quantity most and few as well as all and some, we soon become involved in numerical statements 1 . Pro- positions of the above kind are, therefore, usually relegated to what has been called numerical logic, a topic discussed at length by De Morgan and to some extent by Jevons. Two-thirds of the army are abroad might mean that at least two-thirds of the army are abroad or that exactly two- thirds of the army are abroad 2 . On the first interpretation, the contradictory is Less than two-thirds of the army are abroad. 1 It is obvious that in seeking the contradictory of Few S are P, we cannot treat the proposition as simply equivalent to Some S is not P ; for this would give for the required contradictory All S w P, and this and the original proposition might both be false. 2 We have already indicated a preference for the latter of these interpre- tations if numerically definite propositions are to receive a systematic logical treatment ; but there can be no doubt that in ordinary discourse the former might happen to be the correct interpretation. 62 84 PROPOSITIONS. [PART n. On the second interpretation, it becomes Not exactly two- thirds of the army are abroad, i.e., Either more or less than two-thirds are abroad. None but the brave deserve the fair is, again, a proposition that raises a question of interpretation. If this proposition leaves it an open question whether or not all who are brave are deserving of the fair, then the contradictory is simply Some who are not brave deserve the fair. But if the proposition is exponible and capable of being analysed into the two distinct statements that all who are brave deserve the fair but none who are not brave deserve them, then a disjunctive proposition is required to express its contradictory,, namely, Either some who are brave do not deserve the fair or some who are not brave do deserve the fair. The above examples serve to illustrate the way in which attention is almost inevitably called to any ambiguity in a proposition as soon as we seek 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 we shall find later on that 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. Further light may be thrown upon the nature of contra- dictory opposition by reference to a discussion entered into by Jevons (Studies in 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 cor- responding 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, con- tinues, " But it does not seem to have occurred to logicians in general to inquire how far similar relations could be detected CHAP, n.] CONTRADICTORY OPPOSITION. 85 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 8 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, Jevons is incorrect in speaking of each of these alternatives as being by itself a contradictory of the original proposition. The true logical 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 versa. 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 all P is S. It has but one contradictory, namely, Either some S is not P or some P is not S 1 . 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 Sis all P. These two propositions are indeed inconsistent with one another; but they may both be false. It follows that we must reject Jevons's further statement 1 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. It must always be the case that two different forms of the contradictory of the same proposition are equivalent to one another. 86 PROPOSITIONS. [PART n. 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." No doubt a proposition which is complicated in form may yield an indefinite number of other propositions the truth of any one of which is inconsistent with its own. But it has only one logical contradictory, which contradictory may be still more complicated in form, affirming a number of alternatives one or other of which must hold good if the original proposition is false 1 . In connexion with the same point, Jevons raises another question, in regard to which his view is also very misleading. 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 asser- tion 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 our- selves in hopeless confusion were we to consider the truth or 1 No doubt logicians have often used the word contradict somewhat loosely. For example, in the Port Eoyal 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 strict logical 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. CHAP. II.] CONTRARY OPPOSITION. 87 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. Logical problems would be simply insoluble were we not allowed to proceed, for example, from the falsity of All S is P to the truth of Some S is not P. 55. Contrary Opposition. Generalising the relation be- tween A and E, we should naturally 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, declaring not merely the falsity of the given proposition taken as a whole, but the falsity of every part of it; so that, whilst the contradictory of a proposition denies its entire truth, its contrary may be said to assert its entire falsehood. The notion of contrariety, however, as thus defined cannot very satisfactorily be extended beyond the particular case con- templated in the ordinary square of opposition. Taking, for example, such a proposition as "Two-thirds of the army are abroad," and interpreting it to mean that exactly two-thirds of the army are abroad, it would seem that seeking for an assertion as far as possible removed from the original assertion we might proceed in either of two directions and take our choice between saying that all the army are abroad and that none of the army are abroad. It is certainly difficult to see on what principle we ought to select either of these alternatives in preference to the other as being the diametrical opposite of the given proposition. Hence if it is desired to define contrariety so that the con- ception may be generally applicable, the idea of two proposi- tions standing, as it were, furthest apart from each other must be given up, and any two propositions must be described as contraries if they are inconsistent with one another without at the same time exhausting all possibilities. Contraries must on tins definition always admit of a mean, but they may not always be what we should speak of as diametrical opposites, and any given proposition may have not one only, but an indefinite number of contraries. It will be observed, however, 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. 88 PROPOSITIONS. [PART n. With either definition of contrariety it follows that if the contrary of a given proposition is true, its contradictory must also be true, but not vice versa. Hence in controversy it is under ordinary circumstances better to refute a statement by its contradictory rather than by its contrary. The contra- dictory is sufficient for disproof, and therefore as effective as the contrary for purposes of refutation ; at the same time it commits the controversialist to less, and is not liable as the contrary may be to be upset by evidence which is quite in- adequate so far as establishing the original proposition is con- cerned 1 . 56. The Opposition of Singular Propositions. Taking the proposition Socrates is wise, its contradictory is Socrates is not wise*', and so long as we keep to the same terms, we cannot go beyond this simple denial. The proposition has, therefore, no formal contrary 3 . This opposition of singulars has been called secondary opposition (Hansel's Aldrich, p. 56). If, however, there is secondary quantification in a proposi- tion 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 1 For this reason, Bain characteristically observes that the contradictory "possesses the imposing circumstance of securing great ends by small means" (Deductive Logic, p. 94). 2 This must be regarded as the correct contradictory from the point of view reached in the present chapter. But it will be shewn in section 123, that if the original proposition is understood (as it probably would be understood) to imply the existence of Socrates, then a strict application of the criterion of contradiction requires that our contradictory be written If there is such a man as Socrates, he is unwise. 3 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. CHAP. II.] OPPOSITION OF SINGULARS. 89 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 resolu- tion of the proposition is in any way 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, That flower is blue, and give as their contradictories Browning is not obscure, Epimenides is not a liar, That 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 contra- dictory 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. 1 The contradictory will then be particular or universal accordingly. 57. Possible Relations of Propositions into which the same Terms or their Contradictories enter 3 . If we no longer confine ourselves to propositions having the same subject (S) and predicate (P) respectively, but consider propositions into which two terms or their contradictories (S, not-S, P, not-P) enter, either as subjects or predicates, it becomes necessary to amplify 1 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 the flower is blue. 2 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. 90 PROPOSITIONS. [PART n. the list of formal relations recognised in the square of opposi- tion, and also to extend the meaning of certain terms. We may give the following classification : (1) Two propositions into which only the same terms or their contradictories enter 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. For example, as will presently be shewn, All 8 is P and All not-P is not-S stand to each other in this relation. (2) One of the two propositions may be formally inferable from the other, but not vice versa. Thus the truth of one of the propositions carries the truth of the other with it, but not conversely. Ordinary subaltern propositions with their subal- ternants fall into this class ; and it will be found in our treat- ment of immediate inferences that other pairs of propositions fulfilling the above condition are in all cases equivalent to some pair which have the same subject and predicate respectively. The relation between them may accordingly be said to be im- plicitly subaltern, the term subaltern, however, when used with- out qualification, being still applied only as in the ordinary square of opposition. Thus Some P is S which is equivalent to Some 8 is P is inferable from All 8 is P, but not vice versa. It is, therefore, implicitly subaltern to All 8 is P. Again, All S is P and Some not-S is not P are implicitly subalternant and subalternate. Here it is not so immediately obvious in what direction we are to look for a pair of equivalent propositions the relation between which is explicitly subaltern. No not-P is S and Some not-P is not S will, however, be found to satisfy the required conditions. (3) The propositions may be such that one or other of them must be true while both may be true. A pair of propositions outside the ordinary square of opposition which are thus re- lated for example, Some S is P and Some not-S is P may be said to be implicitly subcontrary. It can be shewn that any pair of implicit subcontraries are equivalent to some pair of explicit subcontraries ; thus, the above pair are equivalent to Some P is S and Some P is not S. (4) The propositions may be such that they can both CHAP. II.] MUTUAL RELATIONS OF PROPOSITIONS. 91 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 two propositions may be contrary to one another, in the sense that they cannot both be true, but can both be false. We may again distinguish between explicit contraries and implicit contraries ; and as before it can be shewn that any pair of implicit contraries are equivalent to some pair of explicit contraries. For example, the implicit contraries All S is P and All not-S is P are equivalent to the explicit contraries No not- P is S and All not-P is 8. (6) The two propositions may be contradictory to one another according to the definition given in section 54, 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, and may on the same principle as before be called implicit contradictories, an equivalent pair of explicit contradic- tories being All 8 is P and Some S is not P. Two propositions, then, into which only the same terms or their contradictories enter may, in respect of their mutual rela- tion, be (1) equivalent, (2) explicitly or implicitly subaltern, (3) explicitly or implicitly subcontrary, (4) independent, (5) ex- plicitly or implicitly contrary, (6) explicitly or implicitly con- tradictory. What pairs of propositions actually fall into these categories respectively will be shewn in sections 72 and 73. EXERCISES. 58. Explain 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.] 59. If some were used in its ordinary colloquial sense, how would the scheme of opposition between propositions have to be modified ? [j.] 92 PROPOSITIONS. [PART n. 60. Give the contradictory of each of the following proposi- tions : 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; WJierever the property A is found, either the property B or the property C will be found with it, but not both of them together 1 . [K.] 61. Shew that any term distributed in a general proposition 2 is undistributed in its contradictory, and vice versa, [K.] 1 For a general discussion of the opposition of complex terms and complex propositions see Part iv. 2 Including under generals both universals and particulars, but excluding singulars. CHAPTER III. IMMEDIATE INFEKENCES 1 . 62. The Conversion of Categorical Propositions. By con- version, in a broad sense, is meant a change in the position of the terms of a proposition 2 . 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 8 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 in- ference of a proposition from a single other proposition 3 . 1 In this chapter, 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 always to have been made implicitly in the traditional treatment of logic. 2 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." 3 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 prepositional forms." It will be found, however, that the new propositions obtained by immediate inference are not always equivalent to the original propositions, 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 94 PROPOSITIONS. [PART n. 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 pro- position having P for its subject and 8 for its predicate. The original proposition may be called the convertend, and the in- ferred proposition the converse 1 . The process will be valid if the two following rules are ob- served : (1) The converse must be the same in quality as the con- vertend (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 8. I. Some 8 is P. I. Some P is S. I. No 8 is P. E. No P is 8. 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 con- nexion with the reading of propositions in extension and inten- eduction from categorical form to categorical, or from hypothetical to hypo- thetical, &c., and a transversion an eduction from categorical form to conditional, or from conditional to categorical, &c. For the present we shall be concerned with eversions only. 1 The process of conversion will be considered in a more generalised form in Part iv. CHAP. III.] CONVERSION OF PROPOSITIONS. 95 sion 1 , namely, that, generally speaking, in any judgment we have naturally before the mind the objects denoted by the subject, but the qualities connoted by the predicate. In con- verting a proposition, however, the extensive force of the predi- cate is made prominent, and an import is given to the predicate similar to that of the subject. In other words, the proposition is taken wholly in extension. It is in passing from the predica- tive 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 clearly that conversion and other im- mediate inferences usually involve a distinct mental act of the above nature. 63. Simple Conversion and Conversion per accidens. It will be observed that in the case of I and E, the converse is of ex- actly the same form as the original proposition ; 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 con- vertend and its converse are accordingly equivalent propositions. The conversion in both these cases is said to be simple. In the case of A, it is different ; we cannot pass by im- mediate inference from All S is P to All P is S, inasmuch as P is distributed in the latter of these propositions but undis- tributed in the former 2 . Hence, although we start with a uni- versal proposition, we obtain by conversion a particular one only 3 , and by no means of operating upon the converse can we regain the original proposition. The convertend and its con- verse are accordingly non-equivalent propositions. The con- 1 See chapter 5. * All S is P and All P is S may of course happen to be true together, as in the case of the two propositions All equilateral triangles are equiangular triangles and All equiangular triangles are equilateral triangles. "But it is only knowledge of the matter of fact contained in the judgment in question which can assure us that the relation, upon which this possibility depends, holds good between S and P in any particular instance" (Lotze, Logic, 80). When it also happens that All P is S, the judgment All S is P is sometimes said to be reciprocal. If this is to be formally expressed in a single judgment, we must make use of the form All S is all P. 3 The failure to recognise or to remember that universal affirmative propo- sitions are not simply convertible is one of the most fertile sources of fallacy. 96 PROPOSITIONS. [PART n. version in this case is called conversion per accidens 1 , or con- version by limitation 2 . 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 logical form All stitches in time are things that save nine stitches. This, being an A proposition, is only con- vertible 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 per- forming other immediate inferences upon any given proposition when once it has been brought into ordinary 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, how- ever, this rule is neglected, mistakes are pretty sure to follow. 64. Inconvertibility of Particular Negative Propositions. It follows immediately from the rules of conversion given in section 62 that Some S is not P does not admit of ordinary con- version ; for S which is undistributed in the convertend would become the predicate of a negative proposition in the converse, and would therefore be distributed 3 . It will be shewn presently, 1 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 J5,' 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. 2 Simple conversion and conversion per accidens are also called respectively conversio pura and conversio impura. Compare Lotze, Logic, 79. 3 As regards the inconvertibility of see also sections 65 and 89. CHAP. III.] CONVERSION OF PROPOSITIONS. 97 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 contra- position). 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 1 . 65. Legitimacy of Conversion. Aristotle proves the con- version of E indirectly, as follows 2 : No S is P, therefore, No P is 8; 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. 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 is inconsistent with the original supposition. The legitimacy of the simple conversion of I follows similarly. It might appear that nothing is required in the above proof beyond the principles of contradiction and excluded middle ; the proof is not however satisfactory, for it may be plausibly maintained that in passing from Some individual P, say Q, is S, to Q is both S and P, we have already practically 1 See section 71. 2 "By the method called &c0e No contrapositive in either of its forms is obtainable, nor any inverse. E. (i) Original proposition, No S is P. (ii) Obverse, All S is not-P. (iii) Converse, No P is S. (iv) Obverted Converse, All P is not-S. (v) Contrapositive, Some not-P is S. (vi) Obverted Contrapositive, Some not-P is not not-S. (vii) Inverse, Some not-S is P. (viii) Obverted Inverse, Some not-S is not not-P. O. (i) Original proposition, Some S is not P. (ii) Obverse, Some S is not-P. (v) Contrapositive, Some not-P is S. (vi) Obverted Contrapositive, Some not-P is not not-S. No converse in either of its forms is obtainable, nor any inverse. All the above is summed up in the following table : A. I. E. 0. i Original proposition SaP SiP SeP SoP ii Obverse SeP SoP SaP' SiP iii Converse PiS PiS PeS iv Obverted Converse PoS' PoS' PaS' v Contrapositive P'eS PiS PiS vi Obverted Contrapositive ... P'aS' P'oS' PoS' vii Inverse S'oP S'iP viii Obverted Inverse S'iP S'oP 110 PROPOSITIONS. [PART u. It may be pointed out that the following rules apply to all the above immediate inferences : 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 8', the quantity must be depressed 1 ; if P, the quantity must be depressed in A and O ; if P', the quantity must be depressed in E and I. 72. Table of Propositions connecting any two terms and their contradictories. Taking any two terms and their contra- dictories, 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 =SeP' = P'eS = P'aS' A' S'aP' = S'eP = PeS' = PaS E SaP' = SeP = PeS = PaS' E' S'aP = S'eP' = P'eS' = P'aS . Particulars O SoP =SiP' = P'iS = P'o8 r S'oP' = S'iP =PiS' = PoS 1 SoP' =SiP =PiS = PoS' I' S'oP = S'iP' = P'iS' = P'oS In this table, columns (i) and (ii) give the propositions in which S or S' is subject, and columns (iii) and (iv) the proposi- tions in which P or P' is subject. Columns (i) and (iv) give the forms which admit of simple contraposition (i.e., A and O), and columns (ii) and (iii) those which admit of simple conver- sion (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. 1 In speaking of the quantity as depressed, it is meant that a universal yields & particular, and a particular yields nothing. CHAP. III.] MUTUAL RELATIONS OF PROPOSITIONS. Ill The forms in black type shew that we may take for our eight non-equivalent propositions the four propositions con- necting 8 and P, and a similar set connecting not-S and not-P 1 . 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 S'aP' and S'eP' ; 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 S'iP' and S'oP' ; 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. 73. Mutual Relations of the non-equivalent propositions connecting any two terms and their contradictories*. We may now investigate the mutual relations of our eight non-equiva- lent propositions. SaP, SeP, SiP, SoP form an ordinary square of opposition ; and so do S'aP', S'eP', S'iP', S'oP'. Reference to columns (iii) and (iv) in the table will shew further that SaP, S'eP', S'iP', SoP are equivalent to another square of opposition ; and that the same is true of S'aP', SeP, SiP, S'oP'. This leaves only the following pairs unaccounted for : SaP, S'aP'; SeP, S'eP'; SoP, S'oP'; SiP, S'iP'; SaP, S'oP'; S'aP', SoP ; SeP, S'iP' ; S'eP', SiP ; and it will be found that in each of these cases we have an independent pair. SaP and S'aP' (which are equivalent to SaP, PaS, and also to PaS', S'aP 1 ) taken together serve to identify the classes S and P, and also the classes S' and P'. They are therefore complementary propositions, in accordance with the definition given in section 66. Similarly, SeP and S'eP' (which are equivalent to SaP', P'aS, and also to PaS', S'aP) are com- plementary ; they serve to identify the classes S and P', and also the classes S' and P. It will be observed that the com- plementary of any universal proposition may be obtained by replacing the subject and predicate respectively by their 1 The former set being denoted by A, E, I, 0, the latter set may be denoted by A', E', I', 0'. 2 This section may be omitted on a first reading. 112 PROPOSITIONS. [PART n. contradictories. No fallacy is more common than 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 contra- dictories of the subject and predicate of the other) will be found to be sub-complementary in accordance with the definition in section 66; this relation holds between $oPand S'oP', and be- tween SiP and S'iP'. SoP and S'oP' (which are equivalent to SoP, PoS, and also to P'oS', S'oP') indicate that the classes S and P are neither coextensive nor either included within the other, and also that the same is true of S f and P'; SiP and S'iP' (which are equivalent to SoP', P'oS, and also to PoS', S'oP) indicate the same thing as regards S and P', S' 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 sub- ordinates two classes to two other classes. Thus, SaP and S'oP' (which are equivalent to SaP, PoS, and also to P'aS', S'oP') taken together shew that the class S is contained in but does not exhaust the class P, and also that the class P' is contained in but does not exhaust the class S' ; S'aP' and SoP (which are equivalent to S'aP', P'oS', and also to PaS, SoP) the same as regards the classes S' and P', and the classes P and S; SeP and S'iP' (which are equivalent to SaP', P'oS, and also to PaS', S'oP) as regards S and P', and P and S' ; and S'eP' and SiP (which are equivalent to S'aP, PoS', and also to P'aS, SoP') as regards S' and P, P' and S. Denoting the complementaries of A and E by A' and E', and the sub-complementaries of I and O by I' and O', the various relations between the non-equivalent propositions connecting any two terms and their contradictories may be exhibited in the following octagon of opposition : CHAP. III.] OCTAGON OF OPPOSITION. A E 113 o' I O 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 re- lation between A and E, A and E', A' and E, A' and E' 1 . 74. The Elimination of Negative Terms 2 . 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 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. It is of course clear that by means of obversion we can get rid of a negative term occurring as the predicate of a proposi- tion. The problem is more difficult when the negative term occurs as subject, but in this case elimination may still be possible ; for example, S'iP = PoS. We may even be able to get rid of two negative terms ; for example, S'aP' = PaS. So long, however, as we are limited to categorical propositions of 1 For the octagon of opposition in the form in which it is here given I am indebted to Mr Johnson. 2 This section may be omitted on a first reading. K. L. 8 114 PROPOSITIONS. [PART n. 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 (6) in a universal negative or a particular affirmative with a negative term as predicate also. The validity of the above results is at once shewn by re- ference to the table of equivalences given in section 72. 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 : S'aP = S'eP = P'eS' = P'aS ; S'oP = S'iP = P'iS' = P'oS. In these cases we may indeed get rid of 8' (as, for example, from S'aP), but it is only by introducing P' (thus, S'aP = P'aS); there is no getting rid of negative terms altogether. We may here refer back to the results obtained in sections 66 and 72 ; 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 ex- istentially negative while particular propositions are existentially affirmative 1 , then negative terms may always be eliminated. Thus, No not-S is not-P is equivalent to the statement that 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 that 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 some- thing 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, 1 It is necessary here to anticipate the results of a discussion that will come at a later stage. See chapter 7. CHAP. III.] ELIMINATION OF NEGATIVE TERMS. 115 Some S is not P, All P is 8, Some P is not S, Everything is either S or P, There is something besides S and P. Following out this line of treatment, the table of equiva- lences given in section 72 may be rewritten as follows [columns (ii) and (iii) being omitted, and columns (v) and (vi) taking their places]: (i) (iv) (v) (vi) SaP = P'aS' = Nothing is SP' = Everything is S' or P. S'aP' = PaS = Nothing is S'P = Everything is S or P'. SaP' = PaS' = Nothing is SP = Everything is S' or P'. S'aP = P'aS = Nothing is S'P' = Everything is S or P. SoP = PoS' = Something is SP' = There is something besides S' and P. S'oP' = PoS = Something is S'P = There is something besides SandF. SoP' = PoS' = Something is SP = There is something besides S' and P. S'oP =P'oS = Something is S'P' = 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 8 and P only 1 . The scheme of propositions given in this section may be brought into interesting relation with the fundamental laws of thought 2 . The scheme is based upon the recognition of the following prepositional 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. 1 The first four propositions in column (v) may be expressed symbolically SP / = Q, &c. ; the second four SP'>0, Ac. ; the first four in column (vi) S' + P=1, &Q.; and the second four S' + P<1, &c. ; where 1= the universe of discourse, and = nonentity, i.e., the contradictory of the universe of discourse. 2 Compare Mrs Ladd Franklin in Mind, January, 1890, p. 87. 82 116 PROPOSITIONS. [PART n. If in the above propositions we now write S for P, we have the following : Every S is 8 ; Every not- 8 is not -8 ; Nothing is both 8 and not-8 ; 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 proposi- tions with which we have been dealing may, therefore, be said to be based upon the recognition of just those prepositional forms which are required in order to express the fundamental laws of thought. Since the pro positional 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 in- ferences involved be granted, then it follows that the funda- mental 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. 75. Other Immediate Inferences. One or two other com- monly recognised forms of immediate inference may be briefly considered. (1) 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 1 . 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 1 See section 33. The inference now under discussion is only a particular case of a wider class of inferences which will be more fully discussed in Part iv. CHAP. III.] INFERENCE BY ADDED DETERMINANTS. 117 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 A P is Q ; moreover, by the law of iden- tity, All AP is A ; therefore, All AP is AQ 1 . The formal validity of immediate inference by added deter- minants 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 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 both in the subject and in the 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. (2) Immediate inference by complex conception is a process of immediate inference which consists in employing the subject and predicate of the original proposition as parts of a more 1 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. Other- wise 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 compli- cations that arise in connexion with the difficult problem of existential import. 118 PROPOSITIONS. [PART n. 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. It must suffice here to call attention to the danger of committing a fallacy, akin to the fallacy of composi- tion (see section 8), by passing from the distributive to the col- lective use of a term. For example, Protestants are Christians, therefore, A majority of Protestants are a majority of Christians. (3) 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 state- ment 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 Conse- quence to logical form The failure of a Material Conse- quence 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 Conse- quence is rightly excluded from Logic Among these material, CHAP. III.] INFERENCE BY CONVERSE RELATION. 119 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. 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 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 in- ferences 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). (4) Immediate inference by modal consequence or, as it is also called, inference by change of modality is somewhat analo- gous to subaltern inference. It consists in nothing more than weakening a statement in respect of its modality ; and hence it 120 PROPOSITIONS. [PART n. 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 versd. 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 versa 1 . 76. Reduction of immediate inferences to the mediate form 2 . 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. We may briefly consider various ways of establishing the validity of immediate inferences by means of mediate inferences. (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 8 ; for, if not, then by the law of contradiction, Some P is S; and we have this syllogism No SisP, Some P is S, therefore, Some P is not P. a reductio ad absurdum*. (2) It may be plausibly maintained that in Aristotle's proof of the conversion of E (given in section 65), 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 : 1 Compare Ueberweg, Logic, 98. 3 Students who have not already a technical knowledge of the syllogism may omit this section until they have read the earlier chapters of Part in. 3 Compare Hansel's Aldrich, p. 62. The conversion of A and the con- version of I may be established similarly. CHAP. III.] SYLLOGISTIC PROOFS OF CONTRAPOSITION. 121 Given All 8 is P, we have also No not-P is P, by the law of contradiction, therefore, No not-P is S 1 . (4) We might also obtain the contrapositive of All 8 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*. (5) The contraposition of A may also be established in- directly by means of a syllogism in Darii : All SisP, 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 3 . 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. 1 Similarly, granting the validity of obversion, the contraposition of 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. 2 Compare Jevons, Principles of Science, chapter 6, 2 ; and Studies in Deductive Logic, p. 44. 3 Compare De Morgan, Formal Logic, p. 25. Granting the validity of obversion, the contraposition of E and the contraposition of may be estab- lished similarly. 122 PROPOSITIONS. [PART n. The following are interesting only as curiosities : (6) Wolf obtains the subaltern of a universal proposition by a syllogism in Darii. Given A II Sis P, we have also Some 8 is S, by the law of identity, therefore, Some S is P. (Compare Mansel, Prolegomena Logica, p. 217.) (7) "Still more absurd is the elaborate system which Krug, after a hint from Wolf, has constructed in which all immediate inferences appear as hypothetical syllogisms ; a major premiss being supplied in the form, ' If all A is B, some A is B.' The author appears to have forgotten, that either this premiss is an additional empirical truth, in which case the immediate reason- ing is not a logical process at all ; or it is a formal inference, presupposing the very reasoning to which it is prefixed, and thus begging the whole question" (Mansel, Prolegomena Logica, p. 217). EXERCISES. 77. 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.] 78. Point out any possible ambiguities in the following proposi- tions, 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.] 79. State in logical form and convert the following proposi- tions : (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.] CHAP. III.] EXERCISES. 123 80. "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.] 81. 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.] 82. 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.] 83. Take the following propositions in pairs, and in regard to each pair state whether the two propositions are consistent or in- consistent 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.] 84. 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.] 85. 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 ; (ii) All organic substances contain carbon ; (iii) Some substances not containing carbon are organic ; (iv) Some inorganic substances do not contain carbon. [c.] f- 86. "All that love virtue love angling." Arrange the following propositions in the three following groups : (a) those which can be inferred from the above proposi- tion ; (/?) those which are consistent with it, but which cannot be inferred from it ; (y) those which are inconsistent with it. (i) None that love not virtue love angling, (ii) All that love angling love virtue. 124 PROPOSITIONS. [PART n. (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.] w 87. 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.] CHAPTER IV. THE DIAGRAMMATIC REPRESENTATION OF PROPOSITIONS. 88. The use of Diagrams in Logic. In representing pro- positions 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 which ought to be satisfied by any diagrammatic scheme is that the interpreta- tion of the diagrams should be intuitively obvious, as soon as the principle upon which they are based has been explained 1 . A second essential requirement is that the diagrams should be adequate ; that is to say, they should give a complete, and not a partial, representation of the relations which they are intended to indicate. Hamilton's use of Euler's diagrams, as 1 Hamilton's "geometric scheme" which he himself describes as "easy, simple, compendious, all-sufficient, consistent, manifest, precise, complete " (Logic, vol. 2, p. 475) fails to satisfy this condition. He represents an affirma- tive 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 in- dicates that the corresponding term is distributed; a comma (,) that it is undistribiited. Thus, for All S is P we have 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. 126 PROPOSITIONS. [PART n. 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 prepositional forms recognised in the schedule of propositions which it is intended to illustrate, e.g., particulars as well as universals. 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, presently to be described, can be adapted to the representation of particulars, but not very conveniently ; where- as they are specially suited to the representation of universals. Their use may, therefore, for the most part be limited to occasions when we are dealing only with 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 prepositional form should be represented by a single diagram, not by a set of alternative diagrams. This is, how- ever, by no means essential. For if we adopt a schedule of propositions which in some cases yields 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 this purpose. This point will be illustrated, with reference to Euler's diagrams, in the following section 1 . 1 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 hi intension. The fundamental objection raised by Mansel against the introduction of diagrammatic aids into logic appears to lose its force, even from the conceptualist standpoint, if attention is paid to this dis- tinction. " If Logic," he says, " is exclusively concerned with Thought, and Thought is exclusively concerned with Concepts, it is impossible to approve of a practice, sanctioned by some eminent Logicians, of representing the relation CHAP, iv.] EULER'S DIAGRAMS. 127 89. Enters 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 1 . 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 prepositional forms is to exclude one or more of these possibilities. All 8 is P limits us to one of the two a, /3 ; Some S is P to one of the four a, /3, 7, 8 ; No S is P to e ; Some S is not P to one of the three 7, S, e. of terms in a syllogism by that of figures in a diagram. To illustrate, for example, the position of the terms in Barbara by a diagram of three circles, one within another, is to lose sight of the distinctive mark of a concept, that it cannot be presented to the sense, and tends to confuse the mental inclusion of one notion in the sphere of another, with the local inclusion of a smaller portion of space in a larger " (Prolegomena Logica, p. 55). It seems clear that in speaking of concepts as incapable of being presented to the sense, Mansel is thinking of the concepts themselves, not of their extension. Even con- ceptualist logicians must, however, recognise and deal with the extension of concepts. It is of course true that the local inclusion of one portion of space in another is not the same thing as the inclusion of one class in another. But the analogy is sufficiently close for purposes of illustration ; and Mansel's argument does not in any degree establish the unfitness of the diagrams for the purpose for which they are intended. 1 Euler lived from 1707 to 1783. His diagrammatic scheme is given in his Lettres a une Princesse d'AUemagne (Letters 102 to 105). 128 PROPOSITIONS. [PART n. It is most misleading to attempt to represent All S is P by a single diagram, thus or Some 8 is P by a single diagram, thus 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 is un- doubtedly sound as regards the use of Euler's circles by Hamilton and Jevons; but it loses most of its force if the diagrams are used in the manner above described. It is true that they 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 pro- positions we may shade the part of the predicate concerning which knowledge is given us. CHAP, iv.] EULER'S DIAGRAMS. We then have A. 129 E. O. 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. Com- paring 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 in- formation 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 how it is that A admits only of conversion K. L. 9 130 PROPOSITIONS. [PART n. per accidens. All S is P limits us to one or other of the following The problem of conversion is What do we know of P in either case ? In the first we have All P is S, in the second Some P is S ; and since we do not know which of the 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 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 contraries, and (ii) is inconsistent with both of them. Hence nothing can be affirmed of P that is true in all three cases indifferently. (4) To illustrate the more complicated forms of imme- diate 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 With regard to not-P, these yield respectively, (i) No not-P is S ; (ii) No not-P is S. And thus we obtain the contraposi- tive of the given proposition. With regard to not-S we have, (i) No not-S is P, (ii) Some CHAP, iv.] EULER'S DIAGRAMS. 