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Title: Studies and Exercises in Formal Logic

Author: John Neville Keynes

Release date: May 24, 2019 [eBook #59590]
Most recently updated: September 6, 2023

Language: English

Credits: Ed Brandon from material at the Internet Archive

*** START OF THE PROJECT GUTENBERG EBOOK STUDIES AND EXERCISES IN FORMAL LOGIC ***

STUDIES AND EXERCISES
IN
FORMAL LOGIC

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

BY
JOHN NEVILLE KEYNES, M.A., Sc.D.
UNIVERSITY LECTURER IN MORAL SCIENCE AND FORMERLY FELLOW OF PEMBROKE COLLEGE IN THE UNIVERSITY OF CAMBRIDGE

FOURTH EDITION RE-WRITTEN AND ENLARGED

𝕷𝖔𝖓𝖉𝖔𝖓

MACMILLAN AND CO., LIMITED

NEW YORK: THE MACMILLAN COMPANY

1906

[The Right of Translation and Reproduction is reserved]

First Edition (Crown 8vo.) printed 1884.

Second Edition (Crown 8vo.) 1887.

Third Edition (Demy 8vo.) 1894.

Fourth Edition (Demy 8vo.) 1906.

PREFACE TO THE FOURTH EDITION.

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

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

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

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

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

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

J. N. KEYNES.

6, HARVEY ROAD,
CAMBRIDGE,
4 September 1906.

PREFACE TO THE FIRST EDITION.1

¹ With some omissions.

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

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

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

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

CAMBRIDGE,
19 January 1884.

viii

PREFACE TO THE SECOND EDITION.

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

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

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

CAMBRIDGE,
22 June 1887.

PREFACE TO THE THIRD EDITION.

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

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

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

CAMBRIDGE,
25 July 1894.

TABLE OF CONTENTS.


INTRODUCTION.
SECTION		PAGE
1.	The General Character of Logic	1
2.	Formal Logic	1
3.	Logic and Language	3
4.	Logic and Psychology	5
5.	The Utility of Logic	6

PART I.

TERMS.

CHAPTER I.

THE LOGIC OF TERMS.

6.	The Three Parts of Logical Doctrine	8
7.	Names and Concepts	10
8.	The Logic of Terms	11
9.	General and Singular Names	11
10.	Proper Names	13
11.	Collective Names	14
12.	Concrete and Abstract Names	16
13.	Can Abstract Names be subdivided into General and Singular?	19
14, 15.	Exercises	21

CHAPTER II.

EXTENSION AND INTENSION.

16.	The Extension and the Intension of Names	22
17.	Connotation, Subjective Intension, and Comprehension.	23
18.	Sigwart’s distinction between Empirical, Metaphysical, and Logical Concepts	27
		xii
19.	Connotation and Etymology	28
20.	Fixity of Connotation	28
21.	Extension and Denotation	29
22.	Dependence of Extension and Intension upon one another	31
23.	Inverse Variation of Extension and Intension	35
24.	Connotative Names	40
25.	Are proper names connotative?	41
26 to 30.	Exercises	47

CHAPTER III.

REAL, VERBAL, AND FORMAL PROPOSITIONS.

31.	Real, Verbal, and Formal Propositions	49
32.	Nature of the Analysis involved in Analytic Propositions	53
33 to 37.	Exercises	56

CHAPTER IV.

NEGATIVE NAMES AND RELATIVE NAMES.

38.	Positive and Negative Names	57
39.	Indefinite Character of Negative Names	59
40.	Contradictory Terms	61
41.	Contrary Terms	62
42.	Relative Names	63
43 to 45.	Exercises	65

PART II.

PROPOSITIONS.

CHAPTER I.

IMPORT OF JUDGMENTS AND PROPOSITIONS.

46.	Judgments and Propositions	66
47.	The Abstract Character of Logic	68
48.	Nature of the Enquiry into the Import of Propositions	70
49.	The Objective Reference in Judgments	74
50.	The Universality of Judgments	76
51.	The Necessity of Judgments	77
52.	Exercise	78
		xiii

CHAPTER II.