131 not-S is not P 1 . Hence in either case we may infer Some not-S is not P. 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 66). Thus, the pair of complementary pro- positions, SaP and PaS, taken together, limit us to the pair of contra-complementary propositions, SaP and PoS, to the pair of sub-complementary propositions, SoP and PoS, to or The application of the diagrams to syllogisms will be shewn 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 really mastering the elementary principles of formal logic, and especially in dealing with immediate in- ferences, will despise any means of illustrating afresh the old truths, and presenting them under a new aspect. 1 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 in- ferences in the preceding chapter has been based. 92 132 PROPOSITIONS. [PART n. The fact that we have not a single pair of circles correspond- ing 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 knowledge given by the propositions themselves, it is rather an advantage than otherwise, inasmuch as it shews how limited in some cases this knowledge actually is 1 . 1 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 dia- grams... 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. Ueberweg (Logic, 71) gives the following : A. I. CHAP, iv.] LAMBERT'S DIAGRAMS. 133 90. Lambert's Diagrams. A scheme of diagrams was employed by Lambert- 1 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. We may first shew how Lambert's lines may be used in such a manner as to be precisely analogous to Euler's circles. Thus, the four fundamental propositions may be represented as follows : P P A. or S 8 P or or or 8 S S S In lieu of the last of the figures on the preceding page, the following may be suggested as simpler, but equally effective : Here we get the three cases yielded by an 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 y, 5, e, on p. 127. 1 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 hi order to obviate certain difficulties involved in Lambert's own diagrams. See the note on p. 135. 134 PROPOSITIONS. [PART n. P E. 8 P P P O. or or 8 S S These diagrams take less room 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 8 where the dotted line indicates that we are uncertain as to whether there is or is not any P which is not S. We ob- viously 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 8 and here we get the four cases previously given for an I propo- sition by (a) filling in the dots to the left and striking out those to the right, (6) 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 8 line, and in the CHAP. IV.] LAMBERT S DIAGRAMS. 135 other no part of any P line, is dotted 1 . These two schemes are given below to the left and right respectively of the preposi- tional forms themselves. 8 8 8 All 8 is P Some 8 is P No 8 is P 8 8 8 8 Some 8 is not P 8 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 2 . An adaptation of Lambert's diagrams in which the contradictories of 8 and P are introduced as well 1 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. 2 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, P S This diagram certainly seems as appropriate to as it does to I ; but the modi- fication introduced in the text, and indeed suggested by Dr Venn himself, is not open to a similar objection. 136 PROPOSITIONS. [PART n. as S and P themselves will be given in section 94. This more elaborated scheme will be found particularly valuable for illus- trating the various processes of immediate inference. 91. 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-8 by S' and what is both 8 and P by SP, &c., it is clear that every- thing must be contained in one or other of the four classes SP, SP', S'P, S'P' ; and the above diagram shews four compartments (one being that which lies outside both the circles) corre- sponding 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 SP', is represented by No S is P by With three terms we shall have three circles and eight compartments, thus : CHAP, iv.] DR VENN'S DIAGRAMS. 137 All S is P or Q is represented by Q 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, equally well adapted to the case of particular propositions. Dr Venn (in Mind, 1883, pp. 599, 600) suggests that we might just draw a bar across the compartment declared to be saved by a particular proposition 1 ; thus, Some S is P would be represented by drawing a bar across the SP compartment. Theoretically this plan might be worked out satisfactorily ; but in representing a combination of propositions in this way special care would have to be taken in the interpretation of the diagrams. For example, if we have the diagram for three terms 8, P, Q, and are given Some S is P, we do not know that both the com- partments SPQ, SPQ', are to be saved, and in a case like this a bar drawn across the SP compartment might easily be misinterpreted. 92. Expression of the possible relations betiueen any two classes by means of the propositional forms A, E, I, O. Any information given with respect to two classes limits the possible 1 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. 138 PROPOSITIONS. [PART n. relations between them to one or more of the five which are indicated diagrammatically by Euler's circles as shewn at the beginning of section 89. 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 , ft, 7, 8, e 1 . 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 a or ft (i.e., the exclusion of 7, S, and e) be denoted by a, ft ; limitation to a, ft, or 7 (i.e., the exclusion of 8 and e) by a, j3, 7 ; and so on. 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 a, ft is expressed by All S is P. Sometimes we require a combination of propo- sitions ; thus a is expressed by the pair of complementary pro- positions All S is P and all P is 8, (since all S is P excludes 7, S, e, and all P is S further excludes ft). Some other cases are still more complicated ; thus the fact that we are limited to a or 8 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 8. Let A All S is P, A=All P is 8, and similarlv for ' 1 * V the other propositions. Also let AA 1 = All S is P and all P is 8, &c. Then the following is a scheme for all possible cases : 1 Thus, the classes being S and P, a denotes that S and P are wholly coincident ; /3 that P contains S and more besides ; y that S contains P and more besides ; 5 that S and P overlap each other, but that each includes some- thing not included by the other ; e that S and P have nothing whatever in common. CHAP. IV.] EXPRESSION OF POSSIBLE CLASS RELATIONS. 139 limitation to denoted by limitation to denoted by a AA, ",Ay A or A : ft A0 t a, ft, 8 A or IO, y A t O ,& A or E 8 IOO, a, y, 8 A t or IO 6 a , y, A, or E */8 A a, 8, e AA : or OO, a , y A, Ay, 3 IO or IO, 0,8 AA, or IOO, fcy, AO, or A,O or E a, e AA, or E AS, ^ o, fty AO, or A,O y> 8, e A3 10, , A y, 8 I ft AO, or E , ^, y, e A or A : or E 7,8 IO a, 0, 8, e A or O, y, e A,O or E a, y, 8, e A, or O *,< 00, A y, 8, O or O, It will be found that any other combinations of propositions than those here given either involve contradictions or redun- dancies, or else no information is given because all the five relations that are cb priori possible still remain possible. For example, AI is clearly redundant ; AO is self-contra- dictory ; A or A t O is redundant (since the same information is given by A or A t ); 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 Ij = I, and E t = E. 140 PROPOSITIONS. [PART n. It should be noticed that if we read the first column downwards and the second column upwards we get pairs of contradictories. 93. 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 8 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 89, the above as- sumption clearly requires that in the cases of a, ft, and l P P (,]} r , P P L In this scheme each line represents the entire universe of dis- course, 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 : S . S' A. r/ E. o. K. L. 10 146 PROPOSITIONS. [PART n. 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 S' (or P or P', 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, S', P, and P all represent existing classes). Thus, from the first diagram we can read off at a glance SaP, PiS, P'aS', S'iP' ; from the second SeP, PeS, P'oS', S'oP; from the third SiP and PiS ; and from the fourth SoP and P'oS'. The last two diagrams are also seen at a glance to be indeterminate in respect to P' and S', P and S', respectively (that is to say, I has no contrapositive and no inverse, O has no converse and no inverse). EXERCISES. 95. 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 .] 96. Take all the ordinary propositions connecting any two terms, combine them as far as possible without contradiction, and represent each combination diagrammatically. [j.] CHAPTER V. PROPOSITIONS IN EXTENSION AND IN INTENSION. 97. Fourfold Implication of Propositions in Connotation and Denotation. In dealing with the question whether pro- positions 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 ex- cluded thereby. It follows, however, from the double aspect of names in extension and intension that the different relations really involve one another, and hence are all of them virtually asserted in any categorical proposition whose subject and pre- dicate are both general names. 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 in- tensively 1 . Both subject and predicate will then have a denotation determined by their connotation, and hence our 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 1 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 compre- hension. 102 148 PROPOSITIONS. [PART n. 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. As an example, we may take the proposition, All men are mortal. 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. 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 (6) 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 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 connota- tion determines denotation and not vice versa] 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 constitutes 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 CHAP. V.] PREDICATIVE READING OF PROPOSITIONS. 149 difficulty may rather be to justify a treatment which in any respect distinguishes between them 1 . (i) 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 undoubtedly corresponds in the great majority of cases with the course of ordinary thought ; 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 con- notative 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 1 The true doctrine is excellently stated by Mrs Ladd Franklin. " The reason that so many different views are possible in regard to the import of propositions is a very simple one. Every term is a double-edged machine it effects the separating out of a certain group of objects and it epitomises a certain complex of marks. From this double nature of the term, it follows with mathematical rigour that a proposition, which contains two terms must have a fourfold implication (though one of the four senses may be at any time uppermost in the mind). Whoever says, for instance, that 'All politicians are statesmen ' must be prepared to maintain that the objects, politicians, are the same as some of the objects, statesmen, and are in possession of all the qualities of statesmen ; and also that the quality-complex, politician, entails the quality-complex, statesman, and is indicative of the presence of some of the objects statesmen... Now it is open to the logician to say that any one of these four implications is the most important or the most prominent implication of the proposition, but it is not open to him to say that less than all four of them is the complete implication... The proposition 'a& is non-existent' does not state that the classes a and b have nothing in common, any more than it states that the qualities a and b are never found in conjunction. Mill's view of the import of the proposition is the third of these that wherever we find certain attributes, there will be found certain other attributes, that the latter set of attributes constantly accompany the former set. The common class view is the first of these. The view that the extent of the subject and the intent of the predicate are most frequently uppermost in the mind is the view that will probably commend itself to the careful psychologist" (Mind, October, 1890, pp. 561, 2). 150 PROPOSITIONS. [PART n. 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 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). 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. (&) 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 classifica- tory sciences. The following propositions may be taken as instances: All palms are endogens, All daisies are composites, CHAP. V.] CLASS READING OF PROPOSITIONS. 151 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). 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. (ii) Subject in denotation, predicate in denotation. If we read both the subject and the predicate of a proposi- tion in denotation, we have a relation between two classes, and hence this is called the class mode of interpreting the proposi- tion. 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 speak- ing 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. (6) 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 denota- tion 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 152 PROPOSITIONS. [PART n. predicate must be read in denotation (or else both in con- notation), 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 proposi- tions 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 erroneous. But the arguments used in support of this view will not bear examination. (1) It is said that to read both subject and predicate in denotation is psychologically false. It has indeed been already pointed out that the class mode of interpretation is not that which as a rule first presents itself to our mind when a proposi- tion is given us; but we have also seen that there are exceptions to this, as, for example, in the propositions All daisies are composite, All Hindoos are Aryans, All Tartars are Turanians. It is, therefore, clearly wrong to describe the reading in question as 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. But it must be added that even when we do not spontaneously adopt the class reading, it is still a reading that is psychologically possible. Mr Bradley writes "Judgment is not inclusion in, or exclusion from, a class. The doctrine that in saying A is equal to B, or B is to the right of G, or To-day precedes Monday, I have in my mind a class, either a collection or description, of things equal to B, or to the right of C, or preceding Monday, is quite opposed to fact " (Principles of Logic, pp. 22, 3). It may be readily admitted that in such cases as these we do not, when the proposition is first given to us, naturally think of the predicate as the name of a class. 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. (2) Mr Welton asks what we mean by a class, for example, by the class of birds when we say All owls are birds. " It is CHAP. V.] CLASS READING OF PROPOSITIONS. 153 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 at- tributes. 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" (Logic, p. 218). In so far as this argu- ment 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, Book i, ch. 5, 3) in order to lead up to his position that the ultimate interpreta- tion of the categorical proposition requires us to read both subject and predicate in connotation, since denotation is determined by connotation. But if we grant this, it does not prove the class reading of the 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. (3) 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," says Mr Welton, "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'" (Logic, p. 218). 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 pre- dicate. 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." 154 PROPOSITIONS. [PART n. (4) It is said that a class interpretation of both 8 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 some is not, No S is P. The point here raised will presently be considered further in connexion with the quanti- fication of the predicate. (iii) Subject in connotation, predicate in connotation. If we read both the subject and the predicate of a proposi- tion 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". 1 Similarly, in the case of Some S is P "the attributes 1 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 distin- guished both from comprehension and from subjective intension. Unfortunately confusion is very 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"). CHAP. V.] CONNOTATIVE BEADING OF PROPOSITIONS. 155 which constitute the connotation of 8 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 unaccom- panied by those which constitute the connotation of P." It will be noticed that in the connotative reading we have 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 pro- positions 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. It equally fails to apply 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. The intensive reading here criticised must be carefully distinguished from the connotative reading given in the text. The latter is not open to similar objections. Both readings must be further distinguished from the reading in comprehension discussed in the following section. It may be added, quite independently of the above criticisms, that Hamilton is in error in speaking of the distinction between judgments in extension and judgments in intension as a division of judgments (Logic, i., p. 231). It is clear that the distinction is really between two different points of view from which the same judgment may be regarded. The view that some propositions will more naturally be interpreted extensively and others intensively may admit of justification (compare Veitch, Institutes of Logic, pp. 72, 3 and 224, 5) ; but if this is all that Hamilton means, he hardly expresses himself accurately. 156 PROPOSITIONS. [PART n. 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 8 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 con- notative 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 pre- sents 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 considera- tion may shew that the two things mutually involve one another 1 . (iv) 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 1 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 nota est nota rei ipsius in its place. CHAP.V.] SUBJECT IN CONNOTATION, PREDICATE IN DENOTATION. 157 attributes connoted by 8 are an indication of the presence of an individual belonging to the class P." This mode of interpreta- tion 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 much better than any of the three other readings which we have been discussing 1 . It is worth while noticing here by way of anticipation that if the view of the existential import of propositions which will be advocated in section 122 is accepted, we shall still have a fourfold reading of categorical propositions in connotation and denotation. The universal negative and the particular affirma- tive may be taken as examples. 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 in- 1 Mr Welton considers it impossible to take the subject of a proposition in connotation and its predicate in denotation. " It is difficult," he says, " to see how a proposition expressed and interpreted strictly in this suggested form could have any meaning whatever. 'The attributes connoted by man are mortal beings ' (or ' are the class, or a portion of the class, mortal beings ') is such an expression, and it is, surely, hopelessly devoid of any intelligible significance. The only meaning it really expresses is, of course, absurdly false ; the attributes are not beings, or a class of beings, of any sort mortal or other- wise" (Logic, p. 235). It may perhaps be sufficient to say in reply to this criticism that it is based on a misapprehension of the kind of relation which is implied in a proposition between the connotation of the subject and the deno- tation of the predicate. A similar misapprehension might reduce to absurdity any one of the interpretations above discussed. Supposing, for example, that we read the subject in denotation and the predicate in connotation, as Mr Welton says we are bound to do, we have an equally unintelligible expression if we say that "the class denoted by man is the attribute mortality." The error consists in imagining that the relation expressed in a proposition must always be the same whether we regard the terms from the extensive or from the intensive standpoint. 158 PROPOSITIONS. [PART n. dividual common to the two classes 8 andP; (3) The attributes connoted by 8 and P respectively are never found conjoined ; (4) There is no individual possessing the attributes connoted by $ and belonging to the class P. Similarly the particular affirmative yields : (1) There are individuals belonging to the class 8 and possessing the attributes connoted by P ; (2) There are individuals common to the two classes 8 and P ; (3) The attributes connoted by 8 and P respectively are sometimes found conjoined ; (4) There are individuals possessing the attributes connoted by 8 and belonging to the class P. The question which has been discussed in this section is, therefore, quite independent of that which will be raised in chapter 7 ; and no solution of the general problem raised in this chapter can afford a complete solution of the problem of the import of categorical propositions. We have here analysed the relation between the things that are signified by the terms of a proposition ; but the real predicative force of the categorical proposition remains to be decided. 98. 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. Asking this question in regard to the universal affirmative proposition All 8 is P, the solution clearly is that the comprehension of 8 includes the comprehension of P. The interpretation in com- prehension is thus precisely the reverse of that in denotation (the denotation of 8 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 same 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 8 and P partly coincide] and corresponding to this we should have the comprehension of 8 and P partly coincide. But this is clearly not what we express CHAP. V.] PROPOSITIONS IN COMPREHENSION. 159 by Some S is P, for it 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 8 and also to the whole of P ; for example, No Pembroke undergraduates are also Trinity undergraduates. Again, given an E proposi- tion 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 comprehen- sion of P ; No S is P, The comprehension of S includes the comprehen- sion 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. CHAPTER VI. LOGICAL EQUATIONS AND THE QUANTIFICATION OF THE PREDICATE. 99. 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 un- altered, 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 equi- angular triangle 1 . It is urged, however, by some writers that, if 1 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. CHAP. VI.] LOGICAL EQUATIONS. 161 this is what our equation comes to, then inasmuch as a statement of mere identity is absolutely empty and meaningless, it strictly speaking leaves us with nothing at all ; it contains no assertion and can represent no judgment 1 . The answer to this criticism is that whilst we have identity in a certain respect, it is erroneous to say that we have mere identity. We have identity of denota- tion 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) 2 . 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 just the same, none will be excluded by our second method of selection although an additional attribute is taken into account. 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 1 Compare Bradley, Principles of Logic, pp. 23 to 27. 3 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. K. L. 11 162 PROPOSITIONS. [PART n. " 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 proposi- tions 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. 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 state- ment, and that must be regarded as for all practical purposes immaterial. 100. Types of Logical Equations 1 . Jevous (Principles of Science, chapter 3) recognises three types of logical equations, which he calls respectively simple identities, partial identities, and limited identities. Simple identities are of the form S = P ; for example, Exogens = dicotyledons. Whilst this is the simplest case equa- tionally, the information given by the equation requires two propositions in order that it may be expressed in ordinary predicative form. Thus, All 8 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. 1 This section may be omitted on a first reading. CHAP. VI.] TYPES OF LOGICAL EQUATIONS. 163 Partial identities are of the form S = SP, and are the expression equationally of ordinary universal affirmative pro- positions. 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. 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 Vs, 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 it is clear that its proper place in his classifica- tion is 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 of course 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, however, 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. It is no part of our present purpose to discuss systems of 112 164 PROPOSITIONS. [PART n. 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 differ- entiate 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 = and VP = 0. V must, therefore, be limited by the antecedent condition that it repre- sents an existing class and a class that contains either S or P r and it is in this condition quite as much as in the equation itself that the real force of the particular proposition is- expressed 1 . If particular propositions are true contradictories of uni- versal propositions, then it would seem to follow that in a system in which universals are expressed as equalities, par- ticulars 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, 8 > SP would imply logically more than mere numerical inequality, it would imply that the class 8 includes the whole of the class SP and more besides. Thus interpreted, 8>SP expresses the particular negative proposition, Some S is not P 2 . If we further introduce the symbol 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 = ; Some S is not P S > SP Sp > ; NoSisP SP = S = Sp; Some S is P SP>0 S>Sp. This scheme, it will be observed, is based on the assumption 1 Compare Venn, Symbolic Logic, pp. 161, 2. 2 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 IV CHAP. VI.] EQUATIONAL READING OF PROPOSITIONS. 165 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 expres- sion 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. 101. 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. 166 PROPOSITIONS. [PART n. 102. The eight prepositional forms resulting from the explicit Quantification of the Predicate. We have seen that m the ordinary fourfold schedule of propositions, the quantity of the predicate is determined by the quality of the proposi- tion, 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 8 is all P, Y. Some S is some P, I. No S is any P, E. No S is some P, 77. Some S is not any P, O. Some S is not some P. w. The symbols attached to the different propositions in the above schedule are those employed by Archbishop Thomson 1 , and they are the ones now commonly adopted so far as the quantification of the predicate is recognised in modern text- books. The symbols used by Hamilton were Afa, Afi, Ifa, Ifi, Ana, Ani, Ina, Ini. Here f indicates an affirmative propo- sition, n a negative ; a means that the corresponding term is distributed, i that it is undistributed. Spalding's symbols (Logic, p. 83) are A 2 , A, P, I, E, \E t 0, |0. Mr Carveth Read (Theory of Logic, p. 193) suggests A\ A, P, I, E, E v 0, 2 . The equivalence of these various symbols is shewn in the following table : 1 Thomson himself, however, ultimately rejects the forms 77 and w. CHAP. VI.] QUANTIFICATION OF THE PREDICATE. 167 Thomson. Hamilton. Spalding. Carveth Bead. All Sis all P U A/a A 9 A* All S is some P A Afi A A Some S is all P Y Ifa P r Some S is some P I W I i No S is any P E Ana E E No S is some P 1 Ani & ti, Some S is not any P Ina Some S is not some P 0) Ini \o o, For the new forms we might also use the symbols SuP, SyP, SrjP, ScaP, on the principle explained in section 40. 103. 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 it will be well to consider very briefly 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. 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 im- plicitly 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 168 PROPOSITIONS. [PART n. 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 very obscurely expressed, but we can have no hesitation in admitting its validity. It is obviously of importance to the logician to clear up all ambiguities and ellipses of language. For this reason it is, amongst other things, desirable that we should as far as possible avoid in logic condensed and elliptical modes of expression. But whether Hamilton's postulate, as now interpreted, supports the doctrine of the quantification of the predicate is another question. This point will be considered in the next two sections. 104. Advantages claimed for the Quantification of the Pre- dicate. 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 ex- pressed in common language because common language is elliptical. Whatever is not really necessary to the clear com- prehension of what is contained in thought, is usually elided in 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." CHAP. VI.] QUANTIFICATION OF THE PREDICATE. 169 Amongst the practical advantages said to result from quan- tifying 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 we adopt the doctrine of the quantification of the predicate, 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 result- ing 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 a> 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). 105. Objections urged against the Quantification of the Pre- dicate. Those who reject Hamilton's doctrine of the quantifica- tion 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 quantity, that is, as an aggregate of objects, at all. We have already in section 97 indicated grounds for the view that in the great majority of instances the subject of a proposition is in ordinary thought naturally interpreted in denotation, but the predicate 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 ? 170 PROPOSITIONS. [PART n. 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 rumi- nates, mean exactly the same thing. The mental process they go through, as 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 pro- position ' 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 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 that is raised to the doctrine of the quantification of the predicate is that some of the quantified forms are composite not simple predications. Thus All 8 is all P is a condensed mode of expression, which may be analysed CHAP. VI.] QUANTIFICATION OF THE PREDICATE. 171 into the two propositions All 8 is P and All Pis 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 proposi- tion resolvable into All S is P and Some P is not S. But, as Professor Fowler observes, " it is the object of logic not to state our thoughts in a condensed form, but to analyse them into their simplest elements " (Deductive Logic, p. 32). As a rule, 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 doc- trine of quantification is sufficiently condemned by its want of internal consistency. Its unphilosophical character in this re- spect will be shewn in the following sections. 106. 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 new 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 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 T, 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 jive propositional forms 172 PROPOSITIONS. [PART n. instead of eight, namely, All S is all P, All S is some P, Some S is all P, Some Sis some P 1 , No S is any P. We have already seen (in section 89) that the only possible relations between two terms in respect of their extension are given by the five diagrams These correspond respectively to the above five propositions 2 ; and it is clear that on the view indicated by Dr Baynes the eight forms are redundant 3 . It is altogether doubtful whether writers who have adopted the eightfold scheme have themselves recognised the pitfalls 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*. Hamilton himself agrees with Thomson and Bowen on these points ; but he is curiously indecisive on the general question here raised. He remarks (Logic, n. p. 282) that some " is held to be a definite some when the other term is definite," i.e., in A and Y, 77 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 to 5 . 1 Using some in the sense here indicated, Some S is some P necessarily implies Some S is not any P and No S is some P. 2 Namely U, A, Y, I, E. and r\ cannot be interpreted as giving pre- cisely determinate information ; allows an alternative between Y and I, and TJ between A and I. For the interpretation of w see note 2 on p. 178. 3 Cf. Venn, Symbolic Logic, chapter i. 4 " 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." 5 Compare Veitch, Institutes of Logic, pp. 307 to 310, and 367, 8. " Hamilton would introduce some only into the theory of propositions, without, however, CHAP. VI.] MEANING OF "SOME." 173 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 affir- mative conclusion ; or that two negative premisses invalidate a syllogism 1 . Thus, the following should be regarded as valid : All P is some M, All M is some 8, 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. 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 relation 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 them- selves" (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." 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 prepositional 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 inter- pretation some at least is steadily adhered to." 1 The anticipation of syllogistic doctrine which follows is necessary in order to illustrate the point which we are just now discussing. 174 PROPOSITIONS. [PART n. 107. 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 area'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, p. 151). It is not, how- ever, the case that we necessarily involve ourselves in self-con- tradiction 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 ; but on the other hand these two propositions are inconsistent with each other. Taking the five prepositional forms which are obtained by attaching this meaning to some, namely, All 8 is all P, All S is some P, Some S is all P, Some S is some P, No S is P, it should be observed that each one of these propositions is incon- sistent with each of the others, whilst at the same time no one is the contradictory of any 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 I and E 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 judg- ments the one that some are, the other that others of the same class are not! The some only would thus appear as the com- posite of two propositions already formed It seems to me that we must, first of all, work out logical principles on the CHAP. VI.] MEANING OF "SOME. 175 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 diffi- culty 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 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. 108. The interpretation of the eight Hamiltonian forms of proposition, "some" being used in its ordinary logical sense 1 . Taking the five possible relations between two terms, as illus- trated by the Eulerian diagrams, and denoting them respectively by a, @, 7, S, e, as in section 89, we may write against each ' of the prepositional forms the relations which are compatible with it, on the supposition that some is used in its ordinary logical sense, that is, as exclusive of none but not of all 2 : u a A a,/8 Y a, y I a, ft, y, 8 E e *} &* O y, 8, u a, ft, y, 8, 1 The corresponding interpretation when some is used in the sense of some only is given in note 2, page 172, and in note 2, p. 178. 2 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. 176 PROPOSITIONS. [PART n. We have then the following pairs of contradictories A, O ; Y, 77 ; I, E. The contradictory of U is obtained by affirming an alternative between y and O. Without the use of quantified predicates, the same informa- tion may be expressed as follows : U=SaP, PaS; Y=PaS; 1 = SiP; 0=SoP. What information, if any, is given by co will be discussed in section 111. 109. The propositions U and Y. It must be admitted that these propositions are met with in ordinary discourse. We may not indeed find propositions which are actually written in the form All 8 is all P ; but we have to all intents and pur- poses 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 1 . Take also such propositions as the following : Christianity and civilization are coextensive ; Europe, Asia, Africa, America, and Australia are all the con- tinents 2 ; 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 3 . 1 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, " Mi- 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. 2 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 proposi- tions. But it is equally clear that the subject denotes the whole of a certain class, however limited that class may be. 3 These are all examples of what Jevons would call simple identities as distinguished from partial identities. Compare section 100. CHAP. VI.] THE PROPOSITIONS U, Y, AND 77. 177 Again, any exclusive proposition (as defined in section 48) may be given as an example of Y; e.g., Only S is P\ Graduates alone are eligible for the appointment; Some passengers are the only survivors. These propositions are respectively equiva- lent to the following : Some 8 is all P ; Some graduates are all who are eligible for the appointment ; Some passengers are all the survivors 1 . Moreover, as was shewn in section 48, this is the only way of treating the propositions which will enable us to retain the original subjects and predicates. We cannot then agree with Professor Fowler that the additional forms " are not merely unusual, but are such as we never do use" (Dediictive Logic, p. 31). U and Y 2 ought certainly to receive some recognition in logic. Still in treating the syllogism &c. on the traditional lines, it is better to retain the traditional schedule of propositions. The addition of 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. 110. The proposition 77. 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 77, provided that we do not regard this proposition as implying that any S is certainly P. Archbishop Thomson remarks that 77 " 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 77 may be true together, U and 1 In these propositions, some is of course to be interpreted in the indefinite sense, and not as exclusive of all. 2 The some of the subject being interpreted as meaning merely tome at leatt. K. L. 12 178 PROPOSITIONS. [PART n. 77 cannot, and Y and 77 are strictly contradictories 1 . The rela- tion of contradiction in which Y and 77 stand to each other is perhaps brought out more clearly if they are written in the forms Only S is P, Not only 8 is P, or 8 alone is P, Not 8 alone is P. It will be observed, moreover, that 77 is the converse of O, and vice versa. If, therefore, 77 has no power of denial, the same will be true of O also. But it certainly is not true of O. 111. The proposition o>. The proposition &>, Some 8 is not some P, is not inconsistent with any of the other prepositional forms, not even with U, All 8 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 o> asserts. Some 8 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 2 . De Morgan 3 (Syllabus, p. 24) points out that its contradictory is " 8 and P are singular and identical ; there is but one 8, there is but one P, and 8 is P." 4 It may be said without hesitation that the proposition eu is of absolutely no logical importance. 1 We are again interpreting some as indefinite. If it means some at most, then the power of denial possessed by i\ is increased. 2 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 is equivalent to U. Some but not all S is not some but not all P informs us that certain S's are not to be found amongst a certain portion of the P's but that they are to be found amongst the remainder of the P's, while the remaining S's are to be found amongst the first set of P's. Hence all S is P; and it follows similarly that all P is S. Some but not all S is not some but not all P is, therefore, equivalent to All S is all P. 3 De Morgan in several passages criticizes with great acuteness the Hamil- tonian scheme of propositions. 4 Professor Veitch remarks that in u " 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 . The appropriateness of this is denied by Mr Monck. "Again, can it be said that the proposition There are patriots and CHAP. VI.] SIXFOLD SCHEDULE OF PROPOSITIONS. 179 112. Sixfold schedule of propositions obtained by recognising Y and 77, in addition to A, E, I, O l . The schedule of proposi- tions obtained by adding Y and 77 to the ordinary schedule presents many interesting features, and is worthy of incidental recognition and discussion 2 . It has been shewn in section 66 that in the ordinary scheme there are six and only six in- dependent propositions connecting any two terms, namely, SaP, PaS, SeP (=PeS), SiP (=PiS), PoS, SoP. If we write the second and the last but one of these in such a form as to keep 8 and P as subject and predicate throughout, we have the schedule which we are now considering, namely, SaP = All S is P; SyP = Only 8 is P ; SeP =No S is P; SiP =Some S is P; SrjP = Not only S is P; SoP = Some S is not P. It will be observed that the pair of propositions, SyP and SrjP, 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 73, 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 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). 1 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. 2 It is not of course intended that this sixfold schedule should supersede the fourfold schedule in the main body of logical doctrine. It is, however, most 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. 122 180 PROPOSITIONS. [PART n. obtain any satisfactory obverse of either Y or t], 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 1 : Converse. Contrapositive. Inverse. SaP = PyS = P'aS' = S'yP' SyP = Pa8 = PyS' = S'aP' SeP = PeS = P'yS = S'yP SiP = PiS = P'rjS = S'fjP SrjP = PoS = P'tjS' = S'oP' SoP = PrjS = P'oS' = S' ' J ,\ClS D. 1 his case of A is B is a case of) But this overlooks the fact that a new judgment is required before we can tell that this is a case of A being B. The mere statement that some cases of A is B are cases of C is D is clearly not equivalent to the conclusion of the hypothetical syllogism 1 . By analogy we should have to argue that the following cate- gorical syllogism in Barbara is an immediate inference : All M is P, This is M, therefore, This is P. Thus the argument again proves too much. The argument in favour of regarding the modus tollens If P is true then Q is true, but Q is not true, therefore, P is not true as mediate inference is still more forcible ; but of course the modus ponens and the modus tollens stand or fall together 2 . Professor Groom Robertson (Mind, 1877, p. 264) has sug- gested an explanation as to the manner in which this contro- versy may have arisen. He distinguishes the hypothetical " if" from the inferential " if," the latter being equivalent to since, seeing that, because. No doubt by the aid of a certain accentua- 1 Professor Bowen obviously has in view a conditional as distinguished from a pure hypothetical major premiss. But this distinction does not materially affect the present argument. 2 In section 244 it will be shewn further that the hypothetical syllogism and the disjunctive syllogism also stand or fall together. 310 SYLLOGISMS. [PART III. tion the word " if " may be made to carry with it this force. Professor Robertson quotes a passage from Clarissa Harlowe in which the remark " If you have the value for my cousin that you say you have, you must needs think her worthy to be your wife," is explained by the speaker to mean, "Since you have &c." Using the word in this sense, the conclusion G is D certainly follows immediately from the bare statement If A is B, is D ; or rather this statement itself affirms the conclusion. When, however, the word " if " carries with it this inferential implication, we cannot regard the proposition in which it occurs as strictly hypothetical. We have rather a condensed mode of expression including two statements in one ; it may indeed be argued that in the single statement thus interpreted we have a hypothetical syllogism expressed elliptically 1 . EXERCISES. 238. Shew how the modus ponens may be reduced to the modus fattens. [K.] 239. Test the following : " If all men were capable of perfection, some would have attained it ; but none having done so, none are capable of it." [v.] 240. Examine technically the following argument : If you needed food, I would give you money ; but as you do not care to work, you cannot need food ; therefore, I will give you 110 money. [j.] 241. Construct conditional syllogisms in Cesare, Bocardo, Fesapo, and reduce them to the first figure. [K.] 242. Name the mood and figure of the following, and shew that either one may be reduced to the other form : (1) If R is true, Q is true, If P is true, Q is not true, therefore, If P is true, R is not true ; 1 Compare Hansel's Aldrich, p. 103. CHAP. V.] EXERCISES. 311 (2) If Z is true, Y is true, If T is true, X is not true, therefore, If X is true, Z is not true. [K.] 243. Let X, Y, Z, P, Q, R be six propositions. Given (1) If X is true, P is true; (2) IfYis true, Q is true ; (3) If Z is true, R is true ; (4) Of X, Y, Z one at least is true ; (5) Of P, Q, R not more than one is true ; prove syllogistically (i) If P is true, X is true ; (ii) If Q is true, Y is true ; (iii) If R is true, Z is true ; (iv) Of P, Q, R, one at least is true ; (v) Of X, Y, Z, not more than one is true. [K.] CHAPTER VI. DISJUNCTIVE SYLLOGISMS. 244. The Disjunctive Syllogism. A disjunctive (or alterna- tive) syllogism may be defined as a formal reasoning in which a categorical premiss is combined with an alternative premiss so as to yield a conclusion which is either categorical or else alternative with fewer alternants than are contained in the alternative premiss 1 . For example, A is either B or C, A is not B, therefore, A is G ; Either P or Q or R is true, P is not true, therefore, Either Q or R is true. The categorical premiss in each of the above syllogisms denies one of the alternants of the alternative premiss, and the conclusion affirms the remaining alternant or alternants. Reasonings of this type are accordingly described as examples of the modus tollendo ponens. It follows from the resolution of alternative propositions given in section 141 that the force of an alternative as a premiss in an argument is equivalent either to that of a con- ditional or to that of a hypothetical proposition. 1 Archbishop Thomson's definition of the disjunctive syllogism "An argu- ment in which there is a disjunctive judgment " (Laws of Thought, p. 197) must be regarded as too wide if, as is usually the case, an affirmative judgment with a disjunctive predicate is considered disjunctive. It would include such a syllogism as the following B is either C or D, A is B, therefore A is either C or D. The argument here in no way turns upon the alternation contained in the major premiss, and the reasoning may be regarded as an ordinary categorical syllogism in Barbara, the major term being complex. A more general treatment of reasonings involving disjunctive judgments is given in Part iv. CHAP. VI.] THE DISJUNCTIVE SYLLOGISM. 313 Thus, Either A is B or G is D, A is not B, therefore, C is D ; may be resolved into the form If A isnotB,CisD, A is not B, therefore, C is D ; or into the form If C is not D, A is B, A is not B, therefore, C is D 1 . A corollary from the above is that those who deny the character of mediate reasoning to the hypothetico-categorical syllogism must also deny it to the disjunctive syllogism, or else must refuse to recognise the resolution of the disjunctive proposition into one or more hypotheticals. In the above example it is not quite clear from the form of the major premiss whether we have a true hypothetical or a conditional. But in the following examples, which are added to illustrate the distinction, it is evident that the alternative propositions are equivalent to a true hypothetical and to a conditional respectively : Either all A's are B's or all A's are CPs, This A is not B, therefore, All A's are C's; All A's are either B or C, This A is not B, therefore, This A is C*. 1 Logicians have not, as a rule, given any distinctive recognition to argu- ments consisting of two disjunctive premisses and a disjunctive conclusion ; and Mr Welton goes so far as to remark that " both premisses of a syllogism cannot be disjunctive since from two assertions as indefinite as disjunctive pro- positions necessarily are, nothing can be inferred " (Logic, p. 327). It is, how- ever, clear that this is erroneous, if an argument consisting of two hypothetical premisses and a hypothetical conclusion is possible, and if a hypothetical can be reduced to the disjunctive form. As an example we may express in disjunc- tives the hypothetical syllogism given on p. 300 : Either Q is not true or R it true, Either P is not true or Q is true, therefore, Either P is not true or R is true. a When the alternative major premiss is complex, not compound (that is, 314 SYLLOGISMS. [PART III. 245. The modus ponendo tollens. In addition to the modus tollendo ponens, some logicians recognise as valid a modus ponendo tollens, in which the categorical premiss affirms one of the alternants of the alternative premiss, and the conclusion denies the other alternant or alternants. Thus, A is either B or G, AisB, therefore, A is not G. The argument here proceeds on the assumption that the alternants are mutually exclusive ; but this, on the interpreta- tion of alternative propositions adopted in section 140, is not necessarily the case. Hence the recognition or denial of the validity of the modus ponendo tollens in its ordinary form de- pends upon our interpretation of the alternative proposition itself 1 . No doubt exclusiveness is often intended to be implied and is understood to be implied. For example, "He was either first or second in the race, He was second, therefore, He was. not first." This reasoning would ordinarily be accepted as valid. But its validity really depends not on the expressed major premiss, but on the understood premiss, " No one can be both first and second in a race." The following reasoning is in fact equally valid with the one stated above, "He was second in the race, therefore, He was not first." The alternative premiss is, therefore, quite immaterial to the reasoning ; we could do- just as well without it, for the really vital premiss, " No one can be both first and second in a race," is true, and would be accepted as such, quite irrespective of the truth of the alterna- tive proposition, "He was either first or second." In other cases the mutual exclusiveness of the alternants may be tacitly equivalent to a conditional, not to a true hypothetical) as in the second of the above examples the syllogism may of course be reduced to pure categorical form. Thus, Every A which is not B is G, This A is an A which is not B, therefore, This A is C. 1 It will be observed that, interpreting the alternants as not necessarily exclusive of one another, the modus ponendo tollens in the above form is equiva- lent to one of the fallacies in the hypothetico-categorical syllogism mentioned in section 235. CHAP. VI.] THE DISJUNCTIVE SYLLOGISM. 315 understood, although not obvious d priori as in the above example. But in no case can a special implication of this kind be recognised when we are dealing with purely symbolic forms. If we hold that the modus ponendo tollens as above stated is formally valid, we must be prepared to interpret the alternants as in every case mutually exclusive. If, however, we take a major premiss which is not alternative at all, but disjunctive, in the stricter sense explained in section 138, then we may have a formally valid reasoning which has every right to be described as a modus ponendo tollens. Thus, P and Q are not both true ; but P is true ; therefore, Q is not true 1 . The following table of the ponendo ponens, &c., in their valid and invalid forms may be useful : Valid Invalid Ponendo Pon&ns If P then Q, but P, .'.Q. If P then Q, but Q, .-. P. Tollendo Tollens If Q then P, but not P, . '. not Q. If Q then P, but not Q, .-. not P. Tollendo Ponens Either P or Q, but not P, .-.Q. Not both P and Q, but not Q, .'. P. Ponendo Tollens Not both P and Q, butP, . . not Q. Either P or Q, but Q, .'. not P. The above valid forms are mutually reducible to one another, and the same is true of the invalid forms. 1 This is in the stricter sense a disjunctive syllogism, the modus tollendo 316 SYLLOGISMS. [PART III. 246. The Dilemma. The proper place of the dilemma amongst hypothetical and disjunctive arguments is difficult to determine, inasmuch as conflicting definitions are given by different logicians. The following definition may be taken as perhaps on the whole the most satisfactory: A dilemma is a formal argument containing a premiss in which two or more hypothetical are conjunctively affirmed, and a second premiss in which the antecedents of these hypothetical are alternatively affirmed or their consequents alternatively denied 1 . These premisses are usually called the major and the minor respec- tively 2 . Dilemmas are called constructive or destructive according as the minor premiss alternatively affirms the antecedents, or denies the consequents, of the major 8 . Since it is a distinguishing characteristic of the dilemma that the minor should be alternative, it follows that the hypotheticals into which the major premiss of a constructive dilemma may be resolved must contain at least two distinct antecedents. They may, however, have a common consequent. The conclusion of the dilemma will then categorically affirm ponens being an alternative syllogism. The reader must, however, be careful to remember that the latter is what is ordinarily meant by the disjunctive syllogism in logical text-books. 1 In the strict use of the term, a dilemma implies only two alternants in the alternative premiss ; if there are more than two alternants we have a trilemma, or a tetralemma, or a polylemma, as the case may be. 2 This application of the terms major and minor is somewhat arbitrary. The dilemmatic force of the argument is indeed made more apparent by stating the alternative premiss (i.e., the so-called minor premiss) first. It will, however, be remembered that, in the view of some logicians, the force of the ordinary cate- gorical syllogism also is made more apparent by stating the minor premiss first. Compare section 147. 3 A further form of argument may be distinguished in which the alternation contained in the so-called minor premiss is affirmed only hypothetically, and in which, therefore, the conclusion also is hypothetical. For example, If A is B, E is F; and if C is D, E is F; If X is Y, either A is B or C is D ; therefore, If X is Y, E is F. This might be called the hypothetical dilemma. It admits of varieties corre- sponding to the varieties of the ordinary dilemma ; but no detailed treatment of it seems called for. CHAP. VI.] THE DILEMMA. 317 this consequent, and will correspond with it in form 1 . The dilemma itself is in this case called simple. If, on the other hand, the major premiss contains more than one consequent, the conclusion will necessarily be alternative, and the dilemma is called complex. Similarly, in a destructive dilemma the hypothetical into which the major can be resolved must have more than one consequent, but they may or may not have a common ante- cedent; and the dilemma will be simple or complex accordingly. We have then four forms of dilemma as follows : (i) The simple constructive dilemma. If A is B, E is F-, and if C is D, E is F; but Either A is B or G is D; therefore, E is F. (ii) The complex constructive dilemma. If A is B, E is F; and if C is D, G is H; but Either A is B or C is D; therefore, Either E is F or G is H*. (iii) The simple destructive dilemma. If A is B, C is D ; and if A is B,EisF; but Either C is not D or E is not F] therefore, A is not B. (iv) The complex destructive dilemma. IfAisB.Eis F; and if C is D, G is E\ but Either E is not F or G is not H ; therefore, Either A is not B or C is not D 3 . 1 It will usually be a simple categorical ; but see the following note. * The following is a simple, not a complex, constructive dilemma : IfAisB,EisForGisH; and if C is D, E is F or G is H; but Either A is B or C i s D ; therefore, Either E is F or G is H. The hypothetical which here constitute the major premiss have a common consequent ; but since this is itself alternative, the conclusion appears in the alternative form. This case is analogous to the following All M is P or Q, All S is M, therefore, All S is P or Q where the conclusion of an intrinsically categorical syllogism also appears in the alternative form. Compare the note on p. 312. 3 The following is a simple, not a complex, destructive dilemma : 318 SYLLOGISMS. [PART III. In the case of dilemmas, as in the case of hypothetico- categorical syllogisms, the constructive form may be reduced to the destructive form, and vice versa. All that has to be done is to contraposit the hypotheticals which constitute the major premiss. One example will suffice. Taking the simple con- structive dilemma above given, and contrapositing the major, we have If E is not F, A is not B ; and if E is not F, C is not D \ but Either A is B or C is D; therefore, E is F; and this is a dilemma in the simple destructive form. The definition of the dilemma above given is practically identical with that given by Dr Fowler (Deductive Logic, p. 116). Mansel (Aldrich, p. 108) defines the dilemma as "a syllogism having a conditional (hypothetical) major premiss with more than one antecedent, and a disjunctive minor." Equi- valent definitions are given by Whately and Jevons. According to this view, while the constructive dilemma may be either simple or complex, the destructive dilemma must always be complex, since in the corresponding simple form (as in the example given on p. 317) there is only one antecedent in the major. This exclusion seems arbitrary and is a ground for rejecting the definition in question. Whately, indeed, regards the name dilemma as necessarily implying two antecedents ; but it should rather be regarded as implying two alternatives, each of which is equally distasteful. Whately goes on to assert that the excluded form is merely a destructive hypothetical syllogism, similar to the following, If A isB,CisD; C is not D; therefore, A is not B. But the two really differ precisely as the simple constructive If both P and Q are true then X is true, and under the same hypothesis Y is true; but Either X or Y is not true ; therefore, Either P or Q is not true. Here we have but a single antecedent. For If both P and Q are true then X is tme cannot be resolved into two distinct hypotheticals. CHAP. VI.] THE DILEMMA. 319 dilemma given on p. 317, differs from the constructive hypo- thetical syllogism, If A is B, E is F; AisB; therefore, E is F. Besides, it is clear that the form under discussion is not merely a destructive hypothetical syllogism such as has been already discussed, since the premiss which is combined with the hypo- thetical premiss is not categorical but alternative \ The following definition is sometimes given : "The dilemma (or trilemma or polylemma) is an argument in which a choice is allowed between two (or three or more) alternatives, but it is shewn that whichever alternative is taken the same conclusion follows." This definition, which no doubt gives point to the expression " the horns of a dilemma," includes the simple con- structive dilemma and the simple destructive dilemma ; but it does not allow that either of the complex dilemmas is properly 1 The argument IfAisB t CisD and E is F; but Either C is not D or E is not F; therefore, A is not B ; must be distinguished from the following IfAisBjC is D and E is F; but C is not D and E is not F; therefore, A is not B. The former is a simple destructive dilemma, but in the latter no alternative is given at all, and the reasoning is equivalent to two simple hypothetical syllo- gisms, yielding the same conclusion, namely, (1) If A isB, CisD; but C is not D ; therefore, A is not B. (2) IfAisB,EisF; but E is not F; therefore, A is not B. Similarly, the simple constructive dilemma given on p. 317 must be distin- guished from the following argument : If A is B t E is F; and if C is D, E is F; but A is B and C is D ; therefore, E is F. Here again we practically have two simple hypothetical syllogisms both yielding the same conclusion. 320 SYLLOGISMS. [PART III. so-called, since in each case we are left with the same number of alternants in the conclusion as are contained in the alter- native premiss. On the other hand, it embraces forms that are excluded by both the preceding definitions ; for example, the following reasoning which should rather be classed simply as a destructive hypothetico-categorical syllogism If A is, either B or C is ; but Neither B nor G is ; therefore, A is not 1 . Jevons (Elements of Logic, p. 168) remarks that " dilem- matic arguments are more often fallacious than not, because it is seldom possible to find instances where two alternatives exhaust all the possible cases, unless indeed one of them be the simple negative of the other." In other words, many dilem- matic arguments will be found to contain a premiss involving a fallacy of incomplete alternation. It should, however, be observed that in strictness a syllogistic argument is not itself to be called fallacious because it contains a false premiss. The fallacy that Jevoris has in view is a material rather than a formal fallacy. 1 Compare Ueberweg, Logic, 123. Hamilton (Logic, i. p. 350) defines the dilemma as "a syllogism in which the sumption (major) is at once hypothetical and disjunctive, and the subsurnp- tion (minor) sublates the whole disjunction, as a consequent, so that the ante- cedent is sublated in the conclusion." This involved definition appears to have chiefly in view the form just given ; but it excludes the following which is one of the typical forms of dilemma according to all the preceding definitions If A is then G is, and if B is then C is ; but either A or B is ; therefore, G is. Thomson (Laws of Thought, p. 203) gives the following, "A dilemma is a syllogism with a conditional (hypothetical) premiss, in which either the antece- dent or the consequent is disjunctive." This definition, however, is probably wider than Thomson himself intended. It would include the following argu- ment If A is B then C is D or E is F, but A is B, therefore, C is D or E is F. CHAP. VI.] EXERCISES. 321 EXERCISES. 247. Express the following argument symbolically, and deter- mine to what type it belongs : The cause must either precede the effect, or be simultaneous with it, or succeed it. The last supposition is absurd ; and the second would render it impossible to distinguish the cause from the effect. On the first supposition the cause must cease before the effect comes into being ; but, surely, that which is not cannot be a cause. Either, then, there is no cause for any effect, or we are unable to discover it. [c.] 248. What can be inferred from the premisses, Either A is B or C is D, Either C is riot D or E is F1 Exhibit the reasoning (a) in the form of a hypothetical syllogism, (b) in the form of a dilemma. [K.] 249. Discuss the logical conclusiveness of fatalistic reasoning like this : If I am fated to be drowned now, there is no use in my struggling ; if not, there is no need of it. But either I am fated to be drowned now or I am not ; so that it is either useless or needless for me to struggle against it. [B.] K. L. 21 CHAPTER VII. IRREGULAR AND COMPOUND SYLLOGISMS. 250. The Enthymeme. By the enthymeme, Aristotle meant what has been called the " rhetorical syllogism " as opposed to the apodeictic, demonstrative, theoretical syllogism. The fol- lowing is from Hansel's notes to Aldrich (pp. 209 to 211): "The enthymeme is defined by Aristotle, i> rj eniftefov. The el/cos and cn^p.elov themselves are propositions; the former stating a general probability, the latter a fact, which is known to be an indication, more or less certain, of the truth of some further statement, whether of a single fact or of a general belief. The former is a proposition nearly, though not quite, universal ; as ' Most men who envy hate': the latter is a singular proposition, which however is not regarded as a sign, except relatively to some other propo- sition, which it is supposed may be inferred from it. The elicof, when employed in an enthymeme, will form the major premiss of a syllogism such as the following : Most men who envy hate, This man envies, therefore, This man (probably) hates. "The reasoning is logically faulty; for, the major premiss not being absolutely universal, the middle term is not dis- tributed. " The o-rj/jielov will form one premiss of a syllogism which may be in any of the three figures, as in the following examples : Figure 1. All ambitious men are liberal, Pittacus is ambitious, therefore, Pittacus is liberal. CHAP. VII.] THE ENTHYMEME. 323 Figure 2. All ambitious men are liberal, Pittacus is liberal, therefore, Pittacus is ambitious. Figure 3. Pittacus is liberal, Pittacus is ambitious, therefore, All ambitious men are liberal. " The syllogism in the first figure alone is logically valid. In the second, there is an undistributed middle term ; in the third, an illicit process of the minor." On this subject the student may be referred to the re- mainder of the note from which the above extract is taken, and to Hamilton, Discussions, pp. 152 to 156 1 . An enthymeme is now usually defined as a syllogism in- completely stated, one of the premisses or the conclusion being understood but not expressed 2 . The arguments of everyday 1 Karslake (Aids to the Study of Logic, Book n.) after mentioning that with Aristotle the conditions of the existence of Demonstration are "that its matter be certain, its method Deduction, and its end scientific certainty" (p. 32), gives the following account of the Aristotelian enthymeme: "Whereas the mathe- matician, e.g., wishing to demonstrate his point, will give all the grounds on which his conclusion rests, and the grounds on which these grounds rest, and so on, till he comes to some primary principles which all will admit ; the rhetorician, on the contrary, will not pursue his proofs in all their ramifications up to their primary source, but will assume a great deal in order to avoid unnecessarily complicating his speech ' What the Example is to Induction, that the Enthymeme is to Deduction' is Aristotle's language always : and since the Example is Induction upon practical matters, the Enthymeme must be Deduction upon practical matters correspondingly. Its matter is practical : its method Deduction : its end persuasion. It is therefore one branch, but the most important branch, of what is called by Aristotle the Dialectical or Topical Syllogism, that is, of reasoning in those cases where the grounds are not certain, but probable only, and where the end sought after is opinion ; and therefore it is pretty much the correlative of Demonstration, extending nearly over the whole sphere of probable, as Demonstration extended over the whole of certain, truth" (pp. 52, 3). 2 This account of the enthymeme appears to have been originally based on the erroneous idea that the name signified the retention of one premiss in the mind, tv 0v/juj>. Thus, in the Port Royal Logic, an enthymeme is described as ' ' a syllogism perfect in the mind, but imperfect in the expression, since some one of the propositions is suppressed as too clear and too well known, and as being easily supplied by the mind of those to whom we speak " (p. 229). As regards the real origin of the name enthymeme, see Mansel's Aldrich, p. 218. 212 324 SYLLOGISMS. [PART III. life are for the most part enthyraematic ; and the same may be said of fallacious arguments, which are seldom completely stated, or their want of cogency would be more quickly re- cognised. An enthymeme is said to be of the first order when the major premiss is suppressed; of the second order when the minor premiss is suppressed ; and of the third order when the conclusion is suppressed. Thus, " Balbus is avaricious, and therefore, he is unhappy," is an enthymeme of the first order; "All avaricious persons are unhappy, and therefore, Balbus is unhappy," is an en- thymeme of the second order; "All avaricious persons are unhappy, and Balbus is avaricious," is an enthymeme of the third order. 251. The Poly syllogism. A chain of syllogisms, that is, a series of syllogisms so linked together that the conclusion of one becomes a premiss of another, is called a poly syllogism. In a polysyllogism, any individual syllogism the conclusion of which becomes the premiss of a succeeding one is called a prosyllogism ; any individual syllogism one of the premisses of which is the conclusion of a preceding syllogism is called an episyllogism. Thus, AllCisD,} All B is C, therefore, All B is D, but All A is B, therefore, All A is D. prosyllogism, episyllogism. The same syllogism may of course be both an episyllogism and a prosyllogism, as would be the case with the above epi- syllogism if the chain were continued further. A chain of reasoning 1 is said to be progressive (or synthetic or episyllogistic) when the progress is from prosyllogism to episyllogism. Here the premisses are first given, and we pass on by successive steps of inference to the conclusions which 1 The distinction which follows is ordinarily applied to chains of reasoning only ; but the reader will observe that it admits of application to the case of the simple syllogism also. CHAP. VII.] THE POLYSYLLOGISM. 325 they yield. A chain of reasoning is, on the other hand, said to be regressive (or analytic or prosyllogistic) when the progress is from episyllogism to prosyllogism. Here the ultimate con- clusion is first given and we pass back by successive steps of proof to the premisses on which it may be based. In the analysis of an argument, it is no doubt important to enquire whether the object had in view is to establish a given thesis or to determine what follows from given premisses. But from the purely formal standpoint, the above distinction resolves itself merely into a difference in order of statement. In the systematic treatment of forms of inference it is con- venient and usual to adopt the progressive order 1 . 252. The Epicheirema. An epicheirema is a polysyllogism with one or more prosyllogisms briefly indicated only. That is, one or more of the syllogisms of which the polysyllogism is composed are enthymematic. The following is an example, All B is D, because it is C, All AisB, therefore, All A is D*. 253. The Sorites. A sorites is a polysyllogism in which all the conclusions are omitted except the final one, the pre- misses being given in such an order that any two successive propositions contain a common term. Two forms of sorites are usually recognised, namely, the so-called Aristotelian sorites and the Godenian sorites. In the former, the premiss stated first contains the subject of the conclusion, while the term 1 On the distinction between progressive and regressive arguments, see Ueberweg, Logic, 124. 2 A distinction has been drawn between single and double epicheiremas according as reasons are enthymematically given in support of one or both of the premisses of the ultimate syllogism. The example given in the text is a single epicheirema ; the following is an example of a double epicheirema : All P is Y, because itisX; All S is P, because allMii P; therefore, All S it Y. The epicheirema is sometimes denned as if it were essentially a regressive chain of reasoning. But this is hardly correct, if, as is usually the case, examples such as the above are given ; for it is clear that in these examples the argument is only partly regressive. 326 SYLLOGISMS. [PART III. common to any two successive premisses occurs first as predicate and then as subject; in the latter, the premiss stated first contains the predicate of the conclusion, while the term common to any two successive premisses occurs first as subject and then as predicate. The following are examples : Aristotelian Sorites All A is B, All B is C, AllC is D, AllD is E, therefore, All A is E. Goclenian Sorites All D is E, All C is D, AllB is C, All A is B, therefore, All A is E. It will be found that, in the case of the Aristotelian sorites, if the argument is drawn out in full, the first premiss and the suppressed conclusions all appear as minor premisses in suc- cessive syllogisms. Thus, the Aristotelian sorites given above may be analysed into the three following syllogisms (1) All B is C, All A is B, therefore, All A is C; (2) AllC is D, All A is C, therefore, All A is D; (3) All D is E, All A is D, therefore, All A is E. Here the premiss originally stated first is the minor premiss of (1), the conclusion of (1) is the minor premiss of (2), that of (2) the minor premiss of (3); and so it would go on if the number of propositions constituting the sorites were increased. In the Goclenian sorites, the premisses are the same, but their order is reversed, and the result of this is that the premiss originally stated first and the suppressed conclusions CHAP. VII.] THE SORITES. 327 become major premisses in successive syllogisms. Thus, the Goclenian sorites given above may be analysed into the three following syllogisms (1) All DisE, AllC is D, therefore, All C is E\ (2) All C is E, All BisC, therefore, All B is E; (3) All BisE, All AisB, therefore, All A is E. Here the premiss originally stated first is the major premiss of (1), the conclusion of (1) is the major premiss of (2); and so on. The so-called Aristotelian sorites 1 is that to which the greater prominence is usually given; but it will be observed that the order of premisses in the Goclenian form is that which really corresponds to the customary order of premisses in a simple syllogism 2 . 1 This form of sorites ought not properly to be called Aristotelian ; but it is generally so described in logical text-books. The name sorites is not to be found in any logical treatise of Aristotle, though in one place he refers vaguely to the form of reasoning which the name is now employed to express. The distinct exposition of this form of reasoning is attributed to the Stoics, and it is called by the name sorites by Cicero ; but it was not till much later that the name came into general use amongst logicians in this sense. The form of sorites called the Goclenian was first given by Professor Rudolf Goclenius of Marburg (1547 to 1628) in his Isagoge in Organum Aristotelis, 1598. Compare Hamilton, Logic, i. p. 375 ; and Ueberweg, Logic, 125. It may be added that the term sorites (which is derived from ] All M 3 is M 2) [therefore, Some S is not M 3 ,] All M t is M 3 , [therefore, Some S is not M t> ] AllP is M t , therefore, Some S is not P. conclusion with its remotest whole, this complex reasoning is called a Chain- Syllogism or Sorites" (Logic, i. p. 366). In connexion with Hamilton's treat- ment of this question, Mill very justly remarks, " If Sir W. Hamilton had found in any other writer such a misuse of logical language as he is here guilty of,, he would have roundly accused him of total ignorance of logical writers " (Examination of Hamilton, p. 515). CHAP. VII.] THE SORITES. 331 This is the only resolution of the sorites possible unless the order of the premisses is transposed, and it will be seen that all the resulting syllogisms are in figure 2 and in the mood Baroco. The sorites may accordingly be said to be in the same mood and figure. It is analogous to the Aristotelian sorites, the subject of the conclusion appearing in the premiss stated first, and the suppressed premisses being all minors in their respective syllogisms. The corresponding analysis of (ii) yields the following : Some M^ is not P, All M t is M 9t [therefore, Some M 3 is not P,] All M 3 is M t , [therefore, Some M 2 is not P,] All M 2 is M lt [therefore, Some M^ is not P,] All M! is S, therefore, Some S is not P. These syllogisms are all in figure 3 and in the mood Bocardo; and the sorites itself may be said to be in the same mood and figure. It is analogous to the Goclenian sorites, the predicate of the conclusion appearing in the premiss stated first, and the suppressed premisses being majors in their respective syllogisms. It will be observed that the rules given in the preceding section have not been satisfied in either of the above sorites, the reason being that the rules in question correspond to the special rules of figure 1, and do not apply unless the sorites is in that figure. For such sorites as are possible in figures 2, 3, and 4, other rules might be framed corresponding to the special rules of these figures in the case of the simple syllogism. It is not maintained that sorites in other figures than the first are likely to be met with in common use, but their con- struction is of some theoretical interest 1 . 1 The examples given in the text have been purposely chosen so as to admit of only one analysis, which was not the case with the examples given in previous editions. The old examples were, however, perfectly valid, and 332 SYLLOGISMS. [PART III. 256. Ultra-total Distribution of the Middle Term. The ordinary syllogistic rule relating to the distribution of the middle term does not contemplate the recognition of any signs of quantity other than all and some ; and if other signs are recognised, the rule must be modified. For example, the admission of the sign most yields the following valid reasoning, although the middle term is not distributed in either of the premisses : Most M is P, Most M is S, therefore, Some 8 is P. Interpreting most in the sense of more than half, it clearly further light may be thrown on the general question by a brief reply to certain criticisms passed upon those examples. The following was given for figure 2 (the suppressed conclusions being inserted in square brackets), and it was said to be analogous to the Aristotelian sorites : All A is B, No C is B, [therefore, No A is C], All D is C, [therefore, No A is D], All E is D, therefore, No A is E. It has, to begin with, been objected that the above is Goclenian, and not Aristotelian, in form, " the subject of each premiss after the first being the predicate of the succeeding one." This overlooks the more fundamental characteristic of the Aristotelian sorites, that the first premiss and the suppressed conclusions are all minors in their respective syllogisms. It has further been objected that the following analysis might serve in lieu of the one above given: AaB, CeB, [.-. CeA,] DaC, [.-. DeA,] EaD, .-. AeE. No doubt this analysis is a possible one, but the objection to it is its heterogeneous character. The first premiss and the first suppressed conclusion are majors, while the last suppressed conclusion is a minor. Again, the first syllogism is in figure 2, the second in figure 1, and the third in figure 4. It must be granted that what has been above called a heterogeneous analysis is in some cases the only one available, but it is better to adopt something more homo- geneous where possible. If the first premiss of a sorites contains the subject, and the last the predicate, of the conclusion, then the last premiss is necessarily the major of the final syllogism; and hence the rule may be laid down that we can work out such a sorites homogeneously only by treating the first premiss and all the suppressed conclusions as minors, and all the remaining premisses as majors, in their respective syllogisms. A corresponding rule may be laid down if the first premiss contains the predicate, and the last the subject, of the conclusion. CHAP. VII.] ULTRA-TOTAL DISTRIBUTION. 333 follows from the above premisses that there must be some M which is both S and P. But we cannot say that in either premiss the term M is distributed. In order to meet cases of this kind, Sir W. Hamilton (Logic, vol. II., p. 362) gives the following modification of the rule relating to the distribution of the middle term : " The quantifications of the middle term, whether as subject or pre- dicate, taken together, must exceed the quantity of that term taken in its whole extent"; in other words, we must have an ultra-total distribution of the middle term in the two premisses taken together. Hamilton then continues somewhat too dog- matically, "The rule of the logicians, that the middle term should be once at least distributed, is untrue. For it is sufficient if, in both the premisses together, its quantification be more than its quantity as a whole (ultra-total). Therefore, a major part (a more or most) in one premiss, and a half in the other, are sufficient to make it effective." De Morgan (Formal Logic, p. 127) writes as follows: "It is said that in every syllogism the middle term must be universal in one of the premisses, in order that we may be sure that the affirmation or denial in the other premiss may be made of some or all of the things about which affirmation or denial has been made in the first. This law, as we shall see, is only a particular case of the truth : it is enough that the two premisses together affirm or deny of more than all the instances of the middle term. If there be a hundred boxes, into which a hundred and one articles of two different kinds are to be put, not more than one of each kind into any one box, some one box, if not more, will have two articles, one of each kind, put into it. The common doctrine has it, that an article of one particular kind must be put into every box, and then some one or more of another kind into one or more of the boxes, before it may be affirmed that one or more of different kinds are found together." De Morgan himself works the question out in detail in his treatment of the numerically definite syllo- gism (Formal Logic, pp. 141 to 170). The following may be taken as an example of numerically definite reasoning: If 70 per cent, of M are P, and 60 per cent are 8, then at least 334 SYLLOGISMS. [PART III. 30 per cent, are both 8 and P\ The argument may be put as follows : On the average, of 100 M's 70 are P and 60 are 8 ; suppose that the 30 M's which are not P are 8, still 30 S's are to be found in the remaining 70 M's which are P's; and this is the desired conclusion. Problems of this kind constitute a borderland between formal logic and algebra. Some further examples will be given in chapter 9 (section 293). 