KINDS OF JUDGMENTS AND PROPOSITIONS.

53.	The Classification of Judgments	79
54.	Kant’s Classification of Judgments	81
55.	Simple Judgments and Compound Judgments	82
56.	The Modality of Judgments	84
57.	Modality in relation to Simple Judgments	85
58.	Subjective Distinctions of Modality	86
59.	Objective Distinctions of Modality	87
60.	Modality in relation to Compound Judgments	90
61.	The Quantity and the Quality of Propositions	91
62.	The traditional Scheme of Propositions	92
63.	The Distribution of Terms in a Proposition	95
64.	The Distinction between Subject and Predicate in the traditional Scheme of Propositions	96
65.	Universal Propositions	97
66.	Particular Propositions	100
67.	Singular Propositions	102
68.	Plurative Propositions and Numerically Definite Propositions	103
69.	Indefinite Propositions	105
70.	Multiple Quantification	105
71.	Infinite or Limitative Propositions	106
72 to 78.	Exercises	107

CHAPTER III.

THE OPPOSITION OF PROPOSITIONS.

79.	The Square of Opposition	109
80.	Contradictory Opposition	111
81.	Contrary Opposition	114
82.	The Opposition of Singular Propositions	115
83.	The Opposition of Modal Propositions	116
84.	Extension of the Doctrine of Opposition	117
85.	The Nature of Significant Denial	119
86 to 95.	Exercises.	124
		xiv
CHAPTER IV.

IMMEDIATE INFERENCES.

96.	The Conversion of Categorical Propositions	126
97.	Simple Conversion and Conversion per accidens.	128
98.	Inconvertibility of Particular Negative Propositions	130
99.	Legitimacy of Conversion	130
100.	Table of Propositions connecting any two terms	132
101.	The Obversion of Categorical Propositions	133
102.	The Contraposition of Categorical Propositions	134
103.	The Inversion of Categorical Propositions	137
104.	The Validity of Inversion	139
105.	Summary of Results	140
106.	Table of Propositions connecting any two terms and their contradictories	141
107.	Mutual Relations of the non-equivalent Propositions connecting any two terms and their contradictories	142
108.	The Elimination of Negative Terms	144
109.	Other Immediate Inferences	147
110.	Reduction of immediate inferences to the mediate form	151
111 to 124.	Exercises	153

CHAPTER V.

THE DIAGRAMMATIC REPRESENTATION OF PROPOSITIONS.

125.	The use of Diagrams in Logic	156
126.	Euler’s Diagrams	157
127.	Lambert’s Diagrams	163
128.	Dr Venn’s Diagrams	166
129.	Expression of the possible relations between any two classes by means of the propositional forms A, E, I, O	168
130.	Euler’s diagrams and the class-relations between S, not-S, P, not-P	170
131.	Lambert’s diagrams and the class-relations between S, not-S, P, not-P	174
132 to 134.	Exercises	176
		xv
CHAPTER VI.

PROPOSITIONS IN EXTENSION AND IN INTENSION.

135.	Fourfold Implication of Propositions in Connotation and Denotation	177
	(1) Subject in denotation, predicate in connotation	179
	(2) Subject in denotation, predicate in denotation	181
	(3) Subject in connotation, predicate in connotation	184
	(4) Subject in connotation, predicate in denotation	186
136.	The Reading of Propositions in Comprehension	187

CHAPTER VII.

LOGICAL EQUATIONS AND THE QUANTIFICATION OF THE PREDICATE.

137.	The employment of the symbol of Equality in Logic	189
138.	Types of Logical Equations	191
139.	The expression of Propositions as Equations	194
140.	The eight propositional forms resulting from the explicit Quantification of the Predicate	195
141.	Sir William Hamilton’s fundamental Postulate of Logic	195
142.	Advantages claimed for the Quantification of the Predicate	196
143.	Objections urged against the Quantification of the Predicate	197
144.	The meaning to be attached to the word some in the eight propositional forms recognised by Sir William Hamilton	199
145.	The use of some in the sense of some only	202
146.	The interpretation of the eight Hamiltonian forms of proposition, some being used in its ordinary logical sense	203
147.	The propositions U and Y	204
148.	The proposition η	206
149.	The proposition ω	206
150.	Sixfold Schedule of Propositions obtained by recognising Y and η, in addition to A, E, I, O	207
151, 152.	Exercises	209

CHAPTER VIII.

THE EXISTENTIAL IMPORT OF CATEGORICAL PROPOSITIONS.

153.	Existence and the Universe of Discourse	210
154.	Formal Logic and the Existential Import of Propositions	215
155.	The Existential Formulation of Propositions	218
156.	Various Suppositions concerning the Existential Import of Categorical Propositions	218
		xvi
157.	Reduction of the traditional forms of proposition to the form of Existential Propositions	221
158.	Immediate Inferences and the Existential Import of Propositions	223
159.	The Doctrine of Opposition and the Existential Import of Propositions	227
160.	The Opposition of Modal Propositions considered in connexion with their Existential Import	231
161.	Jevons’s Criterion of Consistency	232
162.	The Existential Import of the Propositions included in the Traditional Schedule	234
163.	The Existential Import of Modal Propositions	244
164 to 172.	Exercises	245

CHAPTER IX.

CONDITIONAL AND HYPOTHETICAL PROPOSITIONS.

173.	The distinction between Conditional Propositions and Hypothetical Propositions	249
174.	The Import of Conditional Propositions	252
175.	Conditional Propositions and Categorical Propositions	253
176.	The Opposition of Conditional Propositions	256
177.	Immediate Inferences from Conditional Propositions	259
178.	The Import of Hypothetical Propositions	261
179.	The Opposition of Hypothetical Propositions	264
180.	Immediate Inferences from Hypothetical Propositions	268
181.	Hypothetical Propositions and Categorical Propositions	270
182.	Alleged Reciprocal Character of Conditional and Hypothetical Judgments	270
183 to 188.	Exercises	273

CHAPTER X.

DISJUNCTIVE (OR ALTERNATIVE) PROPOSITIONS.

189.	The terms Disjunctive and Alternative as applied to Propositions	275
190.	Two types of Alternative Propositions	276
191.	The Import of Disjunctive (Alternative) Propositions	277
192.	Scheme of Assertoric and Modal Propositions	282
193.	The Relation of Disjunctive (Alternative) Propositions to Conditionals and Hypotheticals	282
194 to 196.	Exercises	284
		xvii
PART III.

SYLLOGISMS.

CHAPTER I.

THE RULES OF THE SYLLOGISM.

197.	The Terms of the Syllogism	285
198.	The Propositions of the Syllogism	287
199.	The Rules of the Syllogism	287
200.	Corollaries from the Rules of the Syllogism	289
201.	Restatement of the Rules of the Syllogism	291
202.	Dependence of the Rules of the Syllogism upon one another	291
203.	Statement of the independent Rules of the Syllogism	293
204.	Proof of the Rule of Quality	294
205.	Two negative premisses may yield a valid conclusion; but not syllogistically	295
206.	Other apparent exceptions to the Rules of the Syllogism	297
207.	Syllogisms with two singular premisses	298
208.	Charge of incompleteness brought against the ordinary syllogistic conclusion	300
209.	The connexion between the Dictum de omni et nullo and the ordinary Rules of the Syllogism	301
210 to 242.	Exercises	302

CHAPTER II.

THE FIGURES AND MOODS OF THE SYLLOGISM.

243.	Figure and Mood	309
244.	The Special Rules of the Figures; and the Determination of the Legitimate Moods in each Figure	309
245.	Weakened Conclusions and Subaltern Moods	313
246.	Strengthened Syllogisms	314
247.	The peculiarities and uses of each of the four figures of the syllogism	315
248 to 255.	Exercises	317
		xviii
CHAPTER III.

THE REDUCTION OF SYLLOGISMS.