257. The Quantification of the Predicate and the Syllo- gism. It will be convenient to consider briefly in this chapter the application of the doctrine of the quantification of the predicate to the syllogism ; the result is the reverse of simpli- fication 2 . The most important points that arise maybe brought out by considering the validity of the following syllogisms : in figure 1, UUU, lUt,, AYI ; in figure 2, r,UO, AUA; in figure 3, YAI. In the next section we will proceed more systematically, U and being left out of account. (1) UUU in figure 1 is valid: 1 Using other letters, this is the example given by Mill, Logic, book 2, chapter 2, 1, note, and quoted by Herbert Spencer, Principles of Psychology, vol. 2, p. 88. The more general problem of which the above is a special instance is as follows : Given that there are n M's in existence, and that a M's are S while b M's are P, to determine what is the least number of S's that are also P's. It is clear that we have no conclusion at all unless a + b>n, i.e., unless there is ultra-total distribution of the middle term. If this condition is satisfied, then supposing the (n - b) M's which are not-P are all of them found amongst the M S's, there will still be some MS's left which are P's, namely, a - (n - b). Hence the least number of S's that are also P's must be a + b - n. 2 In connexion with his doctrine of the quantification of the predicate, Hamilton distinguishes between the figured syllogism and the unfigured syllogism. In the figured syllogism, the distinction between subject and predicate is retained, as in the text. By a rigid quantification of the predicate, however, the distinction between subject and predicate may be dispensed with ; and such being the case there is no ground left for distinction of figure (which depends upon the position of the middle term as subject or predicate in the premisses). This gives what Hamilton calls the unfigured syllogism. For example : Any bashfulness and any praiseworthy are not equivalent, All modesty and some praiseworthy are equivalent, therefore, Any bashfulness and any modesty are not equivalent ; All whales and some mammals are equal, All whales and some water animals are equal, therefore, Some mammals and some water animals are equal. A distinct canon for the unfigured syllogism is given by Hamilton as follows : " In as far as two notions either both agree, or one agreeing the other does not, with a common third notion ; in so far these notions do or do not agree with each other." CHAP. VII.] QUANTIFICATION OF THE PREDICATE. 335 All Mis all P, AllS is all M, therefore, All S is all P. It will be observed that whenever one of the premisses is U, the conclusion may be obtained by substituting S or P (as the case may be) for M in the other premiss. Without the use of quantified predicates, the above reasoning may be expressed by means of the two following syllogisms : AllMisP, AllMisS, All SisM, All Pis M, therefore, All S is P; therefore, All P is S. (2) IUi] in figure 1 is invalid, if some is used in its ordinary logical sense. The premisses are Some M is some P and All S is all M. We may, therefore, obtain the legitimate conclusion by substituting S for M in the major premiss. This yields Some S is some P. If, however, some is here used in the sense of some only, No S is some P follows from Some S is some P, and the original syllogism is valid, although a negative conclusion is obtained from two affirmative premisses. This syllogism is given as valid by Thomson (Laws of Thought, 103) ; but apparently only through a misprint for IEr]. In his scheme of valid syllogisms (thirty-six in each figure) Thomson seems consistently to interpret some in its ordinary logical sense. Using the word in the sense of some only, several other syllogisms would be valid that he does not give as such 1 . (3) AYI in figure 1, some being used in its ordinary logical sense, is equivalent to AAI in figure 3 in the ordinary syllogistic scheme, and it is therefore valid. But it is invalid if some is used in the sense of some only, for the conclusion now implies that S and P are partially excluded from each other as well as partially coincident, whereas this is not implied by the premisses. With this use of some, the correct conclusion can be expressed only by stating an alternative between SuP, SaP, SyP, and SiP. This case may serve to illustrate the 1 Compare section 106. 336 SYLLOGISMS. [PART III. complexities in which we should be involved if we were to attempt to use some consistently in the sense of some only 1 . (4) t]U O i n fig ure 2 is valid : No P is some M, All S is all M, therefore, Some 8 is not any P. Without the use of quantified predicates, we can obtain an equivalent argument in Bocardo, thus Some M is not P, AllMisS, therefore, Some S is not P. (5) AU A in figure 2 runs as follows : All P is some M, All Sis all M, therefore, All S is some P. Here we have neither undistributed middle nor illicit process of major or minor, nor is any rule of quality broken, and yet the syllogism is invalid 2 . Applying the rule given above that "whenever one of the premisses is U, the conclu- sion may be obtained by substituting S or P (as the case may be) for M in the other premiss," we find that the valid con- clusion is Some S is all P. More generally, it follows from this rule of substitution that if one premiss is U while in the other premiss the middle term is undistributed, then the term combined with the middle term in the U premiss must be undis- tributed in the conclusion. This appears to be the one addi- tional syllogistic rule required if we recognise U propositions in syllogistic reasonings. All danger of fallacy is avoided by breaking up the U pro- position into two A propositions. In the case before us we have All P is M, All M is S; All P is M, All S is M. From the first of these pairs of premisses we get the con- 1 Compare Monck, Logic, p. 154. 2 We should have a corresponding case if we were to infer No S is P from the premisses given in the preceding example. CHAP. VII.] QUANTIFICATION OF THE PREDICATE. 337 elusion All P is S; in the second pair the middle term is undistributed, and therefore no conclusion is yielded at all. (6) YAI in figure 3 is valid : Some M is all P, All M is some S, therefore, Some S is some P. The conclusion is however weakened, since from the given premisses we might infer Some S is all P 1 . It will be observed that when we quantify the predicate, the conclusion of a syllogism may be weakened in respect of its predicate as well as in respect of its subject. In the ordinary doctrine of the syllogism this is for obvious reasons not possible. Without quantification of the predicate the above reasoning may be expressed in Bramantip, thus, AllPisM, All MisS, therefore, Some S is P. We could get the full conclusion, All P is S, in Barbara. 258. Table of valid moods resulting from the recognition of Y and i\ in addition to A, E, I, O. If we adopt the sixfold schedule of propositions obtained by adding Only S is P (Y) and Not only S is P (r\) to the ordinary fourfold schedule, as in section 112, every proposition is simply convertible, and, there- fore, a valid mood in any figure is reducible to any other figure by the simple conversion of one or both of the premisses. Hence if the valid moods of any one figure are determined, those of the remaining figures may be immediately deduced therefrom. It will be found that in each figure there are twelve valid moods, which are neither strengthened nor weakened. This result may be established by either of the two alternative methods which follow. I. We may enquire what combinations of premisses will suffice to establish conclusions of the forms A, Y, E, I, O, tj, respectively. 1 Or, retaining the original conclusion, we might replace the major premiss by Some M is some P ; hence, from another point of view, the syllogism may be regarded as strengthened. K. L. 22 338 SYLLOGISMS. [PART III. It will suffice, as we have already seen, to consider some one figure. We may, therefore, take figure 1, so that the position of the terms will be M P 8 M 8 P (i) To prove SaP, both premisses must be affirmative ; and, in order to avoid illicit minor, the minor premiss must be SaM. It follows that the major must be MaP or there would be undistributed middle. Hence AAA is the only valid mood yielding an A conclusion. (ii) To prove SyP, both premisses must be affirmative ; and, in order to avoid illicit major, the major premiss must be MyP. It follows that the minor must be SyM, in order to avoid undistributed middle. Hence YYY is the only valid mood yielding a Y conclusion. (iii) To prove SeP, the major must be (1) MeP or (2) MyP or (3) MoP in order to avoid illicit major. If (1), the minor must be SaM or there would be either two negative premisses or illicit minor; if (2), it must be SeM or there would be undis- tributed middle or illicit minor ; if (3), it must be affirmative and distribute both S and M, which is impossible. Hence EAE and YEE are the only valid moods yielding an E con- clusion. (iv) To prove SiP, both premisses must be affirmative, and since SaM would necessarily be a strengthened premiss, the minor must be (1) SiM or (2) SyM. If (1), the major must be MaP or there would be undistributed middle ; and if (2), it must be MiP or there would be a strengthened premiss. Hence All and IYI are the only valid moods yielding an I conclusion. (v) To prove SoP, the major must be (1) MeP or (2) MyP or (3) MoP or there would be illicit major. If (1), the minor must be SiM or there would be a strengthened premiss ; if (2), it must be SoM or there would be either two affirmative pre- misses with a negative conclusion or undistributed middle or a strengthened premiss ; and if (3), it must be SyM or there would be two negative premisses or undistributed middle. CHAP. VII.] QUANTIFICATION OF THE PREDICATE. 339 Hence EIO, YOO, OYO are the only valid moods yielding an O conclusion. (vi) To prove SrjP, the minor must be (1) SeM or (2) SaM or (3) SrjM or there would be illicit minor. If (1), the major must be MiP or there would be a strengthened premiss ; if (2), the major must be MrjP or there would be undistributed middle or two affirmative premisses with a negative conclusion or a strengthened premiss ; and if (3), the major must be MaP or there would be undistributed middle or two negative premisses. Hence IEtj, i\Ai\, Arp] are the only valid moods yielding an rj conclusion. By converting one or both of the premisses we may at once deduce from the above a table of valid (unstrengthened and unweakened) moods for all four figures as follows : Fig. 1. Fig. 2. Fig. 3. Fig. 4. AAA YAA ATA TTA TTT ATT TAT A AT EAE EAE EYE EYE TEE AEE TEE AEE All Til All Til IYI IYI IAI IAI EIO EIO EIO EIO YOO AOO YrjO ArjO OYO rjYO OAO r,AO T f T" 27* T Z7* T IT* JLJ^rj J.J1/T] iHit] iJiif) OAij yYrj OYij Tim AOr) TO*) II. The above table may also be obtained by (1) taking all the combinations of premisses that are d priori possible, (2) establishing special rules for the particular figure selected, which (taken together with the rules of quality) will enable us to exclude the combinations of premisses which are either invalid or strengthened whatever the conclusion may be, (3) assigning the valid unweakened conclusion in the remaining cases. The following are all possible combinations of premisses, valid and invalid : 222 340 SYLLOGISMS. [PART III. AA (6) YA IA EA (6) OA ^A (6) (c) AY YY(a) IY(a) EY OY (a) i,Y AI YI (a) II (a) El OI (a) t,I (c) AE(6) YE IE [EE](6) [OE] [r,E] (6) AO Y0(a) 10 (a) [EO] [OO](a) [,O] A,(&)(c) Yi, Me) [E,](&) [O,,] W(6)(c) The combinations in square brackets are excluded by the rule that from two negative premisses nothing follows. Taking the third figure, in which the middle term is subject in each premiss, and remembering that the subject is distri- buted in A, E, T] and in these only, while the predicate is distributed in Y, E, O and in these only, the following special rules are obtainable : (a) One premiss must be A, E, or TJ, or the middle term would not be distributed in either premiss ; (6) One premiss must be Y, I, or O, or the middle term would be distributed in both premisses, and there would hence be a strengthened premiss ; (c) If either premiss is negative, one of the premisses must be Y, E, or O, for otherwise (since the conclusion must be negative, distributing one of its terms) there would be illicit process either of major or minor. These rules exclude the combinations of premisses marked respectively (a), (6), (c) above. Assigning the valid unweakened conclusion in the case of each of the twelve combinations which remain, we have the following: AY A, All, AOt,, YAY, YEE, Y^O, IAI, IE,,, EYE, EIO, OAO, ,,YT|. From this, the table of valid (unstrengthened and unweakened) moods for all four figures may be expanded as before \ 1 By adopting a special set of symbols, suggested by Mr Johnson, the solu- tion of the problem discussed in the above section may be generalised, so as to be quite independent of figure. Writing capitals for distributed terms, small letters for undistributed terms ; + for affirmation, - for negation ; and attach- ing to each proposition a number, determined by counting 4 for affirmation, 3 for distribution of the middle term, 2 for distribution of an extreme term ; the six possible major premisses and the six possible minor premisses may be written as follows in descending order of probative strength : CHAP. VII.] QUANTIFICATION OF THE PREDICATE. 341 259. Formal Inferences not reducible to ordinary Syllo- gisms 1 . The following is an example of what is usually called 7 Mp+ 6 mP+ mS+ 5 MP- MS- _J_ m&+ 3 Mp~ Ms~ mS~ The conditions are that in the premisses there shall be one and only one extra distribution (which must include the distribution of the middle term), and one and only one extra affirmation, as compared with the conclusion. Hence the following rule for combining the premisses may be deduced : The sum of the premisses must be an odd number, not less than nine. For the sum of the premisses cannot be even, since the middle term would then be either twice distributed or twice undistributed; the combinations 5 + 2, 3 + 2, 2 + 5, 2 + 3 have to be rejected, because of double negatives ; and the combinations 4 + 3, 3 + 4, because s and p being both undistributed, a negative conclusion is inad- missible. The following combinations then remain valid: 7 + 6; 7 + 4; 7 + 2; 6 + 7; 6 + 5; 6 + 3; 5 + 6; 5 + 4; 4 + 7; 4 + 5; 3 + 6; 2 + 7; and the conclusion may in each case be obtained by multiplying the signs, and dropping M and m. These results are shewn in the following diagram : 1 Mp+ 6 mP+ 5 MP- 4 mp + 3 Mp~ 2 mP~ 7 Ms+ 8P+ sp+ sP~ 6 mS + Sp + SP- s p - 5 MS- SP- Sp~ 4 ms+ sp + sP~ 3 Ms~ sP~ 2 mS~ Sp~ There are thus twelve moods satisfying the required conditions in each figure. For example, 7 + 6 yields in the four figures respectively, AAA, YAA, AYA, YYA. 1 Attempts to reduce immediate inferences to syllogistic form have been already considered in section 76. In the present section, non-syllogistic mediate inferences will be considered. 342 SYLLOGISMS. [PART III. the argument a fortiori : B is greater than C, A is greater than B, therefore, A is greater than C. As this stands, it is clearly not in the ordinary syllogistic form since it contains four terms; some logicians, however, profess to reduce it to the ordinary syllogistic form as follows : B is greater than G, therefore, Whatever is greater than B is greater than C, but A is greater than B, therefore, A is greater than C. With De Morgan, we may treat this as a mere evasion, or as a petitio principii. The principle of the argument a fortiori is really assumed in passing from B is greater than C to What- ever is greater than B is greater than G. The following attempted resolution 1 must be disposed of similarly : Whatever is greater than a greater than C is greater than G, A is greater than a greater than C, therefore, A is greater than C. At any rate, it is clear that this cannot be the whole of the reasoning, since B no longer appears in the premisses at all. The point at issue may perhaps be most clearly indicated by saying that whilst the ordinary syllogism may be based upon the dictum de omni et nullo, the argument a fortiori cannot be made to rest entirely upon this axiom. A new principle is required and one which must be placed on a par with the dictum de omni et nullo, not in subordination to it. This new principle may be expressed in the form, Whatever is greater than a second thing which is greater than a third thing is itself greater than that third thing. Mansel (Aldrich,pp. 199, 200) treats the argument a, fortiori as an example of a material consequence on the ground that it depends upon "some understood proposition or propositions, connecting the terms, by the addition of which the mind is enabled to reduce the consequence to logical form." He would 1 Compare MansePs Aldrich, p. 200. CHAP. VII.] ARGUMENT A FORTIORI. 343 effect the reduction in one of the ways already referred to. This, however, begs the question that the syllogistic is the only logical form. As a matter of fact the cogency of the argument a fortiori is just as intuitively evident as that of a syllogism in Barbara itself. Why should no relation be regarded as formal unless it can be expressed by the word is ? Touching on this case, De Morgan remarks that the formal logician has a right to confine himself to any part of his subject that he pleases; "but he has no right except the right of fallacy to call that part the whole " (Syllabus, p. 42). There are an indefinite number of other arguments which for similar reasons cannot be reduced to syllogistic form. For example, A equals B, B equals C, therefore, A equals C 1 ; X is a contemporary of F, and T of Z, therefore, X is a con- temporary of Z ; A is the brother of B, B is the brother of C, therefore, A is the brother of C ; A is to the right of B, B is to the right of C, therefore, A is to the right of C; A is in tune with B, and B with C, therefore, A is in tune with C. All these arguments depend upon principles which may be placed on a par with the dictum de omni et nullo, and which are equally axiomatic in the particular systems to which they belong. The claims that have been put forward on behalf of the syllogism as the exclusive form of all deductive reasoning must accordingly be rejected. 1 In regard to this argument De Morgan writes, " This is not an instance of common syllogism : the premisses are ' A is an equal of B ; B is an equal of C.' So far as common syllogism is concerned, that ' an equal of B ' is as good for the argument as 'B ' is a material accident of the meaning of ' equal.' The logicians accordingly, to reduce this to a common syllogism, state the effect of composition of relation in a major premiss, and declare that the case before them is an example of that composition in a minor premiss. As in, A is an equal of an equal (of C) ; Every equal of an equal is an equal ; therefore, A is an equal of C. This I treat as a mere evasion. Among various sufficient answers this one is enough: men do not think as above. When A=B, B = C, is made to give A = C, the word equals is a copula in thought, and not a notion attached to a predicate. There are processes which are not those of common syllogism in the logician's major premiss above : but waiving this, logic is an analysis of the form of thought, possible and actual, and the logician has no right to declare that other than the actual is actual" (Syllabus, pp. 31, 2). 344 SYLLOGISMS. [PART III. Such t claims have been made, for example, by Whately. Syllogism, he says, is " the form to which all correct reasoning may be ultimately reduced" (Logic, p. 12). Again, he remarks, "An argument thus stated regularly and at full length, is called a Syllogism ; which, therefore, is evidently not a peculiar kind of argument, but only a peculiar form of expression, in which every argument may be stated" (Logic, p. 26) 1 . Spalding seems to have the same thing in view when he says, " An inference, whose antecedent is constituted by more propositions than one, is a mediate inference. The simplest case, that in which the antecedent propositions are two, is the syllogism. The syllogism is the norm of all inferences whose antecedent is more complex ; and all such inferences may, by those who think it worth while, be resolved into a series of syllogisms" (Logic, p. 158). J. S. Mill endorses these claims. " All valid ratiocination," he observes, "all reasoning by which from general propositions previously admitted, other propositions equally or less general are inferred, may be exhibited in some of the above forms," i.e., the syllogistic moods (Logic, I. p. 191). What is required to fill the logical gap which is created by the admission that the syllogism is not the norm of all valid formal inference has been called the logic of relatives 2 . The function of the logic of relatives is to "take account of relations generally, instead of those merely which are indicated by the ordinary logical copula is" (Venn, Symbolic Logic, p. 400). The line which this branch of logic may take, if it is ever fully worked out, is indicated by the following passage from De Morgan (Syllabus, pp. 30, 31): "A convertible copula is one in which the copular relation exists between two names both 1 Compare also Whately, Logic, pp. 24, 5, and p. 34. Professor Bay expresses himself equally strongly. " The syllogism," he remarks, " is the type of all valid reasoning ; for no reasoning will be valid, unless it can be thrown into the form of a syllogism. As a matter of fact, in daily life, men draw inferences in many different ways, but only those among them will be valid, and properly deserving of the name, which are capable of being ultimately reduced to the syllogistic form, the rest being nothing but suggestions of association, fancy, imagination, &c., wrongly called inferences" (Deductive Logic, p. 255). 2 Compare pp. 118, 119. CHAP. VII.] LOGIC OF RELATIVES. 345 ways : thus ' is fastened to,' ' is joined by a road with,' 'is equal to,' &c. are convertible copulse. If ' X is equal to T' then ' Y is equal to X,' &c. A transitive copula is one in which the copular relation joins X with Z whenever it joins .X" with Y and Y with Z. Thus ' is fastened to ' is usually understood as a transitive copula: ' X is fastened to Y' and ' Fis fastened to Z' give 'X is fastened to Z' " The student may further be referred to Venn, Symbolic Logic, pp. 399 to 404 ; and also to Mr Johnson's articles on the Logical Calculus in Mind, 1892, especially pp. 26 to 28 and 244 to 250. EXERCISES. 260. Take any enthymeme (in the modern sense) and supply premisses so as to expand it into (a) a syllogism, (b) an epicheirema, (c) a sorites ; and name the mood, order, or variety of each product. [*] 261. Shew that if either of two given propositions will suffice to expand a given enthymeme of the first or second order into a valid syllogism, then the two propositions will be equivalent to each other, provided that neither of them constitutes a strengthened premiss. [j.] 262. Discuss the character of the following sorites, in each case indicating how far more than one analysis is possible : (i) Some D is E, All D is C, All C is B, All B is A, therefore, Some A is E; (ii) Some A is , No C is B, All D is C, All E is D, therefore, Some A is not E; (iii) All E is D, All DisC, All C is B, All B is A, therefore, Some A is E ] (iv) No D is E, Some D is C, All C is B, All is A, therefore, Some A is not E. [K.] 263. Examine the validity of the following moods : In figure 1, UAU, YOO, EYO ; In figure 2, A A A, AYY, UO; In figure 3, YEE, OYO, AO. [c.] 264. Enquire in what figures, if any, the following moods are valid, noting cases in which the conclusion is weakened : AUI ; YAY; UOr,; lUij; UEO. [L.] 265. Is it possible that there should be three propositions such that each in turn is deducible from the other two? [v.] 266. Determine special rules for figures 1, 2, and 4, correspond- ing to the special rules for figure 3 given in section 258. [K.] CHAPTER VIII. EXAMPLES OF ARGUMENTS AND FALLACIES. 267. In how many different moods may the argument implied in the following proposition be stated ? "No one can maintain that all persecution is justifiable who admits that persecution is sometimes ineffective." How would the formal correctness of the reasoning be affected by reading "deny" for "maintain" 1 ? [v.] 268. What conclusions (if any) can be drawn from each pair of the following sentences taken two and two together ? (1) None but gentlemen are members of the club ; (2) Some members of the club are not officers ; (3) All members of the club are invited to compete ; (4) All officers are invited to compete. Point out the mood and figure in each case in which you make a valid syllogism ; and state your reasons when you consider that no valid syllogism is possible. [v.] 269. No one can maintain that all republics secure good govern- ment who bears in mind that good government is inconsistent with a licentious press. What premisses must be supplied to express the above reasoning in Ferio, Festino, and Ferison respectively? [v.] 270. Write the following arguments in syllogistic form, and reduce them to the first figure : (a) Falkland was a royalist and a patriot; therefore, some royalists were patriots. ((3) All who are punished should be responsible for their actions; therefore, if some lunatics are not responsible for their actions, they should not be punished. CHAP. VIIL] EXERCISES. 347 (y) All who have passed the Little-Go have a knowledge of Greek ; hence A.B. cannot have passed the Little-Go, for he has no knowledge of Greek. [K.] 271. " It is impossible to maintain that the virtuous alone are happy, and at the same time that selfishness is compatible with happiness but incompatible with virtue." State the above argument syllogistically in as many different moods as possible. [j.] 272. Give the technical name of the following argument : Payment by results sounds extremely promising ; but payment by results necessarily means payment for a minimum of knowledge; payment for a minimum of knowledge means teaching in view of a minimum of knowledge ; teaching in view of a minimum of know- ledge means bad teaching. [K.] 273. From P follows Q ; and from R follows S ; but Q and S cannot both be true ; shew that P and R cannot both be true. [De Morgan.] 274. Every English peer is entitled to sit in the House of Lords, and every member of the House of Commons must be elected to Parliament by a constituency; but no one entitled to a seat in the House of Lords is thus elected to Parliament. What can we conclude from these premisses about an English peer? [M.] 275. If (1) it is false that whenever X is found T is found with it, and (2) not less untrue that X is sometimes found without the accompaniment of Z, are you justified in denying that (3) when- ever Z is found there also you may be sure of finding Y\ And however this may be, can you in the same circumstances judge anything about Y in terms of Z\ [R.] 276. Can the following arguments be reduced to syllogistic form? (1) The sun is a thing insensible ; The Persians worship the sun ; Therefore, the Persians worship a thing insensible. (2) The Divine law commands us to honour kings; Louis xiv. is a king ; Therefore, the Divine law commands us to honour Louis xiv. [Port Royal Logic.] 348 SYLLOGISMS. [PART III. 277. Examine the following arguments ; where they are valid, reduce them if you can to syllogistic form; and where they are invalid, explain the nature of the fallacy : (1) We ought to believe the Scripture ; Tradition is not Scripture; Therefore, we ought not to believe tradition. (2) Every good pastor is ready to give his life for his sheep ; Now, there are few pastors in the present day who are ready to give their lives for their sheep ; Therefore, there are in the present day few good pastors. (3) Those only who are friends of God are happy; Now, there are rich men who are not friends of God ; Therefore, there are rich men who are not happy. (4) The duty of a Christian is not to praise those who commit criminal actions ; Now, those who engage in a duel commit a criminal action ; Therefore, it is the duty of a Christian not to praise those who engage in duels. (5) The gospel promises salvation to Christians ; Some wicked men are Christians ; Therefore, the gospel promises salvation to wicked men. (6) He who says that you are an animal speaks truly ; He who says that you are a goose says that you are an animal ; Therefore, he who says that you are a goose speaks truly. (7) You are not what I am ; I am a man ; Therefore, you are not a man. (8) We can only be happy in this world by abandoning our- selves to our passions, or by combating them ; If we abandon ourselves to them, this is an unhappy state, since it is disgraceful, and we could never be content with it ; If we combat them, this is also an unhappy state, since there is nothing more painful than that inward war which we are continually obliged to carry on with ourselves ; Therefore, we cannot have in this life true happiness. (9) Either our soul perishes with the body, and thus, having no feelings, we shall be incapable of any evil ; or if the soul survives the body, it will be more happy than it was in the body; Therefore, death is not to be feared. [Port Royal Logic.] CHAP. VIII.] EXERCISES. 349 278. Examine the following arguments : (1) "He that is of God heareth my words: ye therefore hear them not, because ye are not of God." (2) All the fish that the net inclosed were an indiscriminate mixture of various kinds : those that were set aside and saved as valuable, were fish that the net inclosed : therefore, those that were set aside and saved as valuable, were an indiscriminate mixture of various kinds. (3) Testimony is a kind of evidence which is very likely to be false : the evidence on which most men believe that there are pyramids in Egypt is testimony : therefore, the evidence on which most men believe that there are pyramids in Egypt is very likely to be false. (4) If Paley's system is to be received, one who has no know- ledge of a future state has no means of distinguishing virtue and vice : now one who has no means of distinguishing virtue and vice can commit no sin : therefore, if Paley's system is to be received, one who has no knowledge of a future state can commit no sin. (5) If Abraham were justified, it must have been either by faith or by works : now he was not justified by faith (according to James), nor by works (according to Paul) : therefore, Abraham was not justified. (6) For those who are bent on cultivating their minds by diligent study, the incitement of academical honours is unnecessary ; and it is ineffectual, for the idle, and such as are indifferent to mental improvement : therefore, the incitement of academical honours is either unnecessary or ineffectual. (7) He who is most hungry eats most ; he who eats least is most hungry : therefore, he who eats least eats most. (8) A monopoly of the sugar-refining business is beneficial to sugar-refiners : and of the corn-trade to corn-growers : and of the silk-manufacture to silk- weavers, &c., &c. ; and thus each class of men are benefited by some restrictions. Now all these classes of men make up the whole community : therefore, a system of restrictions is beneficial to the community. [Whately, Logic.] 279. The following are a few examples in which the reader can try his skill in detecting fallacies, determining the peculiar form of syllogisms, and supplying the suppressed premisses of enthymemes. 350 SYLLOGISMS. [PART III. (1) None but those who are contented with their lot in life can justly be considered happy. But the truly wise man will always make himself contented with his lot in life, and, therefore, he may justly be considered happy. (2) All intelligible propositions must be either true or false. The two propositions "Csesar is living still," and "Csesar is dead," are both intelligible propositions ; therefore, they are both true, or both false. (3) Many things are more difficult than to do nothing. Nothing is more difficult to do than to walk on one's head. Therefore, many things are more difficult than to walk on one's head. (4) None but Whigs vote for Mr B. All who vote for Mr B. are ten-pound householders. Therefore, none but Whigs are ten- pound householders. (5) If the Mosaic account of the cosmogony is strictly correct, the sun was not created till the fourth day. And if the sun was not created till the fourth day, it could not have been the cause of the alternation of day and night for the first three days. But either the word " day " is used in Scripture in a different sense to that in which it is commonly accepted now, or else the sun must have been the cause of the alternation of day and night for the first three days. Hence it follows that either the Mosaic account of the cosmogony is not strictly correct, or else the word "day" is used in Scripture in a different sense to that in which it is commonly accepted now. (6) Suffering is a title to an excellent inheritance ; for God chastens every son whom he receives. (7) It will certainly rain, for the sky looks very black. [Solly, Syllabus of Logic .] 280. Dr Johnson remarked that "a man who sold a penknife was not necessarily an ironmonger." Against what logical fallacy was this remark directed ? [c.] 281. Examine the following arguments, pointing out any fallacies that they contain : (a) The more correct the logic, the more certainly will the conclusion be wrong if the premisses are false. Therefore, where the premisses are wholly uncertain the best logician is the least safe guide. CHAP. VIII.] EXERCISES. 351 (b) The spread of education among the lower orders will make them unfit for their work : for it has always had that effect on those among them who happen to have acquired it in previous times. (c) This pamphlet contains seditious doctrines. The spread of seditious doctrines may be dangerous to the State. Therefore, this pamphlet must be suppressed. [c.] 282. Discuss the nature of the reasoning contained, or apparently intended, in the following sentences : It is impossible to prove that persecution is justifiable if you cannot prove that some non-effective measures are justifiable ; for no persecution has ever been effective. This deed may be genuine though it is not stamped, for some unstamped deeds are genuine. [c.] 283. State the following arguments in logical form, and examine their validity : (1) Poetry must be either true or false : if the latter, it is misleading; if the former, it is disguised history, and savours of imposture as trying to pass itself off for more than it is. Some philosophers have therefore wisely excluded poetry from the ideal commonwealth. (2) If we never find skins except as the teguments of 4 animals, we may safely conclude that animals cannot exist without skins. If colour cannot exist by itself, it follows that neither can anything that is coloured exist without colour. So if language without thought is unreal, thought without language must also be so. (3) Had an armistice been beneficial to France and Germany, it would have been agreed upon by those powers ; but such has not been the case ; it is plain therefore that an armistice would not have been advantageous to either of the belligerents. (4) If we are marked to die, we are enow To do our country loss : and, if to live, The fewer men, the greater share of honour. [o.] 284. Examine logically the following arguments : (a) If truthfulness is never found save with scrupulousness, and if truthfulness is incompatible with stupidity, it follows that stupidity and scrupulousness can never be associated. 352 SYLLOGISMS. [PART III. (6) You say that there is no rule without an exception. I answer that, in that case, what you have just said must have an exception, and so prove that you have contradicted yourself. (c) Knowledge gives power; consequently, since power is desirable, knowledge is desirable. [L.] 285. Examine the following arguments, stating them in syllo- gistic form, and pointing out fallacies, if any : (a) Some who are truly wise are not learned ; but the virtuous alone are truly wise; the learned, therefore, are not always virtuous. (6) If all the accused were innocent, some at least would have been acquitted ; we may infer, then, that none were innocent, since none have been acquitted. (c) Every statement of fact deserves belief; many statements, not unworthy of belief, are asserted in a manner which is anything but strong; we may infer, therefore, that some statements not strongly asserted are statements of fact. (d) That many persons who commit errors are blameworthy is proved by numerous instances in which the commission of errors arises from gross carelessness. [M.] 286. Examine technically the following arguments : (1) Those who hold that the insane should not be punished ought in consistency to admit also that they should not be threatened ; for it is clearly unjust to punish any one without previously threatening him. (2) If he pleads that he did not steal the goods, why, I ask, did he hide them, as no thief ever fails to do 1 (3) Knavery and folly always go together ; so, knowing him to be a fool, I distrusted him. (4) If I deny that poverty and virtue are inconsistent, and you deny that they are inseparable, we can at least agree that some poor are virtuous. (5) How can you deny that the infliction of pain is justifiable if punishment is sometimes justifiable and yet always involves pain ? M 287. Detect the fallacy in the following argument : "A vacuum is impossible, for if there is nothing between two bodies they must touch." [N.] CHAP. VIII.] EXERCISES. 353 288. Examine technically the following arguments : (a) "'Tis only the present that pains, And the present will pass." (6) All legislative restraint is either unjust or unnecessary; since, for the sake of a single man's interests, to restrain all the rest of the community is unjust, and to restrain the man himself is unnecessary. (c) Only Conservatives and not all of them are Protectionists ; only Liberals and not all of them are Home Rulers; but both parties contain supporters of women's franchise. Hence only Unionists and not all of them are Protectionists, while the sup- porters of women's franchise contain both Unionists and Free- traders. (d) No school-boy can be expected to understand Constitutional History, and none but school-boys can be expected to remember dates : so that no one can be expected both to remember dates and to understand Constitutional History. (e) To be wealthy is not to be healthy ; not to be healthy is to be miserable ; therefore, to be wealthy ia to be miserable. CO Whatever any man desires is desirable ; every man desires his own happiness ; therefore, the happiness of every man is desirable. [j.] 289. Examine the validity of the following arguments : (1) I knew he was a Bohemian, for he was a good musician, and Bohemians are always good musicians. (2) Bullies are always cowards, but not always liars ; liars, therefore, are not always cowards. (3) If all the soldiers had been English, they would not all have run away ; but some did run away ; and we may, therefore, infer that some of them at least were not English. (4) None but the good are really to be envied ; all truly wise men are good ; therefore, all truly wise men are to be envied. (5) You cannot affirm that all his acts were virtuous, for you deny that they were all praiseworthy, and you allow that nothing that is not praiseworthy is virtuous. (6) Since the end of poetry is pleasure, that cannot be un- poetical with which all are pleased. K. L. 23 354 SYLLOGISMS. [PART III. (7) Most M is P, Most S is M, therefore, Some S is P. (8) Old Parr, healthy as the wild animals, attained to the age of 152 years; all men might be as healthy as the wild animals; therefore, all men might attain to the age of 152 years. (9) It is quite absurd to say " I would rather not exist than be unhappy," for he who says " I will this, rather than that," chooses something. Non-existence, however, is no something, but nothing, and it is impossible to choose rationally when the object to be chosen is nothing. (10) Because the quality of having warm red blood belongs to all known birds, it must be part of their specific nature ; but unknown birds have the same specific nature as known birds ; therefore, the quality of having warm red blood must belong to the unknown as well as the known birds, i.e., be a universal and essential property of the species. [K.] CHAPTER IX. PROBLEMS ON THE SYLLOGISM. 290. Bearing of the existential import of propositions upon the validity of syllogistic reasonings. We may as before take different suppositions with regard to the existential import of propositions, and proceed to consider how far the validity of the various syllogistic moods is affected by each in turn. (1) Let every proposition imply the existence both of its subject and of its predicate 1 . In this case, the existence of the major, middle, and minor terms is in every case guaranteed by the premisses, and therefore no further assumption with regard to existence is required in order that the conclusion may be legitimately obtained 2 . We may regard the above supposition as that which is tacitly made in the ordinary doctrine of the syllogism. (2) Let every proposition imply the existence of its subject. Under this supposition, as we have already seen, an affirmative proposition ensures the existence of its predicate also ; but not so a negative proposition. It follows that any mood will be valid unless the minor term is in its premiss the predicate of a negative proposition. This cannot happen either in figure 1 or in figure 2, since in these figures the minor is always subject in its premiss ; nor in figure 3, since in this figure the minor premiss is always affirmative. In figure 4, the only moods with 1 It will be observed that this is not quite the same as supposition (1) in sections 117 to 119. 2 If, however, we are to be allowed to proceed as in section 154 (where from all P is M, all S is SI, we inferred some not-S is not-P) we must posit the exist- ence not merely of the terms directly involved, but also of their contradictories. 