256.	The Problem of Reduction	318
257.	Indirect Reduction	318
258.	The mnemonic lines Barbara, Celarent, &c.	319
259.	The direct reduction of Baroco and Bocardo	323
260.	Extension of the Doctrine of Reduction	324
261.	Is Reduction an essential part of the Doctrine of the Syllogism?	325
262.	The Fourth Figure	328
263.	Indirect Moods	329
264.	Further discussion of the process of Indirect Reduction	331
265.	The Antilogism	332
266.	Equivalence of the Moods of the first three Figures shewn by the Method of Indirect Reduction	333
267.	The Moods of Figure 4 in their relation to one another	334
268.	Equivalence of the Special Rules of the First Three Figures	335
269.	Scheme of the Valid Moods of Figure 1	336
270.	Scheme of the Valid Moods of Figure 2	336
271.	Scheme of the Valid Moods of Figure 3	337
272.	Dictum for Figure 4	338
273 to 287.	Exercises	339

CHAPTER IV.

THE DIAGRAMMATIC REPRESENTATION OF SYLLOGISMS.

288.	Euler’s diagrams and syllogistic reasonings	341
289.	Lambert’s diagrams and syllogistic reasonings	344
290.	Dr Venn’s diagrams and syllogistic reasonings	345
291 to 300.	Exercises	347

CHAPTER V.

CONDITIONAL AND HYPOTHETICAL SYLLOGISMS.

301.	The Conditional Syllogism, the Hypothetical Syllogism, and the Hypothetico-Categorical Syllogism	348
302.	Distinctions of Mood and Figure in the case of Conditional and Hypothetical Syllogisms	349
303.	Fallacies in Hypothetical Syllogisms	350
304.	The Reduction of Conditional and Hypothetical Syllogisms	351
		xix
305.	The Moods of the Mixed Hypothetical Syllogism	352
306.	Fallacies in Mixed Hypothetical Syllogisms	353
307.	The Reduction of Mixed Hypothetical Syllogisms	354
308.	Is the reasoning contained in the mixed hypothetical syllogism mediate or immediate?	354
309 to 315.	Exercises	358

CHAPTER VI.

DISJUNCTIVE SYLLOGISMS.

316.	The Disjunctive Syllogism	359
317.	The modus ponendo tollens	361
318.	The Dilemma	363
319 to 321.	Exercises	366

CHAPTER VII.

IRREGULAR AND COMPOUND SYLLOGISMS.

322.	The Enthymeme	367
323.	The Polysyllogism and the Epicheirema	368
324.	The Sorites	370
325.	The Special Rules of the Sorites	372
326.	The possibility of a Sorites in a Figure other than the First	373
327.	Ultra-total Distribution of the Middle Term	376
328.	The Quantification of the Predicate and the Syllogism	378
329.	Table of valid moods resulting from the recognition of Y and η in addition to A, E, I, O	381
330.	Formal Inferences not reducible to ordinary Syllogisms	384
331 to 341.	Exercises	388

CHAPTER VIII.

PROBLEMS ON THE SYLLOGISM.

342.	Bearing of the existential interpretation of propositions upon the validity of syllogistic reasonings	390
343.	Connexion between the truth and falsity of premisses and conclusion in a valid syllogism	394
344.	Arguments from the truth of one premiss and the falsity of the other premiss in a valid syllogism, or from the falsity of one premiss to the truth of the conclusion, or from the truth of one premiss to the falsity of the conclusion	396
345.	Numerical Moods of the Syllogism	400
346 to 375.	Exercises	403
		xx

CHAPTER IX.

THE CHARACTERISTICS OF INFERENCE.

376.	The Nature of Logical Inference	413
377.	The Paradox of Inference	414
378.	The nature of the difference that there must be between premisses and conclusion in an inference	415
379.	The Direct Import and the Implications of a Proposition	420
380.	Syllogisms and Immediate Inferences	423
381.	The charge of petitio principii brought against Syllogistic Reasoning	424

CHAPTER X.

EXAMPLES OF ARGUMENTS AND FALLACIES.

382 to 408.	Exercises	431

APPENDIX A.