232 356 SYLLOGISMS. [PART III. a negative minor are Camenes and its weakened form AEO. Our conclusion then is that on the given supposition every ordinarily recognised mood is valid except these two 1 . (3) Let no proposition imply the existence either of its sub- ject or of its predicate. Taking 8, M, P, as the minor, middle, and major terms respectively, the conclusion will imply that if there is any S there is some P or not-P (according as it is affirmative or negative). Will the premisses also imply this ? If so, then the syllogism is valid ; but not otherwise. It has been shewn in section 160 that a universal affirma- tive conclusion, All S is P, can be proved only by means of the premisses, All M is P, All S is M; and it is clear that these premisses themselves imply that if there is any S there is some P. On our present supposition, then, a syllogism is valid if its conclusion is universal affirmative. Again, as shewn in section 160, a universal negative con- clusion, No S is P, can be proved only in the following ways: (i) No M is P (or No P is M), All S is M, therefore, No S is P; (ii) AllPisM, No S is M (or No M is 8), therefore, No S is P. In (i) the minor premiss implies that if 8 exists then M 1 Beduction to figure 1 appears to be affected by this supposition, since it makes the contraposition of A and the conversion of in general invalid. The contraposition of A is involved in the direct reduction of Baroco (Faksoko). The process is, however, in this particular case valid, as the existence of not-M is given by the minor premiss. The conversion of E is involved in the reduction of Cesare, Camestres, and Festino from figure 2 ; and of Camenes, Fesapo, and Fresison from figure 4. Since, however, one premiss must be affirmative the existence of the middle term is thereby guaranteed, and hence the simple con- version of in the second figure, and in the major of the fourth becomes valid. Also the conversion of the conclusion resulting from the reduction of Camestres is legitimate, since the original minor term is subject in its premiss. Hence Camenes (and its weakened form) are the only moods whose reduction is ren- dered illegitimate by the supposition under consideration. This result agrees with that reached in the text. CHAP. IX.] SYLLOGISMS AND EXISTENTIAL IMPORT. 357 exists, and the major premiss that if M exists then not-P exists. In (ii) the minor premiss implies that if $ exists then not-M exists, and the major premiss that if not-M exists then not-P exists (as shewn in section 118). Hence a syllogism is valid if its conclusion is universal negative. Next, let the conclusion be particular. In figure 1, the implication of the conclusion with regard to existence is con- tained in the premisses themselves, since the minor term is the subject of an affirmative minor premiss, and the middle term the subject of the major premiss. In figure 2, we may con- sider the weakened moods disposed of in what has been already said with regard to universal conclusions ; for under our present supposition subalternation is a valid process. The remaining moods with particular conclusions in this figure are Festino and Baroco. In the former, the minor premiss implies that if $ exists then M exists, and the major that if M exists then not-P exists ; in the latter, the minor premiss implies that if S exists then not-M exists, and the major that if not-M exists then not-P exists. All the ordinarily recognised moods, then, of figures 1 and 2 are valid. But it is otherwise with moods yielding a particular conclusion in figures 3 and 4, with the single exception of the weakened form of Camenes (which is itself the only mood with a universal conclusion in these figures). Subalternation being a valid process, the legitimacy of the latter follows from the legitimacy of Camenes itself. But in all other cases in figures 3 and 4, the minor term is the predicate of an affirmative minor premiss. Its existence, therefore, carries no further implication of existence with it in the premisses. It does so in the con- clusion. Hence all the moods of figures 3 and 4, with the exception of AEE and AEO in the latter figure, are invalid. Take, as an example, a syllogism in Darapti, All M is P, All M is S, therefore, Some S is P. The conclusion implies that if S exists P exists ; but con- sistently with the premisses, S may be existent while M and P 358 SYLLOGISMS. [PART III. are both non-existent. An implication is, therefore, contained in the conclusion which is not justified by the premisses. Hence on the supposition that no proposition implies the existence either of its subject or of its predicate all the ordinarily recognised moods of figures 1 and 2 are valid, but none of those of figures 3 and 4 excepting Camenes and the weakened form of Camenes 1 . (4) Let particulars imply, while universals do not imply, the existence of their subjects. The legitimacy of moods with universal conclusions may be established as in the preceding case. Taking moods with particular conclusions, it is obvious that they will be valid if the minor premiss is particular, having the minor term as its subject ; or if the minor premiss is particular affirmative, whether the minor term is its subject or predicate. Disamis, Bocardo, and Dimaris are also valid, since the major premiss in each case guarantees the existence of M, and the minor implies that if M exists then S exists. The above will be found to cover all the valid moods in which one premiss is particular. There remain only the moods in which from two universals we infer a particular. It is clear that all these moods must be invalid, for their conclusions will imply the existence of the minor term, and this cannot be guaranteed by the premisses 2 . On the supposition then that particulars imply, while univer- sals do not imply, the existence of their subjects, the moods ren- dered invalid are all the weakened moods, together with Darapti, Felapton, Bramantip, and Fesapo 3 , each of which contains a strengthened premiss. More briefly, any ordinarily recognised mood is on this supposition valid, unless it contains either a strengthened premiss or a weakened conclusion. 1 An express statement concerning existence may, however, render the rejected moods legitimate. If, for instance, the existence of the middle term is expressly given, then Darapti becomes valid. Compare note 1 on p. 252. 2 Hypothetical conclusions (of the form If S exists then &c.) will of course still be legitimate. 3 It will be observed that the letter p occurs in the mnemonic for each of these moods, indicating that their reduction to figure 1 involves conversion per accidens. On the supposition under discussion this process is invalid, and we may find here a confirmation of the above result. CHAP. IX.] PROBLEMS. 359 291. Connexion between the truth and falsity of premisses and conclusion in a valid syllogism. By saying that a syllogism is valid we mean that the truth of its conclusion follows from the truth of its premisses ; and it is an immediate inference from this that if the conclusion is false one or both of the premisses must be false. The converse does not, however, hold good in either case. The truth of the premisses does not follow from the truth of the conclusion ; nor does the falsity of the conclusion follow from the falsity of either or both of the premisses. The above statements would probably be accepted as self- evident ; still it is more satisfactory to give a formal proof of them, and such a proof is afforded by means of the three following theorems 1 . (1) Given a valid syllogism, then in no case will the com- bination of either premiss with the conclusion establish the other premiss. We have to shew that if one premiss and the conclusion of a valid syllogism be taken as a new pair of premisses they do not in any case suffice to establish the other premiss. Were it possible for them to do so, then the premiss given true would have to be affirmative, for if it were negative, the original conclusion would be negative, and combining these we should have two negative premisses which could yield no conclusion. Also, the middle term would have to be distributed in the premiss given true. This is clear if it is not distributed in the other premiss; and since the other premiss is the conclusion of the new syllogism, if it is distributed there, it must also be distributed in the premiss given true or we should have an illicit process in the new syllogism. Therefore, the premiss given true, being affirmative, and distributing the middle term, cannot distribute the other term which it contains 2 . Neither therefore can this term be dis- 1 It is assumed throughout this section that our schedule of propositions does not include U. The theorems hold good, however, for the sixfold schedule, including Y and r\, as well as for the ordinary fourfold schedule. 2 This statement, though not holding good for U, holds good for Y as well as A. 360 SYLLOGISMS. [PART III. tributed in the original conclusion. But this is the term which will be the middle term of the new syllogism, and we shall therefore have undistributed middle. Hence the truth of one premiss and the conclusion of a valid syllogism does not establish the truth of the other premiss ; and d fortiori the truth of the conclusion cannot by itself establish the truth of both the premisses 1 . (2) The contradictories of the premisses of a valid syllogism ivill not in any case suffice to establish the contradictory of the original conclusion. .The premisses of the original syllogism must be either (a) both affirmative, or (/3) one affirmative and one negative. In case (a), the contradictories of the original premisses will both be negative ; and from two negatives nothing follows. In case (/3), the contradictories of the original premisses will be one negative and one affirmative ; and if this combina- tion yields any conclusion, it will be negative. But the original conclusion must also be negative, and therefore its contradictory will be affirmative. In neither case then can we establish the contradictory of the original conclusion. (3) One premiss and the contradictory of the other premiss of a valid syllogism will not in any case suffice to establish the contradictory of the original conclusion 2 . This follows at once from the first of the theorems established in this section. Let the premisses of a valid syllogism be P and Q, and the conclusion R. P and the con- tradictory of Q will not prove the contradictory of It ; for if they did, it would follow that P and R would prove Q ; but this has been shewn not to be the case. 1 Other methods of solution more or less distinct from the above might be given. A somewhat similar problem is discussed by Solly, Syllabus of Logic, pp. 123 to 126, 132 to 136. We have shewn that one premiss and the conclu- sion of a valid syllogism will never suffice to prove the other premiss, but it of course does not follow that they will never yield any conclusion at all ; for a consideration of this question, see the following section. 2 It does not follow that one premiss and the contradictory of the other premiss of a valid syllogism will never yield any conclusion at all. See the following section. CHAP. IX.] PROBLEMS. 361 We have now established by strictly formal reasoning Aristotle's dictum that although it is not possible syllogis- tically to get a false conclusion from true premisses, it is quite possible to get a true conclusion from false premisses 1 . In other words, the falsity of one or both of the premisses does not establish the falsity of the conclusion of a syllogism 2 . The second of the above theorems deals with the case in which both the premisses are false ; the third with that in which one only of the premisses is false. 292. 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. In this section we shall consider three problems, mutually involved in one another, which are in a manner related to the theorems contained in the preceding section. It has, for example, been shewn that one premiss and the contradictory of the other premiss will not in any case suffice to establish the contra- dictory of the original conclusion ; the object of the first of the following problems is to enquire in what cases they can establish any conclusion at all. (i) To find a pair of valid syllogisms having a common premiss, such that the remaining premiss of the one contradicts the remaining premiss of the other 3 . 1 Hamilton (Logic, i. p. 450) considers the doctrine " that if the conclusion of a syllogism be true, the premisses may be either true or false, but that if the conclusion be false, one or both of the premisses must be false " to be extra- logical, if it is not absolutely erroneous. He is clearly wrong, since the doctrine in question admits of a purely formal proof. 2 " In all cases where T is not given in direct perception, but deduced from premisses, what really depends on the correctness of those premisses is not the truth of T, but only our insight into that truth. Without correct premisses T cannot indeed be proved, but nevertheless it can be true and its truth is inde- pendent of any errors we may commit, when reflecting about it, and subsists even when conclusively deduced from premisses materially false. This point deserves notice, for it is a common mistake in reasoning to take the invalidity of the proof which is offered for T as a proof of the falsehood of T itself, and to confuse the refutation of an argument with the disproof of a fact" (Lotze, Logic, % 240). 3 This problem was suggested by the following question of Mr O'Sullivan's, which puts the same problem in another form : Given that one premiss of a 362 SYLLOGISMS. [PART III. We have to find cases in which P and Q, P and Q' (the contradictory of Q), are the premisses of two valid syllogisms. In working out this problem and the problems that follow, it must be remembered that if two propositions are contradictories, they will differ in quality, and also in the distribution of their terms, so that any term distributed in either of them is undistributed in the other and vice versa. We may, therefore, assume that Q is affirmative and Q' negative. Let P contain the terms X and F, while Q and Q' contain the terms Y and Z, so that Y is the middle term, and X and Z the extreme terms, of each syllogism. Since Q' is negative, P must be affirmative ; and since Y must be undistributed either in Q or in Q', it must be dis- tributed in P. Hence P=YaX. Q' must distribute Z; for the conclusion (being negative) must distribute one term, and X is undistributed in P. It follows that Z is undistributed in Q. Hence Q = YaZ or YiZ or Zi Y ; Q' = YoZor YeZorZeY. The following syllogisms, therefore, are such that if one premiss (that in black type) is retained, while the other is replaced by its contradictory, a conclusion is still obtainable : In figure 1 : All ; In figure 3 : AAI, AAI, IAI, All, EAO, OAO ; In figure 4 : IAI, EAO. (ii) To find a pair of valid syllogisms, having a common conclusion, such that a premiss in the one contradicts a premiss in the other. Let Q and Q' (which we may assume to be respectively affirmative and negative) be the premisses in question, and P' the conclusion ; also let Q and Q' contain the terms Y and Z, while P 7 contains the terms X and F, so that Z is the middle term, and X and F the extreme terms, of each syllogism. It follows immediately that P' is negative; also that F valid syllogism is false and the other true, determine generally in what cases a conclusion can be drawn from these data. CHAP. IX.] PROBLEMS. olio must be undistributed in P 7 , since it is necessarily undis- tributed either in Q or in Q'. Hence P'=YoX. Since X is distributed in P' it must also be distributed in the premiss which is combined with Q' ; and as this premiss must be affirmative, it cannot also distribute Z, which must therefore be distributed in Q' (and undistributed in Q). Hence Q = YaZ or YiZ or ZiY\ Q'=YoZor YeZorZeY. The following syllogisms, therefore, are such that the same conclusion is obtainable from another pair of premisses, of which one contradicts one of the original premisses (namely, that in black type) : In figure 1 : EAO, EIO ; In figure 2 : EAO, AEO, EIO, AOO ; In figure 3 : EIO ; In figure 4 : AEO, EIO. (iii) To find a pair of valid syllogisms having a common premiss, such ttiat the conclusion of one contradicts the conclusion of the other 1 . Let P be the common premiss, Q and Q' (respectively affirmative and negative) the contradictory conclusions; also let P contain the terms X and I 7 , while Q and Q' contain the terms Y and Z, so that X is the middle term, and Y and Z the extreme terms, of each syllogism. Since Q is affirmative, P must be affirmative ; and since either Q or Q' will distribute F, P must distribute F. Hence P = FoA". The premiss which, combined with P, proves Q must be affirmative and must distribute X\ it cannot therefore dis- tribute Z, and Z must accordingly be undistributed in Q (and distributed in Q'). 1 This problem was suggested by the following question of Mr Panton's, which puts the same problem in another form : If the conclusion be substituted for a premiss in a valid mood, investigate the conditions which must be fulfilled in order that the new premisses should be legitimate. 364 SYLLOGISMS. [PART III. Hence Q = YaZ or YiZ or ZiY; Q'=YoZor YeZorZeY. The following syllogisms, therefore, are such that the con- tradictory of the conclusion is obtainable, although one of the premisses (that in black type) is retained : In figure 1 : AAA, AAI, EAE, EAO ; In figure 2 : EAE, EAO, AEE ; In figure 4: AAI, AEE 1 . The three sets of moods above worked out are mutually derivable from one another. Thus, (i) (ii) (iii) P and Q .: R = Q and R' .: P' = R and P .'. $ P and Q' .'. T = Q' and T .-. P' = T and P :. Q In this table (i) represents the possible cases in which, one premiss being retained, the other premiss may be replaced by its contradictory. We can then deduce (ii) the cases in which, the conclusion being retained, one premiss may be replaced by its contradictory ; and (iii) the cases in which, one premiss being retained, the conclusion may be replaced by its con- tradictory. We might of course equally well start from (ii) or from (iii), and thence deduce the two others. Comparing the first syllogism of (i) with the second syllo- gism of (iii) and vice versa, we see further that (i) gives the cases in which, one premiss being retained, the conclusion may be replaced by the other premiss ; and that (iii) gives the cases in which, one premiss being retained, the other premiss may be replaced by the conclusion. 1 It will be observed that each of the above problems yields nine cases. Between them they cover all the 24 valid moods ; but there are three moods (namely, EAO in figures 1 and 2 and AAI in figure 3) which occur twice over. The 15 unstrengthened and unweakened moods are equally distributed, namely, the four yielding I conclusions (together with OA 0) falling under (i) ; the six yielding conclusions (except OA 0) under (ii) ; the five yielding A or E conclu- sions under (iii). All the moods of figure 1 (except those with an I premiss) fall under (iii) ; all the moods of figure 2 (except those with an E conclusion) under (ii) ; all the moods of figure 3 (except the one not having an A premiss) under (i). CHAP. IX.] PROBLEMS. 365 The following is another method of stating and solving all three problems: To determine in what cases it is possible to obtain two incompatible trios of propositions, each trio containing three and only three terms and each including a proposition which is identical with a proposition in the other and also a proposition which is the contradictory of a proposition in the other. Let the propositions be P, Q, R', and P, Q', T; and let P contain the terms X and Y ; Q and Q', the terms Y and Z ; R and T, the terms Z and X. Suppose Q to be affirmative, and Q' negative. Then since one of each trio of propositions must be negative, and not more than one can be so (as shewn in section 162), P and T must be affirmative, and R' negative. Again, since each of the terms X, Y, Z must be distributed once at least in each trio of propositions (as shewn in section 162), and since F must be undistributed either in Q or in Q', Y must be distributed in P. Hence P=YaX. X, being undistributed in P, must be distributed in R' and T. Hence T= XaZ. Z, being undistributed in T, must be distributed in Q[, and therefore undistributed in Q, and distributed in R'. Hence Q = YaZ or YiZ or ZiY; Q'=YoZor YeZorZeY; R' = XeZ or ZeX. We have then the following solution of our problem : YaX, YaZ or YiZ or ZiY, XeZ or ZeX; YaX, YoZ or YeZ or ZeY, XaZ. 293. Numerical Moods of the Syllogism 1 . The following 1 This section was suggested by the following question of Mr Johnson's : " Shew the validity of the following syllogisms : (i) All ItTs are P's, At least n S's are J/X therefore, At least n S's are P's ; (ii) All P's are JTs, Less than n S's are 3/'s, therefore, Less than n S's are P's ; (iii) Less than n M a are P's, At least n ATs are S's, therefore, Some S's are not P's. Deduce from the above the ordinary non-numerical moods of the first three figures." 366 SYLLOGISMS. [PART III. are examples of numerical moods in the different figures of the syllogism : Figure 1. (i) All M's are P's, At least n S's are M's, therefore, At least n S's are P's; (ii) Less than n M's are P's, All S's are M's, therefore, Less than n S's are P's; (iii) Less than n M's are P's, At least n S's are M's, therefore, Some S's are not P's; Figure 2. (iv) All P's are M's, Less than n S's are M's, therefore, Less than n S's are P's; (v) Less than n P's are M's, All S's are M's, therefore, Less than n S's are P's ; (vi) Less than n P's are M's, At least n S's are M's, therefore, Some S's are not P's; Figure 3. (vii) Less than n M's are P's, At least n M's are S's, therefore, Some S's are not P's; (viii) All M's are P's, At least n M's are S's, therefore, At least n S's are P's; (ix) At least n M's are P's, All M's are S's, therefore, At least n S's are P's ; Figure 4. (x) At least n P's are M's, All M's are S's, therefore, At least n S's are P's; (xi) All P's are M's, Less than n M's are S's, therefore, Less than n S's are P's; CHAP. IX.] NUMERICAL MOODS. 367 (xii) Less than n P's are M's, At least n M's are S's, therefore, Some S's are not P's. The above moods may be established as follows : (i) From All M's are P's, it follows that Every S which is M is also P, and since At least n S's are M's, it follows further that At least n S's are P's. Denoting the major premiss of (i) by A, the minor by B, and the conclusion by C, we obtain immediately the following syllogisms : A, C", C', B, B'; A'; and these are respectively equivalent to (iv) and (vii) 1 . (v) is obtainable from (iv) by transposing the premisses and converting the conclusion ; (ii) from (v) by converting the major premiss ; (iii) from (vii) by converting the minor premiss ; (vi) from (iii) by converting the major premiss ; (viii) from (i) by converting the minor premiss ; (ix) from (viii) by transposing the premisses and converting the conclusion ; (x) from (i) by transposing the premisses and converting the conclusion ; (xi) from (iv) by converting the minor premiss ; (xii) from (vii) by converting the major premiss. 1 The argument here involved may be set out more at length as follows : (iv) All P's are M's, (a) Less than n S's are M 1 s, (b) therefore, Less than n S's are P's ; (c) for, if not, then At least n S's are P's ; (d) and by (i), (a) and (d) yield the conclusion At least n S's are Ws ; but this contradicts (b), and hence we have proved indirectly the desired conclusion. (vii) Less than n M's are P's, (e) At least n Jfs.are S's, (/) therefore, Some S's are not P's ; (g) for, if not, then All S's are P's ; (h) and by (i), (h) and (/) yield the conclusion At least n M's are P's ; but this con- tradicts (e), and hence we have proved indirectly the desired conclusion. 368 SYLLOGISMS. [PART III. The ordinary non-numerical moods of the different figures may be deduced from the above results as follows : Figure 1. (i) Putting n = total number of S's, we have MaP, SaM, .'. SaP, that is, Barbara ; and putting n = 1, we have MaP, SiM, .'. SiP, that is, Darii. (ii) Putting n=l, MeP, SaM, .: SeP (Celarent). (iii) Putting n = 1, JlfeP, ^, .*, oP (Ferio). AAI and ^J.0 follow a fortiori. Figure 2. (iv) Putting w = total number of S's, PaM, SoM, .: SoP (Baroco); putting n = 1, PaM, SeM, .'. SeP (Camestres). (v) Putting n = I, PeM, SaM, .: SeP (Cesare). (vi) Putting n = l, PeM, SiM, .: SoP (Festino). AEO and EAO follow a fortiori. Figure 3. (vii) Putting n = total number of M's, MoP, MaS, .: SoP (Bocardo); putting n = l, MeP, MiS, .'. SoP (Ferison). (viii) Putting n = l, MaP, MiS, SiP (Datisi). (ix) Putting n = l, MiP, MaS, .', SiP (Disamis). Darapti and Felapton follow a fortiori. Figure 4. (x) Putting n = 1, PiM, MaS, .'. SiP (Dimaris). (xi) Putting n = 1, PaM, MeS, /. SeP (Gamenes). (xii) Putting n = 1, PeM, MiS, .'. SoP (Fresison). Bramantip, AEO, and Fesapo follow a fortiori. EXERCISES. 294. " Whatever P and Q may stand for, we may shew a priori that some P is Q. For All PQ is Q by the law of identity, and similarly All PQ is P ; therefore, by a syllogism in Darapti, Some P is Q." How would you deal with this paradox ? [K.] A solution is afforded by the discussion contained in section 290; and this example seems to shew that the enquiry how far assump- tions with regard to existence are involved in syllogistic processes is not irrelevant or unnecessary. CHAP. IX.] PROBLEMS. 369 295. What conclusion can be drawn from the following propositions? The members of the board were all either bondholders or shareholders, but not both ; and the bond- holders, as it happened, were all on the board. [v.] We may take as our premisses : No member of the board is both a bondholder and a shareholder, All bondholders are members of the board ; and these premisses yield a conclusion (in Celarent), No bondholder is both a bondholder and a shareholder, that is, No bondholder is a shareholder. 296. The following rules were drawn up for a club: (i) The financial committee shall be chosen from amongst the general committee; (ii) No one shall be a member both of the general and library committees, unless he be also on the financial committee ; (iii) No member of the library committee shall be on the financial committee. Is there anything self-contradictory or superfluous in these rules ? [VENN, Symbolic Logic, p. 261.] Let F - member of the financial committee, G = member of the general committee, L member of the library committee. The above rules may then be expressed symbolically as follows : (i) All F is G; (ii) If any L is G, that L is F ; (iii) No L is F. From (ii) and (iii) we obtain (iv) No L is G. The rules may therefore be written in the form, (1) All F is G, (2) No L is G, (3) No L is F. But in this form (3) is deducible from (1) and (2). Hence all that is contained in the rules as originally stated may be expressed by (1) and (2) ; that is, the rules as originally stated were partly superfluous, and they may be reduced to (1) The financial committee shall be chosen from amongst the general committee ; (2) No one shall be a member both of the general and library committees. K. L. 24 370 SYLLOGISMS. [PART III. If (ii) is interpreted as implying that there are some individuals who are on both the general and library committees, then it follows that (ii) and (iii) are inconsistent with each other. 297. Given that the middle term is distributed twice in the premisses of a syllogism, determine directly (i.e., without any reference to the mnemonic verses or the special rules of the figures) in what different moods it might possibly be. [K.] The premisses must be either both affirmative, or one affirmative and one negative. In the first case, both premisses being affirmative can distribute their subjects only. The middle term must, therefore, be the subject in each, and both must be universal. This limits us to the one syllogism All M is P, All M is S, therefore, Some S is P. In the second case, one premiss being negative, the conclusion must be negative and will, therefore, distribute the major term. Hence, the major premiss must distribute the major term, and also (by hypothesis) the middle term. This condition can be fulfilled only by its being one or other of the following No M is P or No P is M. The major being negative, the minor must be affirmative, and in order to distribute the middle term must be All M is S. In this case we get two syllogisms, namely, No M is P, AH M is S, therefore, Some S is not P ; No P is M, All M is S, therefore, Some S is not P. The given condition limits us, therefore, to three syllogisms (one affirmative and two negative) ; and by reference to the mnemonic verses we may now identify these with Darapti and Felapton in figure 3, and Fesapo in figure 4. 298. If the major premiss and the conclusion of a valid syllogism agree in quantity, but differ in quality, find the inood and figure. [T.] Since we cannot have a negative premiss with an affirmative CHAP. IX.] PROBLEMS. 371 conclusion, the major premiss must be affirmative and the conclusion negative. It follows immediately that, in order to avoid illicit major, the major premiss must be All P is M (where M is the middle term and P the major term). The conclusion, therefore, must be No S is P (S being the minor term); and this requires that, in order to avoid undistributed middle and illicit minor, the minor premiss should be No S is M or No M is S. Hence the syllogism is in Camestres or in Camenes. 299. Given a valid syllogism with two universal premisses and a particular conclusion, such that the same conclusion cannot be inferred, if for either of the premisses is substituted its subaltern, determine the mood and figure of the syllogism. [K.] Let S, J/", P be respectively the minor, middle, and major terms of the given syllogism. Then, since the conclusion is particular, it must be either Some S is P or Some S is not P. First, if possible, let it be Some S is P. The only term which need be distributed in the premisses is M. But since we have two universal premisses, tioo terms must be dis- tributed in them as subjects 1 . One of these must be superfluous; and, therefore, for one of the premisses we may substitute its subaltern, and still get the same conclusion. The conclusion cannot then be Some S is P. Secondly, if possible, let the conclusion be Some S is not P. If the subject of the minor premiss is S, we may clearly sub- stitute its subaltern without affecting the conclusion. The subject of the minor premiss must therefore be Jf, which will thus be dis- tributed in this premiss. M caunot also be distributed in the major, or else it is clear that its subaltern might be substituted for the minor and nevertheless the same conclusion inferred. The major premiss must, therefore, be affirmative with M for its predicate. This limits us to the syllogism All P is M y No M is S, therefore, Some S is not P; and this syllogism, which is AEO in figure 4, does fulfil the given conditions, for it becomes invalid if either of the premisses is made particular. 1 We here include the case in which the middle term is itself twice distri- buted. 242 372 SYLLOGISMS. [PART IIL The above amounts to a general proof of the proposition laid down in section 192 : Every syllogism in which there are two universal premisses with a particular conclusion is a strengthened syllogism, ivith the single exception o/AEO in figure 4. 300. Given two valid syllogisms in the same figure in which the major, middle, and minor terms are respectively the same, shew, without reference to the mnemonic verses, that if the minor premisses are subcontraries, the conclusions will be identical. [K.] The minor premiss of one of the syllogisms must be O, and the major premiss of this syllogism must, therefore, be A and the con- clusion O. The middle and the major terms having then to be distributed in the premisses, this syllogism is determined, namely, All P is M, Some S is not M, therefore, Some S is not P. Since the other syllogism is to be in the same figure, its minor premiss must be Some S is M; the major must therefore be universal, and in order to distribute the middle term it must be negative. This syllogism therefore is also determined, namely, No P is M, Some S is M, therefore, Some S is not P. The conclusions of the two syllogisms are thus shewn to be identical. 301. Find out in which of the valid syllogisms the com- bination of one premiss with the subcontrary of the conclusion would establish the subcontrary of the other premiss. [j.] In the original syllogism (a) let X (universal) and Y (particular) prove Z (particular), the minor, middle, and major terms being 8 y M, and P, respectively. Then we are to have another syllogism (/3) in which X and Z l (the sub-contrary of Z) prove Y 1 (the sub- contrary of F). In ft, S or P will be the middle term. It is clear that only one term can be distributed in a if the con- clusion is affirmative, and only two if the conclusion is negative. Hence S cannot be distributed in a, and it follows that it cannot be distributed in the premisses of /3. The middle term of /3 must CHAP. IX.] PROBLEMS. 373 therefore be P, and as X must consequently contain P it must be the major premiss of a and Y the minor premiss. Z must be either SiP or SoP. First, let Z = SiP. Then it is clear that X=MaP, Z } =SoP, Y t = SoM, Y=SiM. Secondly, let Z = SoP. Then Z l = SiP, X^PaM or MeP or PeM (since it must distribute P), Y^SiM (if X is affirmative) or SoM (if X is negative), Y=SoM or A^tJ/" accordingly. Hence we have four syllogisms satisfying the required conditions as follows : MaP MeP PeM PaM SiM SiM SiM SoM SiP~ SoP~ SoJ SoP It will be observed that these are all the moods of the first and second figures, in which one premiss is particular. 302. Is it possible that there should be a valid syllogism such that, each of the premisses being converted, a new syllo- gism is obtainable giving a conclusion in which the old major and minor terms have changed places ? Prove the correctness of your answer by general reasoning, and if it is in the affirma- tive, determine the syllogism or syllogisms fulfilling the given conditions. [K.] If such a syllogism be possible, it cannot have two affirmative premisses, or (since A can only be converted per accidens) we should have two particular premisses in the new syllogism. Therefore, the original syllogism must have one negative premiss. This cannot be O, since O is inconvertible. Therefore, one premiss of tJie original syllogism must be E. First, let this be the major premiss. Then the minor premiss must be affirmative, and its converse (being a particular affirmative), will not distribute either of its terms. But this converse will be the major premiss of the new syllogism, which also must have a negative conclusion. We should then have illicit major in the new syllogism; and hence the above supposition will not give us the desired result. Secondly, let the minor premiss of the original syllogism be E. The major premiss in order to distribute the old major term must be A, with the major term as subject. We get then the following, satisfying the given conditions : 374 SYLLOGISMS. [PART III. All P is M, No M is S, or No S is Jf, therefore, No S is P, or Some S is not P ; that is, we really have four syllogisms, such that both premisses being converted, thus, No S is M, or No M is S, Some M is P, we have a new syllogism giving a conclusion in which the old major and minor terms have changed places, namely, Some P is not S. Symbolically, PaM, SeM,\ MeS,) or MeS j or SeMJ MiP, or If it be required to retain the quantity of the original conclusion, this must be SoP ; in this case then we have only two syllogisms fulfilling the given conditions. 303. Shew that if the proportion of 5's out of the class A is greater than that out of the class not- A, then the proportion of A's out of the class B will be greater than that out of the class not-B 1 . [j.] Let the number of A's be denoted by N (A), the number of AJB's by N (AB), &c. Then, since Every A is AB or Ab (by the law of excluded middle) and No A is both AB and Ab (by the law of contradiction), it follows that We have to shew that N(AB) N(Ab) N(B) > N (b) N (AB) N (aB) follows from \ ' > V ' . N (A) N (a) This can be done by substituting N (AB) + N (Ab) for N (A), &c. 1 This and the following problem cannot properly be called problems on the syllogism. They are given as examples in numerical logic. CHAP. IX.] PROBLEMS. 375 Th- N(AB) If (A) N (a) ' J) N W N (aB) N (AB) ' N (aB) + N (ab) N(A B) + N (Ab) N (aB) N (AB) N(ab) N(Ab) N(aB) > N(ABy N(ab) N (aB) N (Ab) > N~(AB)' N(Ab) + N(ab) N (AB) + N (aB) N(Ab) N (AB) N(Ab) N(AB)' N(AB) N(Ab) iV (B) :> N (b) ' 304. Given the number (U) of objects in the Universe, and the number of objects in each of the classes a?,, a? a , # 3 , ... #, shew that the least number of objects in the class (0j2 > s ip > ...gfc) = U-N(x 1 )-N(x 2 )-N(x a )...-N(x n ), where N (x^ means the number of things which are not x l ; N (x 3 ), the number that are not o- 2 ; &c. . [J.] Given ^V (x^, N (x,,), &c., the number of objects in the class (xi or ic 2 ... or x n ) is greatest when no object belongs to any pair of the classes 5^, x a ...; and in this case it = N (x^ + N (x a )...+ ^V (X H ). Hence the least number in the contradictory class, x 1 x a ...x Ht 305. Prove that with three given propositions (of the forms A, E, I, 0) it is never possible to construct more than one valid syllogism. [K.J 306. On the supposition that no proposition implies the ex- istence either of its subject or of its predicate, find in what cases the reduction of syllogisms to figure 1 is invalid. [K.] 376 SYLLOGISMS. [PART III. 307. Some M is not P ; All S is all M. What conclusion follows from the combination of these premisses ? Can you infer anything about either S or P from the knowledge that both the above propositions are false? [K.] 308. (i) Eitk&r all M is all P or Some M is not P ; (ii) Some S is not M. What is all that can be inferred (a) about S in terms of P, (b) about P in terms of S, from the knowledge that both the above statements are false 1 [K.] 309. (a) "A good temper is proof of a good conscience, and the combination of these is proof of a good digestion, which again always produces one or the other." Shew that this is precisely equivalent to the following : "A good temper is proof of a good digestion, and a good digestion of a good conscience." (b) Examine (by diagrams or otherwise) the following argument: "Patriotism and humanitarianism must be either incompatible or inseparable; and though family-affection and humanitarianism are compatible, yet either may exist without the other; hence, family affection may exist without patriotism" Reduce the argument, if you can, to ordinary syllogistic form ; and determine whether the premisses state anything more than is necessary to prove the conclusion. [j.] 310. " All scientific persons are willing to learn ; all unscientific persons are credulous ; therefore, some who are credulous are not willing to learn, and some who are willing to learn are not credulous." Shew that the ordinary rules of immediate and mediate inference justify this reasoning ; but that a certain assumption is involved in thus avoiding the apparent illicit process. Shew also that, accepting the validity of obversion and simple conversion, we have an analogous case in any inference of a particular from a universal. [j.] 311. An invalid syllogism of the second figure with a particular premiss is found to break the general rules of the syllogism in this respect only, that the middle term is undistributed. If the particular premiss is false and the other true, what do we know about the truth or falsity of the conclusion? [K.] 312. A syllogism is found to offend against none of the syllogistic rules except that with two affirmative premisses it has a negative conclusion. Determine the mood and figure of the syllogism. [K.] CHAP. IX.] PROBLEMS. 377 313. Given two valid syllogisms in the same figure in which the major, middle, and minor terms are respectively the same, shew, without reference to the mnemonic verses, that if the minor premisses are contradictories, the conclusions will not be contradictories. [K.] 314. Is it possible that there should be two syllogisms having a common premiss such that their conclusions, being combined as premisses in a new syllogism, may give a universal conclusion ? If so, determine what the two syllogisms must be. [N.] 315. Three given propositions form the premisses and conclusion of a valid syllogism which is neither strengthened nor weakened. Shew that if two of the propositions are replaced by their contra- complementaries, the argument will still be valid, provided that the proposition remaining unaltered is either a universal premiss or a particular conclusion. [j.] 316. Find out the valid syllogisms that may be constructed without using a universal premiss of the same quality as the conclusion. Shew how these syllogisms may be directly reduced to one another ; and represent diagrammatically the combined information that they yield, on the supposition that they have the same minor, middle, and major terms respectively. [j.] PART IV. A GENERALIZATION OF LOGICAL PROCESSES IN THEIR APPLICATION TO COMPLEX PROPOSITIONS 1 . CHAPTER I. THE COMBINATION OF TERMS. 317. Complex Terms. A simple term may be defined as a term which does not consist of a combination of other terms ; for example, A, P, X. The combination of simple terms yields a complex term ; and the combination may be either conjunctive or alternative. A complex term resulting from the conjunctive combination of other terms may be called a conjunctive term 2 , and it will be found convenient to denote such a term by the simple juxta- position of the other terms involved 3 . This kind of combination is sometimes called determination 4 ; and we may speak of the elements combined in a conjunctive term as the determinants 1 The following pages deal with problems that have ordinarily been relegated to symbolic logic. They do not, however, directly treat of symbolic logic itself, if that term is understood in its ordinary sense, namely, as designating the branch of the science in which symbols of operation are used. Of course in one sense all formal logic is symbolic. 2 What is here called a conjunctive term is called by Jevons a combined term (Pure Logic, 40). 3 The conjunctive combination of terms is in symbolic logic usually repre- sented by the sign of multiplication. 4 Compare Schroder, Der Operationskreis des Logikkalkuh, p. 6. CHAP. I.] COMPLEX TERMS. 379 of that term. Thus, A and B are the determinants of the con- junctive term AB\ A complex term resulting from the alternative combination of other terms may be called an alternative term*; and we may speak of the elements combined in such a term as the alter- nants of that term. Thus, A and B are the alternants of the alternative term A or B. In the following pages, in accordance with the view indi- cated in section 140, the alternants in an alternative term are not regarded as necessarily exclusive of one another (except of course where they are formal contradictories). Thus, if we speak of anything as being A or B we do not intend to exclude the possibility of its being both A and B. In other words, A or B does not exclude AB 9 . It is necessary here to discuss briefly the logical significa- tion of the words and, or. In the predicate of a proposition their signification is clear; they indicate conjunctive and 1 Since AD stands for the class made up of all the individuals that belong both to the class A and to the class B, it is represented by the shaded portion of the following diagram : If the classes A and B lie entirely outside one another, then AB is & non- existent class. 2 It is called by Jevons a plural term (Pure Logic, 63). The alternative combination of terms is in symbolic logic usually represented by the sign of addition. 3 On this view, the shaded portion of the following diagram represents A or B: On the other interpretation of alternatives, we should have to shade our diagram as follows in order to represent the same term : 380 COMPLEX TERMS. [PART IV. alternative combination respectively, for example, P is Q and R, P is Q or R. But when they occur in the subject of a proposition there is in each case an ambiguity to which atten- tion must be called. Thus, there would be a gain in brevity if we could write a proposition with an alternative term as subject in the form P or Q is R. The latter expression would, however, more naturally be interpreted to mean P is R or Q is R, the force of the or being understood, not as yielding a single categorical proposition with an alternative subject-term, but as a brief mode of connecting alternatively two propositions with a common predicate. Hence, when we intend the former, the more definite mode of statement, Whatever is either P or Q is R, or Anything that is either P or Q is R, should be adopted. There is also ambiguity in the form P and Q is R. This would naturally be interpreted, not as a single proposition with a conjunctive subject-term (PQ is R), but as a brief mode of connecting conjunctively two propositions with a common predicate, namely, P is R and Q is R. In order, therefore, to express unambiguously a proposition with a conjunctive subject-term, it will be well either to adopt the method of simple juxtaposition without any connecting word as, for example, P Q is R, or else to employ one of the more cumbrous forms, Whatever is both P and Q is R, or Anything that is both PandQ is R 1 . 318. Order of Combination in Complex Terms, The order of combination in a complex term is indifferent whether the combination be conjunctive or alternative 2 . 1 It will be observed that both in this case and in the case of or, we get rid of the ambiguity by making the words occur in the predicate of a subordinate sentence. Mr Johnson expresses the substance of the last three paragraphs in the text by pointing out that " common speech adopts the convention : Subjects are externally synthesized and predicates are internally synthesised " (Mind, 1892, p. 239). In other words, and and or occurring in a predicate are understood as expressing a conjunctive or an alternative term ; but occurring in a subject they are understood as expressing a conjunctive or an alternative proposition. 2 This is sometimes spoken of as the law or property of commutativeness. Compare Boole, Laws of Thought, p. 31, and Jevons, Principles of Science, chapter 2, 8. CHAP. I.] OPPOSITION OF COMPLEX TERMS. 381 Thus, AB and BA have the same signification. It comes to the same thing whether out of the class A we select the B?s or out of the class B we select the A'a. Again, A or B and B o?' A have the same signification. It is a matter of indifference whether we form a class by adding the B's to the A's or by adding the As to the B's. 319. The Opposition of Complex Terms. However complex a term may be, the criterion of contradictory opposition given in section 28 must still apply : " A pair of contradictory terms are 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 at the same time affirmed." In what follows it will be found convenient to denote the contradictory of any simple term by the corre- sponding small letter. Thus for not- A we may write a, and for not-B we may write b. Now whatever is not AB must be either a or 6, whilst nothin that is AB can be either a or 6. Hence (a or b, constitute a pair of contradictories. Similarly, (A or B, [ab, are a pair of contradictories. And the same will hold good if A and B stand for terms which are already themselves complex (although relatively simple as compared with AB or A or B). If, then, two terms are conjunctively combined into a complex term (of which they will constitute the determinants), the contradictory of this complex term is found by alternatively combining the contradictories of the two determinants. And, conversely, if two terms are alternatively combined into a complex term (of which they will constitute the alternants), the contradictory of this complex term is found by conjunc- tively combining the contradictories of the two alternants. In each case, we substitute for the relatively simple terms involved their contradictories, and (as the case may be) change 382 COMPLEX TERMS. [PART IV. conjunctive combination into alternative combination, or alter- native combination into conjunctive combination. But whatever degree of complexity a term may reach, it will consist of a series of conjunctive and alternative combina- tions ; and it may be successively resolved into the combination of pairs of relatively simple terms till it is at last shewn to result from the combination of absolutely simple terms. For example, ABC or DE or FG results from the alternative combination of the pair (ABC or DE, { FG; ABC or DE results from the alternative combination of the pair (ABC, IDE i FG results from the conjunctive combination of the pair til Iff; and ABC, DE, may be resolved similarly. Hence the successive application of the above rule for finding the contradictory of a complex term where we are dealing with a single pair of determinants or alternants will result in our ultimately substituting for each simple term involved its contradictory, and reversing the nature of their combination throughout 1 . We may, therefore, lay down the following rule for obtaining the contradictory of any complex term : Replace each constituent simple term by its contradictory, and throughout substitute conjunctive combination for alternative 1 Thus, taking the term ABC or DE or FG, and in the first instance denoting the contradictory of a complex term by a bar drawn across it, we have successively ABC or DE or FG = ABC (DE or FG) = (ABorc)UE.'FG = (a or b or c) (d or e) (f or g)> CHAP. I.] OPPOSITION OF COMPLEX TERMS. 383 combination and vice versd 1 . This rule is of simple application, and it is of fundamental importance in the treatment of complex propositions adopted in the following pages. Thus, the contradictory of A or BG is a and (b or c), i.e., ab or ac; and the contradictory of ABC or ABD is (a or b or c) and (a or b or d), which, by the aid of rules presently to be given, is reducible to the form a or b or cd. It is possible for two complex terms to be formally incon- sistent or repugnant without being true contradictories. This will be the case if they contain contradictory determinants without between them exhausting the universe of discourse. The terms AB and bO afford an example : nothing can be both AB and bC (for, if this were so, something would be both B and not-B), but we cannot say a priori that everything is either AB or bC (since something may be Abe, which is neither AB nor bC). If two conjunctive terms are such that every determinant in the one has corresponding to it in the other its contradictory, these two terms may be regarded as in the strictest sense logical contraries 2 . Thus, AbC, aBc may be spoken of as con- traries. An alternative term, such as AB or ab, does not seem to admit of a contrary in this distinctive sense. 320. Duality of Formal Equivalences in the case of Complex Terms. It will be shewn in the following sections that certain complex terms are formally equivalent to other com- plex terms or to simple terms (for example, A or aB = A or B, A or AB = A) ; and it is important to notice at the outset that such formal equivalences always go in pairs. For if two 1 Compare Schroder, Der Operationskreis des Logikkalkuk, p. 18. - Compare section 29. 384 COMPLEX TERMS. [PART IV. terms are equivalent, their contradictories must also be equi- valent ; and hence, applying the rule for obtaining contradic- tories given in the preceding section, we are enabled to formulate the simple law that to every formal equivalence there corresponds another formal equivalence in which conjunctive combination is throughout substituted for alternative combination and vice versa 1 . This law may be more precisely established as follows : A formal equivalence that holds good for any given set of simple terms must equally hold good for any other set of simple terms ; and, therefore, whatever holds good for the terms A, B, &c. must equally hold good for their contra- dictories a, b, &c. Hence, given any equivalence, we may first replace each simple term by its contradictory, and then take the contradictory of each side of the equivalence. The result of this double transformation will be that we shall obtain another equivalence in which every conjunctive combination has been replaced by an alternative combination, and conversely, while the term-symbols involved have remained unchanged. This proves what was required. The application of the above law will be fully illustrated in the sections that immediately follow. 321. Laws of Distribution. In order to combine a simple term conjunctively with an alternative term, we must con- junctively combine it with every alternant of the alternative 2 . A and (B or C) 3 denotes whatever is A and at the same time either B or G, and it is, therefore, equivalent to AB or AC. It follows that in order to combine two alternative terms con- junctively, we must conjunctively combine every alternant of the one with every alternant of the other. Thus, (A or B) (C or D) denotes whatever is either A or B and at the same time either C or D, and it is equivalent to A G or AD or BC or BD*. 1 This is pointed out by Schroder, Der Operationskreis des Logikkalkids, p. 3. The two equivalences which are thus mutually deducible the one from the other may be said to be reciprocal. 2 Compare Jevons, Principles of Science, chapter 5, 7. 3 In such a case as this the use of brackets is necessary in order to avoid ambiguity. Thus, A and B or C might mean AB or C, or as above AB or AC. 4 Whether or not we introduce algebraic symbols into logic, there is here a very close analogy with algebraic multiplication which cannot be disguised. CHAP. I.] FORMAL EQUIVALENCES. 885 We have then A (B or C) = AB or AC, and applying the law of duality of formal equivalences given in the preceding section, we have at once another equivalence, namely, A or BC = (A or B) (A or C}\ These two equivalences are called by Schroder the Laws of Distribution 2 . They are of the greatest importance in the manipulation and simplification of complex terms. 322. Laws of Tautology. The following rules may be laid down for the omission of superfluous terms from a complex term : (a) The repetition of any given determinant is superfluous. Out of the class A to select the A's is a process that leaves us just where we began. In other words, what is both A and A is identical with what is A. Thus, such terms as A A, ABB, are tautologous ; the former merely denotes the class A, and the latter the class AB. Hence the above rule, which is called by Jevons the Law of Simplicity 3 . (6) The repetition of any given alternant is superfluous. To say that anything is A or A is equivalent to saying simply that it is A. Hence such terms as A or A, A or BC or BO, are tautologous ; and we have the above rule, which is called by Jevons the Law of Unity*. It will be seen by reference to the rule given in section 320 that the Law of Simplicity (A A = A) and the Law of Unity ( A or A = A) are reciprocal, that is, the former is deducible from the latter and vice versa. For the only difference between them is that conjunctive combination in the one is replaced by alternative combination in the other 8 . 1 This equivalence might also be established independently by the aid of certain of the equivalences given in the following sections. 2 Der Operationskreis des Logikkalkute, pp. 9, 10. 3 See Pure Logic, 42 ; and Principles of Science, chapter 2, 8. The cor- responding equation y?=x is in Boole's system fundamental; see Laws of Thought, p. 31. 4 See Pure Logic, 69 ; and Principles of Science, chapter o, 4. 6 It may assist the reader in following the reasoning in section 320 if we work through this particular case independently. If AA=A, then aa = a, for K. L. 25 386 COMPLEX TERMS. [PART IV. 323. Laws of Development and Reduction. Important formal equivalences are yielded by the laws of contradiction and excluded middle. By the law of contradiction a term containing contradictory determinants (for example, Bb) cannot represent any existing class. Hence the term A or Bb is equivalent to A simply ; in other words, the conjunctive combination of contradictories may be indifferently introduced or omitted as an alternant. Again, by the law of excluded middle a term containing contradictory alternants (for example, B or b) represents the entire universe of discourse. Hence the term A (B or b) is equivalent to A simply ; in other words, the alternative com- bination of contradictories may be indifferently introduced or omitted as a determinant. The above equivalences, namely, A or Bb = A, A (B or b) = A, are reciprocal; that is to say, either is deducible from the other by the rule given in section 320. Applying further the Laws of Distribution given in section 321 we have the following : A = A or Bb = (A or B) (A or b), A=A (B or b) = AB or Ab. These may be taken as formulae for the development and the reduction of terms. Thus, the substitution of (A or B) (A or b) for A may be called the development of a term by means of the law of contradiction ; and the substitution of AB or Ab for A the development of a term by means of the law of excluded middle. In both the above cases the term A is developed with reference to the term B. Similarly by developing A with reference to B and C, we should have (A or B or C) (A or B or c) (A or b or C) (A or b or c) if we make use of the law of contradiction, or ABC or ABc or AbC or Abe if we make use of the law of excluded middle. Development by means of the law of ^whatever is formally valid in the case of A must also be formally valid in the case of any other term. But if two terms are equivalent, their contradictories must be equivalent. Hence from aa = a, it follows that A or A = A. And it is clear that we might pass similarly from A or A = A to AA = A. CHAP. I.] FORMAL EQUIVALENCES. 387 excluded middle is by far the more useful of the two processes in the manipulation of complex terms, and it may be under- stood that this is meant when the development of a term is spoken of without further qualification. Conversely, the process of passing from (A or B} (A or b) to A, or from AB or Ab to A, may be called the reduction of a term by means of the law of contradiction or the law of excluded middle, as the case may be. Following Jevons, we may speak of an alternative term of the type AB or Ab as a dual term 1 , and of the substitution of A for AB or Ab as the reduction of a dual term 2 . 324. Laws of Absorption. It may be shewn that any alternant which is merely a subdivision of another alternant may be indifferently introduced or omitted from a complex term. Thus, AB being a subdivision of A, the terms A or AB and A are equivalent. This rule (which is called by Schroder the Law of Absorption 3 ) may be established as follows : By the development of A with reference to B, A or AB becomes AB or Ab or AB; but, by the law of unity, this is equivalent to AB or Ab', and this is a dual term reducible to A. Applying the rule given in section 320 we obtain a second law of absorption, namely, A (A or B) = A, which is the reciprocal of the first law of absorption, A or AB = A. 325. Laws of Exclusion and Inclusion. The contradictory of any alternant in a complex term may be indifferently introduced or omitted as a determinant of any other alternant ; that is to say, the terms A or aB and A or B are equivalent. This may be established as follows : By the law of absorption A or aB is equivalent to A or AB or aB, aud by the reduction of the dual term here contained, this yields A or B. The above equivalence may be called the Law of Exclusion on the 1 See Pure Logic, 103. Corresponding to this, the law of excluded middle is usually called by Jevons the Law of Duality ; compare Pure Logic, 99, and Principles of Science, chapter 5, 5. 2 Jevons (Principles of Science, chapter 6, 9) also speaks of it as the " abstraction of indifferent circumstances." 3 Der Operationskreis des Logikkalkuls, p. 12. This Law of Absorption is equivalent to one of Boole's " Methods of Abbreviation " (Laws of Thought, p. 130). Compare, also, Jevons, Pure Logic, 70. 25-2 388 COMPLEX TERMS. [PART IV. ground that by passing from A or B to A or aB we make the alternants mutually exclusive. The reciprocal equivalence A (a or B) = AB may be expressed as follows : The contradictory of any determinant in a complex term may be indifferently introduced or omitted as an alternant of any other determinant. This equivalence may be called the Law of Inclusion on the ground that by passing from AB to A (a or B) we make the determinants collectively inclusive of the entire universe of discourse. 326. Summary of Formal Equivalences of Complex Terms. The following is a summary of the formal equivalences con- tained in the five preceding sections (those that are bracketed together being in each case related to one another reciprocally in the manner indicated in section 320): (1) A (B or G) = AB or AG, /o\ r>n /A -n\ / A r>\( Laws of Distribution; (2) A or BC = (A or B) (A or <7),j (3) A A = A, \ Laws of Tautology {Law of Simplicity (4) A or A = A,) and Law of Unity); (5) A = A or Bb ( A or B) (A or b),} Laws of Develop- (6) A = A (B or b) AB or Ab, )ment andReduction; (7) AorAB=A, . , . , Laws of Absorption; (8) A (A or B)=A,} (9) A or B = A or aB,\ Law of Exclusion and Law of (10) AB = A (a or B), J Inclusion. By bearing the above equivalences in mind, the labour of manipulating complex propositions may be very much diminished. 327. The Conjunctive Combination of Alternative Terms. The first law of distribution gives the general rule for the conjunctive combination of alternatives. But with a view to such combination special attention may be called (i) to the second law of distribution, namely, (A or B) (A or C) = A or BC; and (ii) to the equivalence (A or B} (AC or D) = AC or AD or BD, which may be established as follows: By the first law of distribution (A or B) (AC or D) is equivalent to AAC or ABC or AD or BD; but by the law of simplicity AAC = AC, and by the law of absorption AC or ABC = AC] hence our original term is equivalent to AC or AD or BD, which was to be proved. CHAP. I.] FORMAL EQUIVALENCES. From the above equivalences we obtain the two following practical rules which are of the greatest assistance in simplify- ing the process of conjunctively combining alternatives: (1) If two alternatives which are to be conjunctively combined have an alternant in common, this alternant may be at once written down as one alternant of the result, and we need not go through the form of combining it with any of the remaining alternants of either alternative ; (2) If two alternatives are to be conjunctively combined and an alternant of one is a subdivision of an alternant of the other, then the former alternant may be at once written down as one alternant of the result, and we need not go through the form of combining it with the remaining alternants of the other alternative *. EXERCISES. 328. Simplify the following terms : (i) AD or acD; (ii) Ad or Ae or aB or aC or bC or aE or bE or bd or be or cd or ce. [K.] (i) By rule (1) in section 326, AD or acD is equivalent to (A or ac) D ; and this by rule (9) is equivalent to (A or c) D ; which again by rule (1) is equivalent to AD or cD*. (ii) Since bE or be is a dual term, it may be reduced to b, and hence Ad or Ae or aB or aC or bC or aE or bE or bd or be or cd or ce = aB or aC or bC or aE or b or Ad or Ae or bd or cd or ce. By section 326, rule (7), we may now omit all alternants in which b occurs as a determinant, and by rule (9), B may be omitted wherever it occurs as a determinant; accordingly our term is reduced to a or aC or aE or b or Ad or Ae or cd or ce. Since a is now an alternant, a further application of the same rules leaves us with a or b or d or e or cd or ce ; and this is immediately reducible to a or b or d or e. 329. Shew that BG or bD or CD is equivalent to BC or bD. [K.] 1 These rules are equivalent to Boole's second Method of Abbreviation (Laic* of Thought, p. 131). 2 We might also proceed as follows: AD or acD=AD or AcD or acD [by rule (7)] = AD or cD [by rule (5)]. 390 COMPLEX TERMS. [PART IV. 330. Give the contradictories of the following terms in their simplest forms as series of alternants : AB or BC or CD; AB or bC or cD; ABC or aBc; ABcD or Abcde or aBCDe or BCde. [K.] 331. Simplify the following terms : (1) Ab or aC or BCd or Be or bD or CD ; (2) ACD or Ac or Ad or aB or ICD; (3) aBC or aCD or aBe or aDe or AcD or abD or bcD or aDE or cDE; (4) (A or b) (A or c) (a or B) (a or C) (b or C). [K.] 332. Prove the following equivalences : (1) AB or AC or BC or abc or aB or C = a or B or C; (2) aBC or aBd or acd or bed or ABd or Acd or abd or aCd or BCd = Bd or cd or ad. or aBC \ (3) Pqr or pQs or prs or qrs or pq or pS or qR=p or q. [K.] CHAPTER II. COMPLEX PROPOSITIONS AND COMPOUND PROPOSITIONS. 333. Complex Propositions. A complex proposition may be defined as a proposition which has a complex term either for its subject or its predicate. The ordinary distinctions of quantity and quality may be applied to complex propositions ; thus, All AB is C or D is a universal affirmative complex proposition. Some AB is not EF is a particular negative complex proposition. In the following pages propositions written in the indefinite form will be interpreted as universal, so that AB is CD will be understood to mean that all AB is CD. It must be added that in dealing with complex proposi- tions we definitely adopt the view that universals do not imply, while particulars do imply, the existence of their subjects in the universe of discourse. 334. The Opposition of Complex Propositions. The opposi- tion of complex terms has been already dealt with, and the opposition of complex propositions in itself presents no special difficulty. It must, however, be borne in mind that since in dealing with complex propositions we definitely adopt the view that particulars imply the existence of their subjects while universals do not, we have the following divergences from the ordinary doctrine of opposition: (1) we cannot infer I from A, or O from E ; (2) A and E are not necessarily inconsistent with each other; (3) I and O may both be false at the same time. The ordinary doctrine of contradictory opposition re- mains unaffected. The following are examples of contradictory propositions: All X is both A and B, Some X is not both A 392 COMPOUND PROPOSITIONS. [PART IV. and B ; Some X is Y and at the same time either P or Q or R, No X is Y and at the same time either P or Q or R. 335. Compound Propositions. A compound proposition may be defined as a proposition which consists in a combination of other propositions. The combination may be either con- junctive (i.e., when two or more propositions are affirmed to be true together) or alternative (i.e., when an alternative is given between two or more propositions); for example, All AB is C and some P is not either Q or R is a compound conjunctive proposition ; Either all AB is or some P is not either Q or R is a compound alternative proposition. Propositions conjunc- tively combined may be spoken of as determinants of the resulting compound proposition ; and propositions alternatively combined may be spoken of as alternants of the resulting com- pound proposition. Only two types of compound propositions are here recog- nised, the conjunctive and the alternative. Pure hypothetical propositions are compound, but they are, as we have already seen, equivalent to alternative propositions, and they may accordingly be regarded as constituting one mode of expressing an alternative synthesis. Thus (taking x and y as symbols representing propositions, and x and y as their contradictories) the hypothetical proposition If x then y expresses an alternative between x and y and is, therefore, equivalent to the alternative proposition x or y. Combinations of the true disjunctive type (for example, not both x and y} may also be regarded as a mode of expressing an alternative synthesis; thus, the true disjunctive proposition just given is equivalent to the alternative proposi- tion x or y 1 . Mr Johnson shews that any ordinary proposition with a 1 The above may seem to imply that an alternative synthesis may be expressed in a greater number of ways than a conjunctive synthesis. This, however, is not the case. It has been shewn that an alternative synthesis may be expressed by a hypothetical or by the denial of a conjunctive (that is, a true disjunctive). But corresponding to this, a conjunctive synthesis may be ex- pressed by the denial of a hypothetical or by the denial of an alternative. Thus, representing the denial of a proposition by a bar drawn across it, we have = x ory = Ifx, y; = xory=Ifx,y. CHAP. II.] COMPOUND PROPOSITIONS. 393 general term as subject may be regarded as a compound pro- position resulting from the conjunctive or alternative com- bination of singular (molecular) propositions, with a common predication, but different subjects. Let 8 lt S t , ... 8^ represent a number of different individual subjects ; and let 8 represent the aggregate collection of individuals 8 lt S 3 , ... 5^. Then S l and S % and S a ... and S^ = Every 8 ; $j or S t or S a or S^ = Some 8. " Thus we arrive at the common logical forms, Every Sis P> Some S is P. The former is an abbreviation for a determinative, the latter for an alternative, synthesis of molecular proposi- tions." 1 In other words, Every S is P = S l is P and S a is P and S 3 is P ... and S^is P; Some SisP=S l isPorS !t isPorS a isP...orS tt> isP. 1X9 00 336. The Opposition of Compound Propositions. The rule for obtaining the contradictory of a complex term given in section 319 may be applied also to compound propositions. Thus, the contradictory of a compound proposition is obtained by replacing the constituent propositions by their contradictories and everywhere changing the manner of their combination, that is to say, substituting conjunctive combination for alternative and vice versa*. The following are examples: All A is B and some P is Q has for its contradictory Either some A is not B or no P is Q', Either some A is both B and C, or all B is either C or both D and E has for its contradictory No A is both B and C, and some B is not either C or both D and E. It follows, as in section 320, that there is a duality of formal equivalences in the case of compound propositions, each equiva- lence yielding a reciprocal equivalence in which conjunctive 1 Mind, 1892, p. 25. Mr Johnson of coarse recognises that a quantified subject-term (all S) is not usually a mere enumeration of individuals first appre- hended and named. But he points out that " however the aggregate of things, to which the universal name applies, is mentally reached, the prepositional force for purposes of inference or synthesis in general is the same" (p. 28). 2 It has been shewn in the preceding section that the words all and some are abbreviations of conjunctive and alternative synthesis respectively. Hence the rule that, in the ordinarily recognised prepositional forms, contra- dictories differ in quantity as well as in quality is itself only a particular application of the general law here laid down. 394 COMPOUND PROPOSITIONS. [PART IV. combination is throughout substituted for alternative combina- tion and vice versa. 337. Formal Equivalences of Compound Propositions. The laws relating to the conjunctive or alternative synthesis of propositions are practically identical with those relating to the conjunctive or alternative combination of terms; and we have accordingly the following prepositional equivalences correspond- ing to the equivalences of terms given in section 326. The symbols here stand for propositions, not terms; and negation is represented by a bar over the proposition denied. (1) x (y or z) = xy or xz, \ T /. n - ^ -i. * /ftx . . , . \ Laws of Distribution ; (2) x or yz = (x or y) (x or z),\ (3) xx = x, I Laws of Tautology (Law of Simplicity and (4) x or x = x,} Law of Unity); (5) x = x or yy = (x or y) (x or y},\ Laws of Development (6) x = x(yory} = xy or xy, ) and Reduction ; (7) x or xy = x, \ T . . } Laws of Absorption ; (8) x (x or y) = x,} (9) x or y = x or xy,] Law of Exclusion and Law of (10) xy = x (x or y), } Inclusion 1 . 1 It is not maintained that all the above laws are ultimate or even inde- pendent of one another. The synthesis of propositions is admirably worked out by Mr Johnson in his articles on the Logical Calculus (Mind, 1892). He gives Jive independent laws which are necessary and sufficient for propositional synthesis. These laws are briefly enumerated below; for a more complete exposition the reader must be referred to Mr Johnson's own treatment of them. (i) The Commutative Law: The order of pure synthesis is indifferent (xy = yx). (ii) The Associative Law : The mode of grouping in pure synthesis is indif- ferent (xy . z = x . yz). (iii) The Law of Tautology : The mere repetition of a proposition does not in any way add to or alter its force (xx=x). (iv) The Law of Reciprocity : The denial of the denial of a proposition is equivalent to its affirmation (3=a;). "In this principle are included the so- called Laws of Contradiction and Excluded Middle, viz., ' If x, then not not-x ', and ' If-not not-rr, then x '." (v) The Law of Dichotomy : The denial of any proposition is equivalent to the denial of its conjunction with any other proposition together with the denial of its conjunction with the contradictory of that other proposition (x=xy xy). " This is a further extension of the Law of Excluded Middle, when applied to the combination of propositions with one another. The denial that x is con- joined with y combined with the denial that x is conjoined with not-y is equiva- CHAP. II.] COMPLEX PROPOSITIONS. 395 338. The Simplification of Complex Propositions. The terms of a complex proposition may of course often be simplified by means of the rules given in the preceding chapter, and the force of the assertion will remain unaffected. For the further simplification of complex propositions the following rules may be added : (1) In a universal negative or a particular affirmative pro- position any determinant of the subject may be indifferently introduced or omitted as a determinant of the predicate, and vice versa. To say that No AB is AC is the same as to say that No AB is C, or that No B is AC. For to say that No AB is AC is the same thing as to deny that anything is ABAC; but, as shewn in section 322, the repetition of the determinant A is superfluous, and the statement may therefore be reduced to the denial that anything is ABC. And this may equally well be expressed by saying No AB is C, or No B is AC 1 . Similarly, No AB is AC or AD may be reduced to No AB is C or D, or to No B is AC or AD. Again, Some AB is AC may be shewn to be equivalent to Some AB is C, or to Some B is AC; for it simply affirms that something is ABAC, and the proof follows as above. (2) In a universal affirmative or a particular negative proposition any determinant of the subject may be indifferently introduced or omitted as a determinant of any alternant of the predicate. All A is AB may obviously be resolved into the two pro- positions All A is A, All A is B 2 . But the former of these is a merely identical proposition and gives no information. All A is AB is, therefore, equivalent to the simple proposition All A is B. lent to the denial of x absolutely. For, if x were true, it must be conjoined either with y or with not-y. This law, which (it must be admitted) looks at first a little complicated, is the special instrument of the logical calculus. By its means we may always resolve a proposition into two determinants, or conversely we may compound certain pairs of determinants into a single proposition." 1 See also the sections in chapter 3 relating to the conversion of propositions. 2 The resolution of complex propositions into a combination of relatively simple ones will be considered further in the following section. 396 COMPLEX PROPOSITIONS. [PART IV. Similarly, All AB is AC or DE is equivalent to All AB is G or DE. Again, Some A is not AB affirms that Some A is a or b 1 ; but by the law of contradiction No A is a; therefore, Some A is not B, and obviously we can also pass back from this pro- position to the one from which we started. Similarly, Some AB is not either AC or DE is equivalent to Some AB is not either C or DE. (3) In a universal affirmative or a particular negative proposition any alternant of the predicate may be indifferently introduced or omitted as an alternant of the subject. If All A is B or C, then by the law of identity it follows that Whatever is A or B is B or (7 ; it is also obvious that we can pass back from this to the original proposition. Again, if Some A or B is not either B or C, then since by the law of identity All B is B it follows that Some A is not either B or C ; and it is also obvious that we can pass back from this to the original proposition 2 . (4) In a universal affirmative or a particular negative proposition the contradictory of any determinant of the subject may be indifferently introduced or omitted as an alternant of the predicate, and vice versd. By this rule the three following propositions are affirmed to be equivalent to one another : (All AB is a or C; All B is a or C; AllABisC; and also the three following : (Some AB is not either a or C; J Some B is not either a or C ; [Some AB is not C. The rule follows directly from rule (1) by aid of the process of obversion (see chapter 3). (5) In a universal negative or a particular affirmative 1 The process of obversion will be considered in detail in chapter 3. z What follows to the end of the section may be omitted till the chapter on immediate inferences from complex propositions has been read. CHAP. II.] COMPLEX PROPOSITIONS. 397 proposition the contradictory of any determinant of the subject may be indifferently introduced or omitted as an alternant of the predicate. By this rule the two following propositions are affirmed to be equivalent to one another : (No AB is a or C; (NoABisC; and also the two following : (Some AB is a or C ; [Some AB is C. The rule follows directly from rule (2) by ob version. (6) In a universal negative or a particular affirmative proposition the contradictory of any determinant of the predicate may be indifferently introduced or omitted as an alternant of the subject. This rule follows from rule (3) by obversion. 339. The Resolution of Universal Complex Propositions into Equivalent Compound Propositions. We may enquire how far complex propositions are immediately resolvable into a conjunctive or alternative combination of relatively simple propositions. Universal propositions will be considered in this section, and particulars in the next. Universal Affirmatives. Universal affirmative complex propositions may be immediately resolved into a conjunction of relatively simple ones, so far as there is alternative combina- tion in the subject or conjunctive combination in the predicate. Thus, (1) Whatever is P or Q is R = All P is R and all Q is R ; (2) All P is QR = All P is Q and all P is R. Universal Negatives. Universal negative complex proposi- tions may be immediately resolved into a conjunction of relatively simple ones, so far as there is alternative combina- tion either in the subject or in the predicate. Thus, (3) Nothing that is P or Q is R = No P is R and no Q is R', (4) No P is either Q or R = No P is Q and no P is R. So far as there is conjunctive combination in the subject or alternative combination in the predicate of universal affirma- tive propositions, or conjunctive combination either in the 398 COMPLEX PROPOSITIONS. [PART IV. subject or in the predicate of universal negative propositions, they cannot be immediately 1 resolved into either a conjunctive or an alternative combination of simpler propositions. It may, however, be added that propositions falling into this latter category are immediately implied by certain compound alterna- tives. Thus, (i) All PQ is R is implied by All P is R or all Q is R ; (ii) All P is Q or R is implied by All P is Q or all P is R; (iii) No PQ is R is implied by No P is R or no Q is R ; (iv) No P is QR is implied by No P is Q or no P is R. 340. The Resolution of Particular Complex Propositions into Equivalent Compound Propositions. Complex particular propositions cannot be resolved into compound conjunctives, but they may under certain conditions be immediately resolved into equivalent compound alternative propositions in which the alternants are relatively simple. This is the case so far as there is alternative combination in the subject or conjunctive combination in the predicate of a particular negative, or alterna- tive combination either in the subject or in the predicate of a particular affirmative. Thus, (1) Some P or Q is not R = Some P is not R or some Q is not R ; (2) Some P is not QR = Some P is not Q or some P is not R; (3) Some P or Q is R = Some P is R or some Q is R; (4) Some P is Q or R = Some P is Q or some P is R. Particular complex propositions cannot be immediately resolved into compound propositions (either conjunctive or alternative) so far as there is conjunctive combination in the subject or alternative combination in the predicate if the proposition is negative, or so far as there is conjunctive combination either in the subject or in the predicate if the proposition is affirmative. In these cases, however, the com- plex proposition implies a compound conjunctive proposition, though we cannot pass back from the latter to the former. 1 It will be shewn subsequently that even in these cases universal complex propositions may be resolved into a conjunction of relatively simpler ones by the aid of certain immediate inferences. CHAP. II.] COMPLEX PROPOSITIONS. 399 Thus, (i) Some PQ is not R implies Some P is not R and some Q is not R ; (ii) Some P is not either Q or R implies Some P is not Q and some P is not R ; (iii) Some PQ is R implies Some P is R and some Q is R; (iv) Some P is QR implies Some P is Q and some P is R. It must be particularly noticed that although in these cases the compound proposition can be inferred from the complex proposition, still the two are not equivalent. For example, from Some P is Q and some P is R it does not follow that Some P is QR, for we cannot be sure that the same P's are referred to in the two cases. All the results of this section follow from those of the preceding by applying the rule of contradiction to the pro- positions themselves and the rule of contraposition to the relations of implication between them. 341. The Omission of Terms from a Complex Proposition. From the two preceding sections we may obtain immediately the following rules for inferring from a given proposition another proposition in which certain terms contained in the original proposition are omitted : (1) A determinant may at any time be omitted from an undistributed term 1 ] (2) An alternant may at any time be omitted from a dis- tributed term*. For example, Whatever is A or B is CD, therefore, All A is C; Some AB is CD, therefore, Some A is C ; Nothing that is A or B is C or D, therefore, No A is C; Some AB is not either C or D, therefore, Some A is not C. The above rules may also be justified independently, as will be shewn in the following section. The results which they yield must be distinguished from those obtained in section 338. In the cases discussed in that section, the terms omitted were superfluous in the sense that their omission left us with pro- positions equivalent to our original propositions; but in the 1 The subject of a particular or the predicate of an affirmative proposition. 2 The subject of a universal or the predicate of a negative proposition. 400 COMPLEX PROPOSITIONS. [PART IV. above inferences we cannot pass back from conclusion to premiss. From Some A is C, for example, we cannot infer that Some AB is G. 342. The Introduction of Terms into a Complex Proposition. Corresponding to the rules laid down in the preceding section we have also the following : (1) A determinant may at any time be introduced into a distributed term ; (2) An alternant may at any time be introduced into an undistributed term. These rules, and also the rules given in the preceding section, may be established by the aid of the following axioms : What is true of all (distributively) is true of every part ; What is true of part of a part is true of a part of the larger whole. When we add a determinant to a term, or remove an alternant, we potentially narrow the extension of the term ; when, on the other hand, we add an alternant, or remove a determi- nant, we potentially widen its extension. Hence it follows that if a term is distributed we may add a determinant or remove an alternant, whilst if a term is undistributed we may add an alternant or remove a determinant. Thus, All A is CD, therefore, All AB is C; No A is C, therefore, No AB is CD ; Some AB is C, therefore, Some A is C or D; Some AB is not either C or D, therefore, Some A is not C. From the above rules taken in connexion with the rules given in section 338 we may obtain the following corollaries : (3) In universal affirmatives, any determinant may be introduced into the predicate, if it is also introduced into the subject; and any alternant may be introduced into the subject if it is also introduced into the predicate. Given All A is C, then All AB is C by rule (1) above ; and from this we obtain All AB is BC by rule (2) of section 338. Again, given All A is C, then All A is B or C; and there- fore, by rule (3) of section 338, Whatever is A or B is B or C. (4). In universal negatives any alternant may be intro- duced into subject or predicate, if its contradictory is introduced into the other term as a determinant. CHAP. II.] ANOMALOUS FORMS. 401 Given No A is C, then No AB is C ; and, therefore, by rule (5) of section 338, No AB is b or C. Again, given No A is C, then No A is BC; and, therefore, by rule (6) of section 338, No A or bis BC. In none of the inferences considered in this section is it possible to pass back from the conclusion to the original proposition. 343. Interpretation of Anomalous Forms. It will be found that propositions which apparently involve a contradiction in terms and are thus in direct contravention of the fundamental laws of thought for example, No AB is B, All Ab is B sometimes result from the manipulation of complex proposi- tions. In interpreting such propositions as these, a distinction must be drawn between universals and particulars, at any rate if it is held that particulars imply, while universals do not imply, the existence of their subjects. It can be shewn that a universal proposition of the form No AB is B or All Ab is B must be interpreted as affirming the non-existence of the subject of the proposition. For a universal negative denies the existence of anything that comes under both its subject and its predicate ; thus, No AB is B denies the existence of ABB, that is, it denies the existence of AB. Again, a universal affirmative denies the existence of anything that comes under its subject without also coming under its predicate ; thus, All Ab is B denies the existence of anything that is Ab and at the same time not-B, that is, b; but Ab is Ab and also b, and hence the existence of Ab is denied. Since the existence of its subject is held to be part of the implication of a particular proposition, the above interpretation is obviously inapplicable in the case of particulars. Hence if a proposition of the form Some Ab is B is obtained, we are thrown back on the alternative that there is some inconsistency in the premisses; either some one individual premiss is self- contradictory, or the premisses are inconsistent with one another. K. L. 26 402 COMPLEX PROPOSITIONS. [PART IV. EXERCISES. 344. Shew that if No A is be or Cd, then No A is bd. [K.] 345. Give the contradictory of each of the following proposi- tions: (1) Flowering plants are either endogens or exogens, but not both ; (2) Flowering plants are vascular, and either endogens or exogens, but not both. [M.] 346. Simplify the following propositions : (1) All AS is EG or be or CD or cE or DE; (2) Nothing that is either PQ or PR is Pqr or pQs or prs or qrs or pq or pS or qR. [K.] CHAPTER III. IMMEDIATE INFERENCES FROM COMPLEX PROPOSITIONS. 347. The Obversion of Complex Propositions. The doctrine of obversion is immediately applicable to complex propositions ; and no modification of the definition of obversion already given is necessary. From any given proposition we may infer a new one by changing its quality and taking as a new predicate the contradictory of the original predicate. The proposition thus obtained is called the obverse of the original proposition. The only difficulty connected with the obversion of complex propositions consists in finding the contradictory of a complex term ; but a simple rule for performing this process has been given in section 319 : Replace all the simple terms involved by their contradictories, and throughout substitute alternative combi- nation for conjunctive and vice versd. Applying this rule to AB or ab, we have (a or 6) and (A or B), that is, Aa or Ab or aB or Bb; but since the alternants Aa and Bb involve self-contradiction, they may by rule (5) of section 326, be omitted. The obverse, therefore, of All X is AB or ab is No X is Ab or aB. As additional examples we may find the obverse of the following propositions : (1) All A is BC or DE; (2) No A is BcE or BCF\ (3) Some A is not either B or bcDEf or bcdEF. (1) All A is BG or DE yields No A is (b or c) and at the same time (d or e), or by the reduction of the predicate to a series of alternants, No A is bd or be or cd or ce. (2) No A is BcE or BCF. Here the contradictory of the predicate is (b or C or e) and (b or c or /), which yields b or Cc 262 404 COMPLEX PROPOSITIONS. [PART IV. or Cf or ce or ef. Cc may be omitted by rule (5) of section 326 ; also ef by rule (7), since ef is either Cef or cef. Hence the required obverse is All A is b or Cf or ce. (3) Some A is not either B or bcDEf or bcdEF. The obverse is Some A is b and (B or G or d or e or F) and (B or C or D or e or /) ; and by the application of the rules summa- rised in section 326 this will be found to be equivalent to Some A is bC or bDF or be or bdf. 348. The Conversion of Complex Propositions. Generalising, we may say that we have a process of conversion whenever from a given proposition we infer a new one in which any term that appeared in the predicate of the original proposition now appears in the subject, or vice versa. Thus the inference from No A is BC to No B is AC is of the nature of conversion. The process may be simply analysed as follows : No A is both B and C, therefore, Nothing is at the same time A, B, and C, therefore, No B is both A and C. The reasoning may also be resolved into a series of ordinary conversions : No A is BC, therefore (by conversion), No BC is A, that is, within the sphere of C, no B is A, therefore (by conversion), within the sphere of C, no A is B, that is, No AC is B, therefore (by conversion), No B is AC. Or, it may be treated thus, No A is BC, therefore, by section 338, rule (1), No AC is BC, therefore, also by section 338, rule (1), No AC is B, therefore (by conversion), No B is A C. Similarly it may be shewn that from Some A is BC we may infer Some B is AC. Hence we obtain the following rule : In a universal negative or a particular affirmative proposition any determinant of the subject may be transferred to the predicate or vice versa without affecting the force of the assertion. We have just shewn how from No A is BC, CHAP. III.] IMMEDIATE INFERENCES. 405 we may obtain by conversion No B is AC. Similarly, we may infer NoCis AB, No AB is C, No AC is B, No EG is A. The proposition may also be written in the form There is no ABC, or, Nothing is at the same time A, B, and C. The last of these is a specially useful form to which to bring universal negatives for the purpose of logical manipulation. In the same way from Some A is BC or BD we may infer Some AB is C or D, Some AC or AD is B, Some B is AC or AD, Some C or D is AB, Some BC or BD is A, Something is ABC or ABD. There is no inference by conversion from a universal affir- mative or from a particular negative. 349. The Contraposition of Complex Propositions. Ac- cording to our original definition of contraposition, we con- traposit a proposition when we infer from it a new proposition having the contradictory of the old predicate for its subject. Adopting this definition, the contrapositive of All A is B or C is All be is a. The process can be applied to universal affirmatives and to particular negatives. By obversion, conversion, and then again obversion, it is clear that in each of these cases we may obtain a legitimate contrapositive in its obverted form by taking as a new subject the contradictory of the old predicate, and as a new predicate the contradictory of the old subject, the proposition retaining its original quality. For example : All A is BC, therefore, Whatever is b or c is a ; Some A is not either B or C, therefore, Some be is not a. The above may be called the full contrapositive of a complex proposition. It should be observed that any proposi- 406 COMPLEX PROPOSITIONS. [PART IV. tion and its full contrapositive are equivalent to each other; in other words, we can pass back from a contrapositive to the original proposition. In dealing with complex propositions, however, it is con- venient to give to the term contraposition an extended meaning. We may say that we have a process of contraposition when from a given proposition we infer a new one in which the contradictory of any term that appeared in the predicate of the original proposition now appears in the subject, or the contradictory of any term that appeared in the subject of the original proposition now appears in the predicate. Three operations may be distinguished all of which are included under the above definition, and all of which leave us with a full equivalent of the original proposition, so that there is no loss of logical power. (1) The operation of obtaining the full contrapositive of a given proposition, as above described and defined 1 . (2) An operation which may be described as the generalisa- tion of the subject of a proposition by the addition of one or more alternants in the predicate. Thus, from All AB is G we may infer All A is b or C ; from Some AB is not either C or D we may infer Some A is not either b or C or D. For inferences of this type the following general rule may be given : Any determinant may be dropped from the subject of a universal affirmative or a particular negative proposition, if its contradictory is at the same time added as an alternant in the predicate. This rule may be established as follows: Given All AB is C (or Some AB is not (7) and these may be taken, so far as the rule in question is concerned, as types of universal affirmatives and particular negatives respectively we have by obversion No AB is c (or Some AB is c), and thence, by the rule fin- con version given in section 348, No A is Be (or Some A is Be) ; then again obverting we have All A is either b or C (or Some A is not either b or C), the required result. 1 In some cases we may desire to drop part of the information given by the complete contrapositive. Thus, from All A is BC or E we may infer Whatever is be or ce is a ; but in a given application it may be sufficient for us to know that All be is a. CHAP. III.] IMMEDIATE INFERENCES. 407 It will be observed that, as stated at the outset, these operations leave us with a proposition that is equivalent to our original proposition. There is, therefore, no loss of logical power. By the application of the above rule with regard to all the explicit determinants of the subject any universal affirmative proposition may be brought to the form Everything is X l or X 3 ...or X n \ and it will be found that by means of this transfor- mation, complex inferences are in many cases simplified and rendered easy. (3) An operation which may be described as the omission of one or more of a series of alternants in the predicate by a further particularisation of the subject. Thus, from All A is B or C we may infer All Ab is C ; from Some A is not either B or C we may infer Some Ab is not C. For inferences of this type the following general rule may be given : Any alternant may be dropped from the predicate of a universal affirmative or a particular negative proposition, if its contradictory is at the same time introduced as a determinant of the subject 1 . This rule is the converse of that given under the preceding head ; and it follows from the fact that the application of that rule leaves us with an equivalent proposition. The following may be taken as typical examples of the different operations included above under the name contra- 1 The application of this rule again leaves us with a proposition equivalent to our original proposition. The following rule, which may be regarded as a corollary from the above rule, or which may be arrived at independently, does not necessarily leave us with an equivalent : If a new determinant is introduced into tlie subject of a universal affirmative proposition (see section 342) every alternant in the predicate which contains the contradictory of this determinant may be omitted. Thus, from Whatever is A or B is C or DX or Ex, we may infer Whatever is AX or HX is C or D. The application of this rule may sometimes result in the disappearance of all the alternants from the predicate ; and the meaning of such a result is that we now have a non-existent subject. Thus, given All P is ABCD or Abed or aBCd, if we particularise the subject by making it PbC, we find that all the alternants in the predicate disappear. The interpretation is that the class PbC is non-existent, that is, No P it bC ; a conclusion which might of course have been obtained directly from the given proposition. 408 COMPLEX PROPOSITIONS. [PART IV. position : All AS is CD or de ; therefore, (1) Anything that is either cD or dE is a or b 1 ; (2) All A isbor CD or de; (3) Whatever is ABD or ABE is CD. Combinations of the second and third operations give Anything that is Ac or Ad is b or de ; Anything that is BD or BE is a or CD ; &c. In all the above cases one or more terms disappear from the subject or the predicate of the original proposition, and are replaced by their contradictories in the predicate or the subject accordingly. Only in the full contrapositive, however, is every term thus transposed. The importance of contraposition as we are now dealing with it in connexion with complex propositions is that by its means, given a universal affirmative proposition of any complexity, we may obtain separate information with regard to any term that appears in the subject, or with regard to the contradictory of any term that appears in the predicate, or with regard to any combination of such terms. Thus, given All AB is C or De, by the process described as the generalisation of the subject we have All A is b or C or De, All B is a or C or De, Everything is a or b or C or De ; the particularisation of the subject yields All A Be is De, What- ever is ABd or ABE is C, &c. ; and by the combination of these processes we have All Ac is b or De, &c. Again, the full coiitrapositive of the original proposition is Whatever is cd or cE is a or b; from which we have All c is a or b or De, Whatever is d or E is a or b or C, &c. 350. ' Summary of the results obtainable by Obversion, Con- version, and Contraposition. The following is a summary of the results obtainable by the aid of the processes discussed in the three preceding sections : (1) By obversion any proposition may be changed from the affirmative to the negative form, or vice versa. For example, All AB is CD or EF, therefore, No AB is 1 From this, the propositions, All cD is a or b, All dE is a or b, are immediately deducible. CHAP. III.] IMMEDIATE INFERENCES. 409 ce or cf or de or df; Some P is not QR, therefore, Some P is either q or r. (2) By the conversion of a universal negative proposition separate information may be obtained with regard to any term that appears either in the subject or in the predicate, or with regard to any combination of these terms. For example, from No AB is CD or EF we may infer No A is BCD or BEF, No C is ABD or ABEF, No BD is AC or AEF, &c. Also by conversion any universal negative proposition may be reduced to the form Nothing is either X l or X r ..or X n . For example, the above proposition is equivalent to the following: Nothing is either A BCD or ABEF. (3) By the conversion of a particular affirmative proposition separate information may be obtained with regard to any determinant of the subject or of the predicate, or with regard to any combination of such determinants. For example, from Some AB or AC is DE or DF we may infer Some A is BDE or BDF or CDE or CDF, Some D is ABE or ABF or ACE or ACF, Some AD is BE or BF or CE or CF, &c. Also by conversion any particular affirmative proposition may be reduced to the form Something is either X l or X r .. or X n . For example, the above proposition is equivalent to the following: Something is either A BDE or A BDF or ACDE or A CDF. (4) By the contraposition of a universal affirmative pro- position separate information may be obtained with regard to any term that appears in the subject, or with regard to the contradictory of any term that appears in the predicate, or with regard to any combination of these terms. For example, from All AB is CD or EF we may infer All A is b or CD or EF, All c is a or b or EF, All Be is a or CD, All ce is a or b, All Adfis b, &c. Also by contraposition any universal affirmative proposition may be reduced to the form Everything is either X l or X a ... orX.. 410 COMPLEX PROPOSITIONS. [PART IV. For example, the above proposition is equivalent to the following : Everything is a or b or CD or EF. (5) By the contraposition of a particular negative proposi- tion separate information may be obtained with regard to any determinant of the subject or with regard to the contradictory of any alternant of the predicate or with regard to any combi- nation of these. For example, from Some AB or AC is not either D or EF we may infer Some A is not either be or D or EF, Some d is not either a or be or EF, Some Ae or Af is not either be or D, &c. Also by contraposition any particular negative proposition may be reduced to the form Something is not either X l or X 2 . . . orX n . For example, the above proposition is equivalent to the following : Something is not either a or be or D or EF. EXERCISES. 351. No citizen is at once a voter, a householder, and a lodger ; nor is there any citizen who is neither of the three. Every citizen is either a voter but not a householder, or a householder and not a lodger, or a lodger without a vote. Are these statements precisely equivalent ? [v.] It may be shewn that each of these statements is the logical obverse of the other. They are, therefore, precisely equivalent. Let V = voter, v = not voter ; H= householder, h = not householder ; L = lodger, I = not lodger. The first of the given statements is No citizen is VHL or vhl ; therefore (by obversion), Every citizen is either v or h or I and is also either V or H or L; therefore (combining these possibilities), Every citizen is either Hv or Lv or Vh or Lh or VI or HI. But (by the law of excluded middle), Hv is either HLv or Hlv ; therefore, Hv is Lv or HI. Similarly, Lh is Vh or Lv; and VI is HI or Vh. Therefore, Every citizen is Vh or HI or Lv, which is the second of the given statements. CHAP. III.] EXERCISES. 411 Again, starting from this second statement, it follows (by obversion) that No citizen is at the same time v or H, h or L, I or V ; therefore, No citizen is vh or vL or HL, and at the same time I or V; therefore, No citizen is vhl or VHL, which brings us back to the first of the given statements. 352. Given " All D that is either B or C is A," shew that " Everything that is not-.A is either not-B and not-(7 or else it is not-Z)." [De Morgan.] This example and those given in section 359 are adapted from De Morgan, Syllabus, p. 42. They are also given by Jevona, Studies, p. 241, in connexion with his Equational Logic. They are all simple exercises in contraposition. We have What is either BD or CD is A ; therefore, All a is (b or d) and (c or d); therefore, All a is be or d. 353. Infer all that you possibly can by way of contra- position or otherwise, from the assertion, All A that is neither B nor C is X. [R.] The given proposition may be thrown into the form Everything is either a or or C or X ; and it is seen to be symmetrical with regard to the terms a, B, C, X, and therefore with regard to the terms A, b, c, x. We are sure then that anything that is true of A is true 'mutatis mutandis of 6, c, and x, that anything that is true of Ab is true mutatis mutandis of any pair of the terms, and similarly for combinations three and three together. We have at once the four symmetrical propositions : All A is B or or X; (1) All b is a or C or X ; (2) All c is a or B or X ; (3) All x is a or B or C. (4) Then from (1) by particularisation of the subject : AllAbisCorX; (i) with the five corresponding propositions : All Ac is B or X ; (ii) AH Ax is B or C ; (iii) All be is a or X ; (iv) All bx is a or C ; (v) AU ex is a or B. (vi) 412 COMPLEX PROPOSITIONS. [PART IV. By a repetition of the same process, we have All Abe is X (which is the original proposition over again); (a) and corresponding to this : All Abx is C ; (/3) All Acx is B ; (y) All box is a. (8) It will be observed that the following are pairs of full contra- positives :-(!) (8), (2) ( y ), (3) (/?), (4) (a), (i) (vi), (ii) (v), (iii) (iv). A further series of propositions may be obtained by obverting all the above; and as there has been no loss of logical power in any of the processes employed we have in all thirty propositions that are equivalent to one another. 354. If AB is either Cd or cDe, and also either eF or H, and if the same is true of BH, what do we know of that which is El [K.] Whatever is AB or BH is (Cd or cDe) and (eF or H}; therefore, Whatever is AB or BH is CdeF or cDeF or CdH or cDeH", therefore, Whatever is ABE or BEE is CdH ; therefore, All E is CdH or b or ah. 355. Given A is EG or BDE or BDF, infer descriptions of the terms Ace, Acf, ABcD. [Jevons, Studies, pp. 237, 238.] In accordance with rules already laid down, we have immediately Ace is BDF; Acf is BDE ; ABcD is E or F. 356. Find the obverse of each of the following propositions : (1) Nothing is A, B, or C ; (2) All A is Be or bD ; (3) No Ab is CDEf or Cd or cDf or cdE ; (4) No A is BCD or Bed ; (5) Some A is not either bed or Cd or cD. [K.] 357. Shew that the two following propositions are equivalent to each other: No A is B or CD or CE or EF; All A is bCde or bcEf or bee. [K.] 358. Contraposit the proposition, All A that is neither B nor C is both X and Y. [L.] CHAP. III.] EXERCISES. 413 359. Find the full contrapositive of each of the following pro- positions : (1 ) AH A is eit/ier EC or BD ; (2) Whatever is B or CD or CE is A (3) Whatever is either B or C and at the same time either D or E is A; (4) Whatever is A or BC and at the same time either D or EFis X. [De Morgan.] 360. Find the full contrapositive of each of the following propositions : All A is BCDe or bcDe Some AS is not either CD or cDE or de ; WJiatever is AB or bC is aCd or Acd ; Where A is present along with either B or C, D is present and C absent or D and E are both absent ; Some ABC or abc is not either DEF or def. [K.] 361. What information can you obtain about c, Be, Af, D, from the proposition All AB is CD or EF'\ [M.] 362. Establish the following : Where B is absent, either A and C are both present or A and D are both absent ; therefore, where C is absent, either B is present or D is absent. [K.] 363. Establish the following : Where A is present, either B and C are both present or C is present D being absent or C is present F being absent or H is present ; therefore, where C is absent, A cannot be present H being absent. [K.] 364. Given that WJiatever is PQ or AP is bCD or abdE or aBCdE or Abed, shew that (1) All abP is CD or dE or q; (2) All DP isbC or aq ; (3) WJiatever is B or Cd or cD is a or p; (4) All B is C or p or aq; (5) All Cd is a or p ; (6) All AB is p ; (7) If ae is c or d it is p or q; (8) If BP is c or D or e it is aq. [K.] 365. Bring the following propositions to the form Everything is eitJier X t or X 3 ... or X n : Whatever is Ac or ab or aC is bdf or deF; Nothing that is A and at the same time either B or C is D or dE. [K.] 366. Shew that the results in section 340 follow from those in section 339 by the rules of contradiction and contraposition. [K.] CHAPTER IV. THE COMBINATION OF COMPLEX PROPOSITIONS. 367. The Problem of combining Complex Propositions. Two or more complex propositions given in simple combination, either conjunctive or alternative, constitute a compound propo- sition. Hence the problem of dealing with a combination of complex propositions so as to obtain from them a single equivalent complex proposition, which is the problem to be considered in the present chapter, is identical with that of passing from a compound proposition to an equivalent complex proposition ; and it is, therefore, the converse of the problem which was partially discussed in sections 339, 340. The latter problem, namely, that of passing from a complex to an equi- valent compound proposition, will be further discussed in chapter 6. 368. The Conjunctive Combination of Universal Affirma- tives. We may here distinguish two cases according as the propositions to be combined have or have not the same subject. (1) Universal affirmatives having the same subject. AllXisP^or P 2 ...... or P m , AllXisQ l0 rQ 2 ...... orQ n , may for our present purpose be taken as types of universal affirmative propositions having the same subject. By con- junctively combining their predicates, thus, All X is (P l or P 8 ... or P m ) and also (Q l or Q. 2 ... or Q H ), that is, All X is P 1 Q 1 or P& ...or P,Q n or orP m Q l0 rP m Q 2 ...orP m Q n , CHAP. IV.] COMBINATION OF COMPLEX PROPOSITIONS. 415 we may obtain a new proposition which is equivalent to the conjunctive combination of the two original propositions; it sums up all the information which they jointly contain, and we can pass back from it to them. In almost all cases of the conjunctive combination of terms there are numerous opportunities of simplification ; and after a little practice, the student will find it unnecessary to write out all the alternants of the new predicate in full. The following are examples : (i) All X is A B or bee, AllXisaBCorDE; therefore, All X is ABDE. It will be found that all the other combinations in the predicate contain contradictories. (ii) All X is A or Be or D, All X is aB or Be or Cd ; therefore, All X is Be or aBD or A Cd. (iii) Everything is A or bd or cE, Everything is AC or aBe or d; therefore, Everything is AC or bd or Ad or cdE. (2) Universal affirmatives having different subjects. Given the conjunctive combination of two universal affir- mative propositions with different subjects a new complex proposition may be obtained by conjunctively combining both their subjects and their predicates. Thus, if All X is P, or P 8 and All Y is Q, or Q 8> it follows that All XY is P$, or P& or P 2 Qj or P 2 Q 2 . But in this case the new proposition thus obtained is not equivalent to the conjunctive combination of the original propositions ; and we cannot pass back from it to them. A single complex proposition which sums up all the infor- mation contained in the original propositions may, however, be obtained by first reducing each of them to the form Everything is X l or X a ... or X n , and then conjunctively combining their predicates. 369. The Conjunctive Combination of Universal Negatives. Here again we may distinguish two cases according as the propositions to be combined have or have not the same subject. 416 COMPLEX PROPOSITIONS. [PART IV. (1) Universal negatives having the same subject. No X is P l or P, orP m , NoXisQ l0 rQ 2 or Q n , may for our present purpose be taken as types of universal negative propositions having the same subject. Given these two propositions in conjunctive combination, a new complex proposition may be obtained by alternatively combining their predicates. Thus, No X is P 1 or P 2 or P m or Q t or Q 2 or Q n . This new proposition is equivalent to the two original propo- sitions taken together, so that we can pass back from it to them. The process of combining the predicates is again likely to give opportunities of simplification of which advantage should be taken. The following are examples : (i) No A is be, No A is Cd ; therefore, No A is be or Cd. (ii) No X is either aB or aC or bC or aE or bE, No X is either Ad or Ae or bd or be or cd or ce ; therefore, No X is either a or b or d or e 1 . (ni) Nothing is aBG or aCD or aBe or aDe, Nothing is AcD or abD or bcD or aDE or cDE ; therefore, Nothing is aBC or aD or cD or aBe. (2) Universal negatives having different subjects. Given the conjunctive combination of two universal negative propositions with different subjects a new complex proposition may be obtained by conjunctively combining their subjects and alternatively combining their predicates. Thus, if No X is P^ or P 2 and No Y is Q 1 or Q z , it follows that No XY is P t or P 2 or Q 1 or Q 2 . In this case, as in that considered in the pre- ceding section, the inferred proposition is not equivalent to the premisses ; and we cannot pass back from it to them. A single complex proposition which sums up all the infor- mation contained in the original propositions may, however, be obtained by first reducing each of them to the form Nothing is X 1 or X. 2 ... or X n , and then alternatively combining their predicates. 1 Compare section 328. CHAP. IV.] COMBINATION OF COMPLEX PROPOSITIONS. 417 370. The Conjunctive Combination of Universals with Par- ticulars of the same Quality. We may here consider, first, affirmatives, and then, negatives. (1) Affirmatives. From the conjunctive combination of a universal affirmative and a particular affirmative having the same subject, a new particular affirmative proposition may be obtained by conjunctively combining their predicates. If All X is P l or P 2 and Some X is Q l or Q 2 , it follows that Some X is P,Qj or P,Q 2 or P 2 Q, or P 2 Q 2 . It will be observed that the particular premiss affirms the existence of either XQ V or XQ, and therefore certainly guarantees the existence of X ; and the universal premiss implies that if X exists then either XP l or XP 2 exists. We can pass back from the conclusion to the particular premiss, but not to the universal premiss. The conclusion is, therefore, not equivalent to the two premisses taken together. A new complex proposition cannot be directly obtained from the conjunctive combination of a universal affirmative and a particular affirmative having different subjects. The propositions may, however, be reduced respectively to the forms Everything is P l or P 2 ... or P m , Something is Q, or Q 2 ... or Q n , and their predicates may then be conjunctively combined in accordance with the above rule. (2) Negatives. From the conjunctive combination of a universal negative and a particular negative having the same subject, a new particular negative proposition may be obtained by the alternative combination of their predicates. If No X is either P l or P 2 and Some X is not either Q, or Q 2 , it follows that Some X is not either P l or P 2 or Q l or Q 2 . The validity of this process is obvious since the particular premiss guarantees the existence of X. By obversion it can also be exhibited as a corollary from the rule given above in regard to affirmatives. We can again pass back from the conclusion to the particular premiss, but not to the universal premiss. With regard to the conjunctive combination of universal negatives and particular negatives having different subjects, the remarks made concerning affirmatives apply mutatis mutandis. K. L. 27 418 COMPLEX PROPOSITIONS. [PART IV. 371. The Conjunctive Combination of Affirmatives with Negatives. By first obverting one of the propositions, the conjunctive combination of an affirmative with a negative may be made to yield a new complex proposition in accordance with the rules given in the preceding sections. For example, (1) All X is A or B, No X is aC, therefore, All X is A or Be; (2) Everything is P or Q, Nothing is Pq or pR, therefore, Nothing is pR or q ; (3) All X is AB or bee, Some X is not either aBC or DE, therefore, Some X is ABd or A Be or bee. 372. The Conjunctive Combination of Particulars with Particulars. Particulars cannot to any purpose be con- junctively combined with particulars so as to yield a new complex proposition. It is true that from Some X is P, or P 2 and some X is Q 1 or Q 2 , we can pass to Some X is P l or P 2 or Qi or Qv But this is a mere weakening of the information given by either of the premisses singly; and by the rule that an alternant may at any time be introduced into an undistributed term (section 342), it could equally well be inferred from either premiss taken by itself. Again from Some X is not either P x or P 2 and some X is not either Q l or Q 2 , we can pass to Some X is not either P^, or P^ 2 or P 2 Q, or P 2 Q 2 . But similar remarks again apply, since we have already found that a determinant may at any time be introduced into a distributed term. 373. The Alternative Combination of Universal Proposi- tions. Given a number of universal propositions as alternants in a compound alternative proposition we cannot obtain a single equivalent complex proposition 1 . From the compound proposition Either all A is P, or P 2 or all A is Q 1 or Q 2 we can indeed infer All A is P l or P 2 or Q^ or Q 2 ; but we cannot pass back from this to the original proposition. 374. The Alternative Combination of Particular Proposi- tions. It follows from the equivalences shewn in section 340 1 Compare section 339. CHAP. IV.] COMBINATION OF COMPLEX PROPOSITIONS. 419 that a compound alternative proposition in which all the alternants are particular can be reduced to the form of a single complex proposition. If all the alternants of the compound proposition have the same subject and are all affirmative, their predicates must be alternatively combined in the complex proposition; if they all have the same subject and are all negative, their predicates must be conjunctively combined in the complex proposition. If the alternants have different subjects, they must all be reduced to the form Something is ... before their predicates are combined ; if they differ in quality, recourse must be had to the process of obversion. It is un- necessary to discuss these different cases in detail, but the following may be taken as examples : (i) Some X is P or some X is Q = Some X is P or Q; (ii) Some X is not P or some X is not Q = Soriie X is not PQ ; (iii) Some X is P or some Y is Q = Something is XP or YQ ; (iv) Some X is P or some Y is not Q = Something is XP or Yq. 375. The Alternative Combination of Particulars with Universals. From a compound alternative proposition in which some of the alternants are particular and some universal, we can infer a particular complex proposition; but in this case we cannot pass back from the complex proposition to the compound proposition. The following are examples : (1) All A is P or some A is Q, therefore, Something is a or PorQ 1 ; (2) All A is P or some B is not Q, therefore, Something is a or P or Bq. EXERCISES. 376. Reduce the propositions All P is Q, No Q is R, to such a form that the universe of discourse appears as the subject of each ; and then combine them into a single complex proposition. How is your result related to the ordinary syllogistic conclusion No P is R ? w 1 We cannot infer Some A is P or Q, since this implies the existence of A, whereas the non-existence of A is compatible with the premiss. 272 420 COMPLEX PROPOSITIONS. [PART IV. 377. Combine the following propositions into a single equivalent complex proposition: All X is either A or b; No X is either AC or acD or CD; All a is B or x. [K.] 378. Every voter is both a ratepayer and occupier, or not a ratepayer at all; If any voter who pays rates is an occupier, then he is on the list; No voter on the list is both a ratepayer and an occupier. Examine the results of combining these three statements. [v.] 379. Every A is BC, except when it is D ; everything which is not A is D ; what is both C and D is B ; and every D is C. What can be determined from these premisses as to the contents of our universe of discourse ] [M.] CHAPTER V. INFERENCES FROM COMBINATIONS OF COMPLEX PROPOSITIONS. 380. Conditions under which a universal proposition affords information in regard to any given term. The problem to be solved in order to determine these conditions may be formu- lated as follows: Given any universal proposition, and any term X, to discriminate between the cases in which the propo- sition does and those in which it does not afford information with regard to this term. In the first place, it is clear that if the proposition is to afford information in regard to any term whatever it must be non-formal. If it is negative, let it by obversion be made affirmative. Then it may be written in the form Whatever is A^ A^ . . . or B^B^ ...or &c. is P,P 2 . . . or Q^ . . . or &c., where A lt B I} P lt Q t , &c. are all simple terms 1 . As shewn in section 339, this may be resolved into the independent propositions : All AiA z ...is PiP 2 ... or QiQ 2 ... or &c.\ All B,B 2 ...is PjP., ...or 44 or &c. ; &c. &c. &c. ; in none of which is there any alternation in the subject. These propositions may be dealt with separately, and if any one of them affords information with regard to X, then of course the original proposition does so. We have then to consider a proposition of the form All A t A, ...A n is P,P 2 ... or Q& ... or <&c.', 1 So that both subject and predicate consist of a series of alternants which themselves contain only simple determinants ; that is, there is no alternant of the form (A or B) (C or D). 422 COMPLEX PROPOSITIONS. [PART IV. and this proposition may by contraposition be reduced to the form Everything is c^ or a 2 . . . or a n or PJP?. ...or Q } Q 2 ...or &c. ; from which may be inferred All X is <*! or a. 2 . . . or a n or PiP 2 ... or Q^ 2 ...or &c. Any alternant in the predicate of this proposition which contains x may clearly be omitted. If all the alternants contain #, then the information afforded with regard to X is that it is non-existent. If some alternants are left, then the proposition will afford information concerning X unless, when the predicate has been simplified to the fullest possible extent 1 , one of the alternants is itself X uncombined with any other term, in which case it is clear that we are left with a merely formal proposition. Now one of these alternants will be X in the following cases, and only in these cases : First, If one of the alternants in the predicate of the original proposition, when reduced to the affirmative form, is X. Secondly, If any set of alternants in the predicate of the original proposition, when reduced to the affirmative form, constitutes a development of X, since any such development (for example, AX or aX, ABX or AbX or aBX or abX) is equivalent to X simply 2 . Thirdly, If one of the alternants in the predicate of the original proposition, when reduced to the affirmative form, contains X in combination solely with some determinant that is also a determinant of the subject or the contradictory of some other alternant of the predicate ; since in either of these cases such alternant is equivalent to X simply 3 . Fourthly, If one of the determinants of the subject is cc ; since in that case we shall after contraposition have X as one of the alternants of the predicate. The above may be summed up in the following proposi- 1 All superfluous terms being omitted, but the predicate still consisting of a series of alternants which themselves contain only simple determinants. 2 See section 323. 3 By section 338, rule (2), All AB is AX or D is equivalent to All AB is X or D ; and by the law of exclusion (section 325) A or aX is equivalent to A or X. CHAP. V.] COMPLEX PROPOSITIONS. 423 tion : Any non-formal universal proposition will afford informa- tion with regard to any term X, unless, after it has been brought to the affirmative form, (1) one of the alternants of the predicate is X, or (2) any set of alternants in the predicate constitutes a development of X, or (3) any alternant of the predicate contains X in combination solely with some determinant that is also a determinant of the subject or the contradictory of some other alternant of the predicate, or (4) x is a determinant of the subject. If, after the proposition has been reduced to the affirmative form, all superfluous terms are omitted in accordance with the rules given in chapters 1 and 2, then the criterion becomes more simple : Any non-formal universal proposition will afford information with regard to any term X, unless (after it has been brought to the affirmative form and its predicate has been so simplified that it contains no superfluous terms) X is itself an alternant of the predicate or a: is a determinant of the subject 1 . If instead of X we have a complex term XYZ, then no determinant of this term must appear by itself as an alternant of the predicate, and there must be at least one alternant in the subject which does not contain as a determinant the con- tradictory of any determinant of this complex term ; i.e., no alternant in the predicate must be X, Y, or Z, or any combina- tion of these, and some alternant of the subject must contain neither x, y, nor z. The above criterion is of simple application. 381. Information jointly afforded by a series of universal propositions with regard to any given term. The great majority of direct problems 2 involving complex propositions may be brought under the general form, Given any number of universal propositions involving any number of terms, to determine what is all the information that they jointly afford with regard to any given term or combination of terms. If the student turns to 1 It may be added that every universal proposition, unless it be purely formal, will afford information either mth regard to X or with regard to x. For if both A' and x appear as alternants of the predicate, or as determinants of the subject of a universal affirmative proposition, then the proposition will neces- sarily be formal. 2 Inverse problems will be discussed in the following chapter. 424 COMPLEX PROPOSITIONS. [PART IV. Boole, Jevons, or Venn, he will find that this problem is treated by them as the central problem of symbolic logic 1 . A general method of solution is as follows : Let X be the term concerning which information is desired. Find what information each proposition gives separately with regard to X, thus obtaining a new set of propositions of the form All X is P t or P 2 ... or P n . This is always possible by the aid of the rules for ob version and contraposition given in chapter 3. By the rule given in the preceding section those propositions which do not afford any information at all with regard to X may at once be left out of account. Next let the propositions thus obtained be combined in the manner indicated in section 368. This will give the desired solution. If information is desired with regard to several terms, it will be convenient to bring all the propositions to the form Everything is P a or P 2 . . . or P n ; and to combine them at once, thus summing up in a single proposition all the information given by the separate proposi- tions taken together. From this proposition all that is known concerning X may immediately be deduced by omitting every alternant that contains x, all that is known concerning Y by omitting every alternant that contains y, and so on. The method may be varied by bringing the propositions to the form No X is Qi or Q 2 ... or Q n , or to the form Nothing is Q or Q 2 ... or Q n , then combining them as in section 369, and (if an affirmative 1 "Boole," says Jevons, "first put forth the problem of Logical Science in its complete generality: Given certain logical premisses or conditions, to deter- mine the description of any class of objects under those conditions. Such was the general problem of which the ancient logic had solved but a few isolated cases the nineteen moods of the syllogism, the sorites, the dilemma, the disjunctive syllogism, and a few other forms. Boole shewed incoutestably that it was possible, by the aid of a system of mathematical signs, to deduce the conclusions of all these ancient modes of reasoning, and an indefinite number of other conclusions. Any conclusion, in short, that it was possible to deduce from any set of premisses or conditions, however numerous and complicated, could be calculated by his method" (Philosophical Transactions, 1870). Compare also Principles of Science, chapter 6, 5. CHAP. V.] INFERENCES FROM COMPLEX PROPOSITIONS. 425 solution is desired) finally obverting the result. It will depend on the form of the original propositions whether this variation is desirable 1 . In an equational system of symbolic logic, a solution with regard to any term X generally involves a partial solution with regard to x also. In employing the above methods, x must be found separately. It may be added that the complete solution for X and that for x sum up between them the whole of the information given by the original data ; in other words, they are, taken together, equivalent to the given premisses*. The following may be taken as a simple example of the first of the above methods. It is adapted from Boole (Laws of Thought, p. 118). " Given 1st, that wherever the properties A and B are combined, either the property C, or the property D, is present also, but they are not jointly present ; 2nd, that wherever the properties B and G are combined, the properties A and D are either both present with them, or both absent; 3rd, that wherever the properties A and B are both absent, the pro- perties C and D are both absent also; and vice versa, where the properties C and D are both absent, A and B are both absent also. Find what can be inferred from the presence of A with regard to the presence or absence of 5, G, and D." The premisses may be written as follows : (1) All AB is Cd or cD; (2) All BG is AD or ad; (3) All ab is cd; (4) All cd is ab. Then, from (1), All A is b or Cd or cD ; and from (2), All A is b or c or D ; therefore (by combining these), All A is b or cD ; (3) gives no information regarding A (see the preceding section) ; but by (4), All A is C or D; 1 This second method is analogous to that which is usually employed by Dr Venn in his Symbolic Logic. Both methods bear a certain resemblance to Jevons's Indirect Method ; but neither of them is identical with that method. 2 Having determined that All X i P and that All x is q, we may by contraposition bring the latter proposition to the form All Q is X, when it may be found that P and Q have some alternants in common. These alternants are the terms which (in Boole's system) are taken in their whole extent in the equation giving .Y; and the solution thus obtained is closely analogous to that given by any equational system of symbolic logic. 426 COMPLEX PROPOSITIONS. [PART IV. therefore, All A is bC or bD or cD ; and, since bD is by development either bCD or bcD, this becomes All A is bC or cD 1 . This solves the problem as set. Proceeding also to deter- mine a, we find that (1) gives no information with regard to this term ; but by (2), All a is b or c or d; and by (3), All a is B or cd; therefore, All a is Be or Bd or cd. Again by (4), All a is b or C or D. Therefore, All a is BCd or BcD or bed ; and by contraposition, Whatever is Bed or bC or bD or CD is A 2 . 382. The Problem of Elimination. By elimination in logic is meant the omission of certain elements from a pro- position or set of propositions with the object of expressing more directly and concisely the connexion between the elements which remain. An example of the process is afforded by the ordinary categorical syllogism, where the so-called middle term is eliminated. Thus, given the premisses All M is P, All S is M, we may infer All 8 is MP ; but if we desire to know the relation between 8 and P independently of M we are content with the less precise but sufficient statement All 8 is P ; in other words, we eliminate M. Elimination has been considered by some writers to be absolutely essential to logical reasoning. It is not, however, necessarily involved either in the process of contraposition or in the process discussed in the preceding section ; and from the point of view of the formal logician it is difficult to see how the name of inference can be denied to these processes. We must, therefore, refuse to regard elimination as of the essence of reasoning, although it may usually be involved therein 3 . 383. Elimination from Universal Affirmatives. Any uni- versal affirmative proposition (or, by combination, any set of universal affirmative propositions) involving the term X and 1 We shall find that by the elimination of D, in accordance with the rule given in the next section but one, we have All A is bC or c. This is Boole's result. 2 Taking into account the result arrived at above with regard to A, it will be seen that this may be resolved into Whatever is bC or bD is A and Nothing is BCD or Bed. These two propositions taken together with the solution for A are equivalent to the original premisses. 3 Compare sections 155, 156. CHAP. V.] ELIMINATION OF TERMS. 427 its contradictory x may by contraposition be reduced to the form Everything is PX or Qx or R, where P, Q, R are them- selves simple or complex terms not involving either X or x; and since by the rule given in section 341 a determinant may at any time be omitted from an undistributed term, we may eliminate X (and x) from this proposition by simply omitting them, and reducing the proposition to the form Everything is PorQorR 1 . We must, however, here admit the possibility of P, Q, R being of the forms A or a, Aa. These are equivalent respec- tively to the entire universe of discourse and to nothing. Thus, if P is of the form A or a, and Q is of the form Aa, our proposition will before elimination more naturally be written Evert/thing is X or R', if Q is of the form A or a, and R of the form Aa, it will more naturally be written Everything is PX or x. It follows that if either P or Q is of the form A or a (that is, if either P or Q is equivalent to the entire universe of discourse), the proposition resulting from elimination will not afford any real information, since it is always true a priori that Everything is A or a or &c. Thus we are practically unable to eliminate X from such a proposition as All A is X or EG. The following may be given as an example of elimination from universal affirmatives. Let it be required to eliminate X (together with x) from the propositions All P is XQ or xR, Whatever is XorRispor XQR. Combining these propositions, we have Everything is XQR or p; therefore, by elimination, Everything is QR or p, that is, All P is QR. It will be observed that P (together with p) cannot be eliminated from the above propositions. 384. Elimination from Universal Negatives. Any universal negative proposition (or, by combination, any set of universal negative propositions) containing the term X and its contradic- tory a; may by conversion be reduced to the form Nothing is PX or Qx or R, where P, Q, R are interpreted as in the preceding 1 We might also proceed as follows: Solve for A' and for x, as in section 381, so that we have All X is A, All x is B, where A and B are simple or complex terms not involving either A' or x. Then, since Everything is X or x, we shall have Everything is A or B, and this will be a proposition containing neither X nor x. 428 COMPLEX PROPOSITIONS. [PART IV. section. Here we might, in accordance with the rule given in section 341, simply omit the alternants PX, Qx, leaving us with the proposition Nothing is R. This, however, is but part of the information obtainable by the elimination of X. We have also No X is P, and No Q is x, that is, All Q is X] whence by a syllogism in Celarent we may infer No Q is P. The full result of the elimination is, therefore, given by the proposition Nothing is PQ or R\ The following is an example: Let it be required to eliminate X from the propositions No P is Xq or xr, No X or R is xP or Pq or Pr. Combining these propositions we have Nothing is XPq or XPr or xP or PqR\ therefore, by elimination in ac- cordance with the above rule, Nothing is Pq or Pr, that is, No P is q or r. 385. Elimination from Particular Affirmatives. Any particular affirmative proposition involving the term X may by conversion be reduced to the form Something is either PX or Qx or R, where P, Q, R are interpreted as in section 383. We may here immediately apply the rule given in section 341 that a determinant may at any time be omitted from an undistributed term ; and the result of eliminating X is accordingly Something is either P or Q or R*. 1 Compare Mrs Ladd Franklin's Essay on The Algebra of Logic (Studies in Logic by Members of the Johns Hopkins University). The same conclusion may be deduced by obversion from the result obtained in the preceding section. Nothing is PX or Qx or R becomes by obversion Everything is prX or qrx. Therefore, by the elimination of X, Everything is pr or qr ; and this proposition becomes by obversion Nothing is PQ or R. Another method by which the same result may be obtained is as follows: By developing the first alternant with reference to Q and the second with reference to P, Nothing is PX or Qx or R becomes Nothing is PQX or PqX or PQx or pQx or R. But PQX or PQx is reducible to PQ, and omitting PqX and pQx, we have Nothing is PQ or R. It is interesting to observe that the above rule for elimination from negatives is equivalent to Boole's famous rule for elimination. In order to eliminate X from the equation F(A') = 0, he gives the formula F(l) F(0) = 0. Now any equation containing X can be brought to the form AX+Bx+ (7=0, where A, B, C are independent of X. Applying Boole's rule we have (A + C) (B + C) = 0, that is, AB + (7=0; and this is precisely equivalent to the rule given in the text. 2 Thus the rule for elimination from particular affirmatives is practically identical with the rule for elimination from universal affirmatives. CHAP. V.] ELIMINATION OF TERMS. 429 386. Elimination from Particular Negatives. Any parti- cular negative proposition involving the term X may by con- traposition be reduced to the form Something is not either PX or Qx or R. By the development of the first alternant with reference to Q and that of the second alternant with reference to P, this proposition becomes Something is not either PQX or PqX or PQx or pQx or R. But PQX or PQx is reducible to PQ, and the alternants PqX, pQx may by the rule given in section 341 be omitted. Hence we get the proposition Something is not either PQ or R, from which X has been eliminated 1 . 387. Order of procedure in the process of elimination. Schroder (Der Operationskreis des Logikkalkuls, p. 23) points out that first to eliminate and then combine is not the same thing as first to combine and then eliminate. For, as a rule, if a term X is eliminated from several isolated propositions the combined results give less information than is afforded by first combining the given propositions and then effecting the required elimination. There are indeed many cases in which we cannot eliminate at all unless we first combine the given propositions. This is of course obvious in syllogisms ; and we have a similar case if we take the premisses Everything is A or X, Everything is B or x. We cannot eliminate X from either of these proposi- tions taken by itself, since in each of them X (or x) appears as an isolated alternant. But by combination we have Every- thing is Ax or BX; and this by the elimination of X becomes Everything is A or B*. There are other cases in which elimination from the separate propositions is possible, but where this order of procedure leads to a weakened conclusion. Take the propositions Everything is 1 Thus the rule for elimination from particular negatives is practically identical with the rule for elimination from universal negatives. The same rule may be deduced by obversion from the result obtained in the preceding section. Something ii not either PX or Qx or R; therefore, Something is either prX or qrx ; therefore, Something is either pr or qr ; therefore, Something is not either PQ or E. 2 Working with negatives we get the same result. Taking the proposi- tions Nothing is ax, Nothing is bX, separately, we cannot eliminate X from either of them. But combining them in the proposition Nothing is ax or bX t we are able to infer Nothing is ab. 430 COMPLEX PROPOSITIONS. [PART IV. AX or Bx, Everything is CX or Dx. By first eliminating X and then combining, we have Everything is AC or AD or BC or BD. But by first combining and then eliminating X, our conclusion becomes Everything is AC or BD which gives more information than is afforded by the previous conclusion. EXERCISES. 388. Suppose that an analysis of the properties of a particular class of substances has led to the following general conclusions, namely : 1st, That wherever the properties A and B are combined, either the property C, or the property D, is present also ; but they are not jointly present; 2nd, That wherever the properties B and C are combined, the properties A and D are either both present with them , or both absent ; 3rd, That wherever the properties A and B are both absent, the properties C and D are both absent also; and vice versa, where the properties C and D are both absent, A and B are both absent also. Shew that wherever the property A is present, the properties B and C are not both present ; also that wherever B is absent while C is present, A is present. [Boole, Laws of Thought, pp. 118 to 120; compare also Venn, Symbolic Logic, pp. 276 to 278.] A solution of this problem has already been given in section 381. We may also proceed as follows. The premisses are : All AB is Cd or cD, (i) All BC is AD or ad, (ii) All ab is cd, (iii) All cd is ab. (iv) By (i), No A Bis CD, therefore, No A is BCD. (1) By (ii), No BC is Ad, therefore, No A is BCd. (2) Combining (1) and (2), it follows immediately that No A is BC. Boole also shews that All bC is A. This is a partial contra- positive of (iii). We have so far not required to make use of (iv) at all. CHAP. V.] EXERCISES. 431 389. Given the same premisses as in the preceding section, prove that : (1) Wherever the property C is found, either the property A or the property B will be found with it, but not both of them together ; (2) If the property B is absent, either A and C will be jointly present, or C will be absent ; (3) If A and C are jointly present, B will be absent. [Boole, Laws of Thought, p. 12.9.] First, By (i), All C is a or b or d; by (ii), All C is a or b or D; therefore, All C is a or b. Also, by (iii), All C is A or B; therefore, All C is Ab or aB. (1) Secondly, By (iii), All b is A or c, therefore, by section 325, All b is AC or c. (2) Thirdly, from (1) it follows immediately that All AC is b. (3) The given premisses may all be summed up in the proposition : Everything is AbC or AbD or aBCd or abed or BcD. From this, the above special results and others follow immediately. 390. Given that everything is either Q or R, and that all R is Q, unless it is not P, prove that all P is Q. . [K.] The premisses may be written as follows: (1) All r is Q, (2) All PR is Q. By (1), All Pr is Q, and by (2), Att PRisQ; but All P is Pr or PR; therefore, All P is Q. 391. Where A is present, B and C are either both present at once or absent at once; and where G is present, A is present. Describe the class not-B under these conditions. [Jevons, Studies, p. 204.] The premisses are (1) All A is BC or be, (2) All C is A. By (1) All b is a or c, and by (2) All b is A or c; therefore, All b is c. 392. It is known of certain things that (1) where the quality A is, B is not; (2) where B is, and only where B is, C and D are. Derive from these conditions a description of the class of things in which A is not present, but C is. [Jevons, Studies, p. 200.] 432 COMPLEX PROPOSITIONS. [PART IV. The premisses are: (1) All A is b; (2) All B is CD; (3) Att CD is B. No information regarding aC is given by (1). But by (2), All aC is b or D ; and by (3), All aC is B or d. Therefore, All aC is BD or bd. 393. Taking the same premisses as in the previous section, draw descriptions of the classes Ac, ab, and cD. [Jevons, Studies, p. 244.] By (1), Everything is a or b; and by (2), Everything is b or CD. Therefore, Everything is aCD or b; and by (3), Everything is B or c oi' d. Therefore, Everything is aBCD or be or bd. Hence we infer immediately All Ac is b, All ab is c or d, All cD is b. 394. Every A is one only of the two B or C; D is both B and G except when B is E and then it is neither ; therefore, no A is D. [De Morgan, Formal Logic, p. 124.] This example, originally given by De Morgan (using, however, different letters), and taken by Professor Jevons to illustrate his symbolic method (Principles of Science, chapter 6, 10; Studies in Deductive Logic, p. 203), is chosen by Professor Groom Robertson to shew that "the most complex problems can, as special logical questions, be more easily and shortly dealt with upon the principles and with the recognised methods of the traditional logic" than by Jevons's system. " The mention of E as E has no bearing on the special conclusion A is not D and may be dropt, while the implication is kept in view ; otherwise, for simplification, let BC stand for 'both B and C,' and be for ' neither B nor (?.' The premisses then are, (1) D is either BC or be, (2) A is neither BC nor be, which is a well-recognised form of Dilemma with the conclusion A is not I . Or, by expressing (2) as A is not either BC or be, the con- clusion may be got in Camestres. If it be objected that we have here by the traditional processes got only a special conclusion, it is a sufficient reply that any conclusion by itself must be special. What other conclusion from these premisses is the common logic powerless to obtain?" (Mind, 1876, p. 222.) The solution is also obtainable as follows : By the first premiss, All A is Be or bC, and by the second, All A is BC or be or d; there- fore, All A is Bed or bCd, therefore, All A is d. CHAP. V.] EXERCISES. 433 395. There is a certain class of things from which A picks out the ' X that is Z, and the Y that is not Z,' and B picks out from the remainder 'the Z which is F and the X that is not F' It is then found that nothing is left but the class 'Z which is not X.' The whole of this class is however left. What can be determined about the class originally ? [Venn, Symbolic Logic, pp. 267, 8.] The chief difficulty in this problem consists in the accurate statement of the premisses. Call the original class W. We then have All W is XZ or Yz or YZ or Xy or xZ, that is, All W is X or Y or Z ; (1) AllxZ is W; (2) No xZ is WXZ or WYz or WYZ or WXy, that is, No xZ is WYZ. (3) We may now proceed as follows: By (1), All W is X or Y or Z; and by (3), All W is X or y or z. Therefore, All W is X or Yz or yZ. (2) affords no information regarding the class W, except that everything that is Z but not X is contained within it. 396. Shew what may be inferred as a possible description of warm-blooded vertebrates from the following, and state whether any of the information there given is superfluous for the purpose : (1) All vertebrates may be divided into warm- blooded, and cold-blooded, and all produce their young in but one of two ways, i.e., are either viviparous or oviparous ; (2) No feathered vertebrate is both viviparous and warm-blooded ; (3) No oviparous vertebrate that is cold-blooded has feathers ; (4) Every viviparous vertebrate is either feathered or warm- blooded. [L.] Taking vertebrate as the universe of discourse, let P= warm- blooded, Q viviparous, R = feathered. Then by (1), p = cold- blooded, q oviparous. The remaining premisses are as follows: (!} Xo R is PQ, (3) No pq is R, (4) All Q is P or R. Required a description of P. By the obversion of (2), All R is p or q; therefore, All P is q or r. (3) and (4) give no information with regard to P, and are therefore superfluous for our purpose. Hence the required descrip- tion is that All warm-blooded vertebrates are either oviparous or featherless. K. L. 28 434 COMPLEX PROPOSITIONS. [PART IV. 397. In a certain town the old buildings are either ecclesiastical and built entirely of stone, or, if not ecclesiastical are built entirely of brick ; the brick-and-stone buildings are all modern as well as secular or they are neither; but there are no modern buildings at once secular and built entirely of stone. State what assumptions you make in interpreting the above, and determine (a) in what cases brick may be found in the buildings of this town and in what cases it cannot be, (6) what old buildings it would be useless to look for. [L:] Let A = old, B = ecclesiastical, P = containing brick, Q = contain- ing stone. Then assuming that old and modern, ecclesiastical and secular, are respectively contradictories, a modern, b = secular. The premisses are (1) All A is BpQ or bPq, (2) All PQ is AB or ab, (3) No a is bpQ. We may interpret the question as follows : (a) after eliminating Q, determine P both positively and negatively ; (b) determine A negatively. (3) gives no information with regard to P ; but by (1) All P is a or bq, and by (2) All P is AS or ab or q ; therefore, All P is ab or aq or bq. Eliminating Q, All P is a or b. Therefore, Brick is to be found only in secular or modern buildings ; and by obversion, No brick is to be found in old ecclesiastical buildings. (3) gives no information with regard to A, and (2) adds no information to that contained in (1). Hence the second part of the question is answered by simply obvertiug (1). No old buildings are ecclesiastical and built of brick, or ecclesiastical and not built of stone, or secular and built of stone, or secular and not built of brick. 398. (1) If a nation has natural resources, and a good government, it will be prosperous. (2) If it has natural re- sources without a good government, or a good government without natural resources, it will be contented, but not prosper- ous. (3) If it has neither natural resources nor a good govern- ment it will be neither contented nor prosperous. Shew that these statements may be reduced to two propo- sitions of the form of Hamilton's U. [' s -] Let a nation with natural resources be denoted by 7?, a nation with a good government by G, a prosperous nation by P, and a contented nation by C. Then the given statements may be expressed CHAP. V.] EXERCISES. 435 as follows : (1) All RG is P ; (2) All Rg or rG is Cp; (3) All rg is cp. By contraposition, (2) may be resolved into the two propositions, All cp is RG or rg, All P is RG or rg. But by (1) No cp is RG; and by (3) No P is rg. Hence the two propositions into which (2) was resolved may be reduced to the form, All cp is rg, All P is RG. The three original statements are accordingly equivalent to the two U propositions All RG is all P, All rg is all cp. 399. Let the observation of a class of natural productions be supposed to have led to the following general results. 1st. That in whichsoever of these productions the pro- perties A and C are missing, the property E is found, together with one of the properties B and D, but not with both. 2nd. That wherever the properties A and D are found while E is missing, the properties B and G will either both be found, or both be missing. 3rd. That wherever the property A is found in con- junction with either B or E, or both of them, there either the property C or the property D will be found, but not both of them. And conversely, wherever the property C or D is found singly, there the property A will be found in conjunction with either B or E, or both of them. Shew that it follows that In ivhatever substances the pro- perty A is found, there will also be found either the property C or the property D, but not both, or else the properties B, G, and D will all be wanting. And conversely, Where either the property G or the property D is found singly, or the properties B, C, and D are together missing, there the property A will be found. Shew also that If the property A is absent and G present, D is present. [Boole, Laws of Thought, pp. 146 148. Venn, Symbolic Logic, pp. 280, 281. Johns Hopkins Studies in Logic, pp. 57, 58, 82, 83.] The premisses are as follows : 1st, All ac is BdE or bDE ; (i) 2nd, All ADe is EC or be ; (ii) 3rd, Wliatever is AB or AE is Cd or cD ; (iii) Whatever is Cd or cD is AB or AE ; (iv) 282 436 COMPLEX PROPOSITIONS. [PART IV. We are required to prove : All A is Cd or cD or bed ; (a) All Cd is A ; ((3) All cD is A-, (y) All bed is A ; (8) All aC is D. () First, By (iii), All A is Cd or cD or be. But by (ii), All Abe is c or d; and by (iv), All Abe is CD or cd; therefore, All Abe is cd. Hence, All A is Cd or cD or bed. (a) Secondly, ((3) and (y) follow immediately from (iv). Thirdly, from (i), we have directly, No ac is bd- therefore (by conversion), No bed is a ; therefore, All bed is A. (8) Lastly, by (iv), All Cd is A ; therefore, by contraposition, All aC is D. (e) We may obtain a complete solution so far as A is concerned as follows : By (ii) 1 , All A is BC or be or dor E; by (iii), All A is be or Cd or cD ; therefore, All A is Cd or cDE or bcD or bee or bde ; by (iv), All A is or E or CD or cd ; therefore, All A is cDE or bcde or BCd or CdE. This includes the partial solution with regard to A, All A is Cd or cD or bed. Boole contents himself with this because he has started with the intention of eliminating E from his conclusion. We may now solve for a. (ii) and (iii) give no information with, regard to this term. But by (i), All a is BdE or bDE or C ; and by (iv), All a is CD or cd. Therefore All a is BcdE or CD. And this yields by conti-aposition, Whatever is be or Cd or cD or ce is A. 400. Given the same premisses as in the preceding section, shew that, 1st. If the property B be present in one of the productions, either the properties A, C, and D are all absent, or some one alone of them is absent. And conversely, if they are all absent it may be concluded that the property B is present. 2nd. If A and C are both present or both absent, D will be absent, quite independently of the presence or absence of B. [Boole, Laws of Thought, p. 149.] 1 No information whatever with regard to A is given by (i), since a appears as a determinant of tlie subject. See section 380. CHAP. V.] EXERCISES. 437 We may proceed here by combining all the given premisses in the manner indicated in section 368. From the result thus obtained the above conclusions as well as those contained in the preceding section will immediately follow. By (iii), Everything is a or be or Cd or cD ; and by (iv), Everything is AB or AE or CD or cd; therefore, Everything is ABCd or ABcD or ACdE or AcDE or aCD or acd or bCDe or bcde ; therefore by (i), Everything is ABCd or ABcD or Abode or ACdE or AcDE or aBcdE or aCD or bCDe therefore by (ii), Everything is ABCd or Abcde or ACdE or AcDE or aBcdE or aCD. (v) Hence, All B is ACd or AcDE or acdE or aCD; All acd is BE-, All AC is Bd or dE ; All ac is BdE. Eliminating E from each of the above we have the results arrived at by Boole. Eliminating both A and E from (v) we have Everything is BCd or bed or Cd or cD or Bed or CD ; that is, Everything is C or D or cd, which is an identity. This is equivalent to Boole's conclusion that "there is no independent relation among the properties B, C, and D " (Laws of T/iought, p. 148). Any further results that may be desired are obtainable im- mediately from (v). 401. Given XY= A, 7Z= C, find XZ in terms of A and C. [Venn, Symbolic Logic, pp. 279, 310 312. Johns Hopkins Studies in Logic, pp. 53, 54.] The premisses may be written as follows : Everything is AXY or ax or ay ; (1) Everything is CYZ or cy or cz. (2) By (1), All XZ is AY or ay, and by (2), All XZ isCYorcy; therefore, All XZ is AC Y or acy. Hence, eliminating Y, All XZ is AC or ac. This solves the problem as set. But in order to get a complete solution equivalent to that which would be obtained by Boole, the following may be added : Solving as above for x or z, and eliminat- ing Y, we have All that is either x or z is AcXz or aCxZ or ac. Whence, by contraposition, Whatever is AC or Ax or AZ or CX or 438 COMPLEX PROPOSITIONS. [PART IV. Cz is XZ. In other words, Whatever is AC or AZ or CX is XZ ; and Nothing is Ax or Cz. 402. Shew the equivalence between the three following systems of propositions: (1) All Ab is cd; All aB is Ce; All D is E; (2) All A is B or cor D; All BE is A ; All Be is Ad or Cd; All bD is aE; (3) Whatever is A or e is B or d; All a is IE or bd or BCe ; All bC is a; All D is E. [K.] By obversion, the first set of propositions become No Ab is C or D; No aB is c or E; No D is e; and these propositions are combined in the statement Nothing is either AbC or AbD or aBc or aBE or De. (1) By obverting and combining the second set of propositions, we have Nothing is AbCd or aBE or aBce or BDe or AbD or bDe. (2) But AbCd or AbD is equivalent to AbC or AbD ; aBE or aBce to aBE or aBc; BDe or bDe to De. Hence (1) and (2) are equivalent. Again, by obverting and combining the third set of propositions, we have Nothing is AbD or bDe or aBc or aBE or abDe or acDe or AbC or De. (3) But since bDe, abDe, acDe are all subdivisions of De, (3) im- mediately resolves itself into (1). 403. Given (i) All Pqr is ST; (ii) Q and R are always present or absent together; (iii) All QRS is PT or pt; (iv) All QRs is Pt; (v) All pqrS is T; then it follows that (1) All Pq is rST; (2) All Ps is QRt ; (3) All pQ is RSt ; (4) All pT is qr ; (5) All Qs is PRt ; (6) All QT is PRS ; (7) All qS is rT ; (8) All qs is pr ; (9) All qt is prs ; (10) All sT is pqr. [K.] By (i), Everything is p or Q or R or ST ; by (ii), Everything is QR or qr ; therefore, Everything is QR or pqr or qrST ; by (iii), Everything is q or r or s or PT or pt ; therefore, Everything is pqr or qrST or QRs or PQRT or pQRt ; by (iv), Everything is q or r or S or Pt; therefore, Everything is pqr or qrST or PQRst or PQRST or pQRSt; by (v), Everything is s or P or Q or R or T ; therefore, Everything is pqrs or pqrT or qrST or PQRst or PQRST orpQRSt. The desired results follow from this immediately. 404. From the premisses (1) No Ax is cd or cy, (2) No CHAP. V.] EXERCISES. 439 BX is cde or cey, (3) No ab is cdx or cEx, (4) No A or B or C is xy, deduce a proposition containing neither X nor T. [Johns Hopkins Studies, p. 53.] By (2), No X is Bcde, and by (1) and (3), No x is Acd or abed or abcE ; therefore, by section 384, No Acd or abed or abcE is Bcde ; therefore, No Acd is Be. It will be observed that since T does not appear in the premisses, y can be eliminated only by omitting all the terms containing it. 405. The members of a scientific society are divided into three sections, which are denoted by A, B, G. Every member must join one, at least, of these sections, subject to the follow- ing conditions: (1) any one who is a member of A but not of B, of B but not of C, or of C but not of A, may deliver a lecture to the members if he has paid his subscription, but otherwise not ; (2) one who is a member of A but not of G, of C but not of A, or of B but not of A, may exhibit an experiment to the members if he has paid his subscription, but otherwise not; but (3) every member must either deliver a lecture or perform an experiment annually before the other members. Find the least addition to these rules which will compel every member to pay his subscription or forfeit his membership. [Johns Hopkins Studies, p. 54.] Let A = member of section A, ^hich still involve alternative combination may be dealt with in the same way, until no alternative com- bination remains. We shall now be left with a set of propositions which will 1 The proposition in its original form may admit of simplification in accord- ance with the rules laid down in chapter i. It will generally speaking be found advantageous to have recourse to such simplification before proceeding further with the solution. 2 See section 339. CHAP. VI.] THE INVERSE PROBLEM. 449 satisfy the required conditions. The possibility of various simplifications has, however, to be considered. Thus, it will be necessary to make sure that each of the propositions is itself expressed in its simplest form 1 ; and to observe whether any two or more of the propositions admit of a simple recom- bination 2 . It may also be found that some of the propositions can be altogether omitted, inasmuch as they add nothing to the information jointly afforded by the remainder; or that, considered in their relation to the remaining propositions, they may, at any rate, be simplified by the omission of one or more of the terms which they contain 8 . When these simplifications have been carried as far as is possible we shall have our final solution*. The solution may, if we wish, be verified by recombining into a single complex proposition the propositions that have been obtained, an operation by which we shall arrive again at a series of alternants substantially identical with those originally given us. Such verification is, however, not essential to the validity of our process, which, if it has been correctly performed, contains no possible source of error. The following examples will serve to illustrate the above method. I. For our first example we may take one of those chosen by Jevons in the extract quoted in the preceding section. Given the proposition, Everything is either ABC or Abe or aBC or cibC, we are to find a set of propositions not involving alternative combination which shall be equivalent to it. By the reduction of dual terms and contraposition we have 1 For example, All AB is EC may be reduced to All AB is C. 2 For example, All ac is d and All Be is d may be combined into All cD is Ab. 3 Thus, for the propositions All AB is CD and All Ab is C we may substi- tute the propositions All A B is D and All A is C. 4 It may be observed that it is no part of our object to obtain a set of propo- sitions which are mutually independent. As a matter of fact, it will generally be found that the maximum simplification involves the repetition of some items of information. Thus, in the example given in the preceding note the propo- sitions All AB is CD and All Ab is C are quite independent of one another; but the proposition All A is C repeats part of the information given by the proposition All AB is D. K. L. 29 450 COMPLEX PROPOSITIONS. [PART IV. What is neither ABC nor Abe is aO; therefore, What is a or Be or bC is aC; and this may be resolved into the three propositions : (All a is C, 1 Be is non-existent, (All bC is a. Be is non-existent is reducible to All B is C\ and this pro- position and All a is C may be combined into All c is Ab. Hence we have for our solution the two propositions : ll c is Ab, (All bC is a. It will be found that by the recombination of these propo- sitions we regain the original proposition. II. We may next take the more complex example con- tained in the same extract from Jevons. The given alternants are ACe, aBCe, aBcdE, abCe, abcE; and by the reduction of dual terms, they become aBcdE, abcE, Ce. Therefore, What is not aBcdE or abcE is Ce; and this proposition may be resolved into the four propositions : (All A is Ce; (1) \AllBDisCe; (2) \AllCise; (3) ( All e is C. (4) But since by (3) All C is e, (1) may be reduced to All A is (7; and this proposition may be combined with (4) yielding All c is aE. Also by (3), (2) may be reduced to All BD is C. Hence our solution becomes (AllBDisC; \All C is e; [All c is aE. III. The following problem is from Jevons, Principles of Science, 2nd ed., p. 127 (Problem v.). The given alternants are ABCD, ABCd, ABcd, AbCD, AbcD, aBCD, aBcD, aBcd, abCd. By the reduction of duals these alternants may be written as follows : ABC or ABcd or AbD or aBCD or aBc or abCd. Therefore by contraposition, Whatever is not ABC or AbD or aBc is ABcd or aBCD or abCd. CHAP. VI.] THE INVERSE PROBLEM. 451 But Whatever is not ABC or AID or aBc is equivalent to Whatever is ABc or aBC or ab or bd. Hence we have for our solution the following set of propositions : (1) All ABc is d, (2) All aBC is D, (3) All ab is Cd, (4) All bd is a 1 . This is equivalent to the solution given in Jevons, Studies, p. 256. If we wish to verify our solution we may proceed as follows: By (3), . Everything is A or B or Cd ; By (4), Everything is a or B or D ; therefore, Everything is AD or aCd or B. By (1), Everything is a or b or C or d; therefore, Everything is AbD or ACD or aB or aCd or BC or Bd', By (2), Everything is A or b or c or D ; therefore, Everything is ABC or ABd or AbD or ACD or aBc or aBD or abCd or BCD or Bed ; But, AbD is AbCD or AbcD ; and expanding all the terms similarly, we have Everything is ABCD or ABCd or A Bed or AbCD or AbcD or aBCD or aBcD or aBcd or abCd. These are the alternants originally given. IV. The following example is also from Jevons, Principles of Science, 2nd edition, p. 127 (Problem viii). In his Studies, p. 256, he speaks of the solution as unknown. A fairly simple solution may, however, be obtained by the application of the general rule formulated in this section. The given alternants are ABODE, ABCDe, ABCde, ABcde, AbCDE, AbcdE, Abcde, aBCDe, aBCde, aBcDe, abCDe, abCdE, abcDe, abcdE. By the reduction of duals these alternants may be written : ABCe or ABcde or Abed or ACDE or aBCde or abdE or aDe. Therefore by contraposition, Whatever is not either ABCe or ABcde or Abed or abdE or aDe is ACDE or aBCde. But it will be found that, by the application of the ordinary 1 We first obtain All bd it aC; but since by (3) All abd it C, this may be reduced to All bd it a. 292 452 COMPLEX PROPOSITIONS. [PART IV. rule for obtaining the contradictory of a given term, Whatever is not either ABCe or ABcde or Abed or abdE or aDe is equivalent to Whatever is AbC or ade or BE or AcD or DE. Hence our proposition is resolvable into the following : (i) All AbC is DE- (ii) All ade is BC; (iii) All BE is ACD; (iv) AcD is non-existent ; (v) All DE is AC. I But by (v) All BE is AC or d; therefore, (iii) may be reduced to All BE is D. Again by (iv), All DE is a or C ; therefore, (v) may be reduced to All DE is A. Hence we have the following as our final solution : (1) All AbC is DE; (2) All ade is BC; (3) All BE is D; (4) All cD is a ; (6) All DE is A. 431. Another Method of Solution of the Inverse Problem. Another method of solving the inverse problem, suggested to me (in a slightly different form) by Dr Venn, is to write down the original complex proposition in the negative form, i.e., to obvert it, before resolving it. It has been already shewn that a negative proposition with an alternative predicate may be immediately broken -up into a set of simpler propositions. In some cases, especially where the number of destroyed combinations as compared with those that are saved is small, this plan is of easier application than that given in the pre- ceding section. To illustrate this method we may take two or three of the examples already discussed. I. Everything is ABC or Abe or aBC or abC; therefore, by obversion, Nothing is AbC or Be or ac; and this proposition is at once resolvable into (All Ab is c. (All c is Ab 1 . 1 The equivalence between this and our former solution is immediately obvious. Equationally it would be written Ab=c. CHAP. VI.] THE INVERSE PROBLEM. 453 II. Everything is ACe or aBCe or aBcdE or abCe or abcE; therefore, by obversion, Nothing is CE or Ac or BcD or ce. This proposition may be successively resolved as follows : (NoEisC; { No c is A or e ; (No BD is c. (AllEisc; I All c is aE', [All BD is a III. Everything is ABCD or ABCd or ABcd or AbCD or AbcD or aBCD or aBcD or aBcd or abCd; therefore, by obver- sion, Nothing is Abd or A BcD or abc or abD or aBCd; and this proposition may be successively resolved as follows : No bd is A; No ABc is D ; No ab is c or D; NoaBCisd. (All bd is a; I All ABc is d ; 1 All ab is Cd; [AllaBOis D. It is rather interesting to find that notwithstanding the indeterminateness of the problem we obtain by independent methods the same result in each of the above cases. 432. A Third Method of Solution of the Inverse Problem. The following is a third independent method of solution of the inverse problem, and it is in some cases easier of application than either of the two preceding methods. Any proposition of the form Everything is may be resolved into the two propositions : (All A is [All a is ...... which taken together are equivalent to it; similarly All A is may be resolved into the two All AB is , All Ab is ; and it is clear that by taking pairs of contradictories in this way we may resolve any given complex proposition into a set 454 COMPLEX PROPOSITIONS. [PART IV. of propositions containing no alternative terms. Redundancies must of course as before be as far as possible avoided. To illustrate this method we may again take the first three examples given in section 430. I. Everything is ABC or Abe or aBC or abC may be resolved successively as follows : (All C is AB or aB or ab ; (All c is Ab. (AllbCisa 1 ; (All c is Ab. II. Everything is ACe or aBCe or aBcdE or abCe or abcE may be resolved successively as follows : f All G is Ae or aBe or abe ; (All c is aBdE or abE. (AllCise;