THE DOCTRINE OF DIVISION.

409.	Logical Division	441
410.	Physical Division, Metaphysical Division, and Verbal Division	442
411.	Rules of Logical Division	443
412.	Division by Dichotomy	445
413.	The place of the Doctrine of Division in Logic	446

APPENDIX B.

THE FUNDAMENTAL LAWS OF THOUGHT.

414.	The Three Laws of Thought	450
415.	The Law of Identity	451
416.	The Law of Contradiction	454
417.	The Sophism of “The Liar”	457
418.	The Law of Excluded Middle	458
419.	Grounds on which the absolute universality and necessity of the law of excluded middle have been denied	460
420.	Are the Laws of Thought also Laws of Things?	463
421.	Mutual Relations of the three Laws of Thought	464
422.	The Laws of Thought in relation to Immediate Inferences	464
423.	The Laws of Thought and Formal Mediate Inferences	466
		xxi
APPENDIX C.

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

CHAPTER I.

THE COMBINATION OF TERMS.

424.	Complex Terms	468
425.	Order of Combination in Complex Terms	469
426.	The Opposition of Complex Terms	470
427.	Duality of Formal Equivalences in the case of Complex Terms	472
428.	Laws of Distribution	472
429.	Laws of Tautology	473
430.	Laws of Development and Reduction	474
431.	Laws of Absorption	475
432.	Laws of Exclusion and Inclusion	475
433.	Summary of Formal Equivalences of Complex Terms	475
434.	The Conjunctive Combination of Alternative Terms	478
435 to 439.	Exercises	477

CHAPTER II.

COMPLEX PROPOSITIONS AND COMPOUND PROPOSITIONS.

440.	Complex Propositions	478
441.	The Opposition of Complex Propositions	478
442.	Compound Propositions	478
443.	The Opposition of Compound Propositions	480
444.	Formal Equivalences of Compound Propositions	480
445.	The Simplification of Complex Propositions	481
446.	The Resolution of Universal Complex Propositions into Equivalent Compound Propositions	483
447.	The Resolution of Particular Complex Propositions into Equivalent Compound Propositions	484
448.	The Omission of Terms from Complex Propositions	485
449.	The Introduction of Terms into Complex Propositions	485
450.	Interpretation of Anomalous Forms	486
451 to 453.	Exercises	487
		xxii
CHAPTER III.

IMMEDIATE INFERENCES FROM COMPLEX PROPOSITIONS.

454.	The Obversion of Complex Propositions	488
455.	The Conversion of Complex Propositions	489
456.	The Contraposition of Complex Propositions	490
457.	Summary of the results obtainable by Obversion, Conversion, and Contraposition	493
458 to 473.	Exercises	494

CHAPTER IV.

THE COMBINATION OF COMPLEX PROPOSITIONS.

474.	The Problem of combining Complex Propositions	498
475.	The Conjunctive Combination of Universal Affirmatives	498
476.	The Conjunctive Combination of Universal Negatives	499
477.	The Conjunctive Combination of Universals with Particulars of the same Quality	500
478.	The Conjunctive Combination of Affirmatives with Negatives	501
479.	The Conjunctive Combination of Particulars with Particulars	501
480.	The Alternative Combination of Universal Propositions	502
481.	The Alternative Combination of Particular Propositions	502
482.	The Alternative Combination of Particulars with Universals	502
483 to 486.	Exercises	503

CHAPTER V.

INFERENCES FROM COMBINATIONS OF COMPLEX PROPOSITIONS.

487.	Conditions under which a universal proposition affords information in regard to any given term	504
488.	Information jointly afforded by a series of universal propositions with regard to any given term	506
489.	The Problem of Elimination	508
490.	Elimination from Universal Affirmatives	509
491.	Elimination from Universal Negatives	510
492.	Elimination from Particular Affirmatives	511
493.	Elimination from Particular Negatives	511
494.	Order of procedure in the process of elimination	511
495 to 533.	Exercises	512
		xxiii

CHAPTER VI.

THE INVERSE PROBLEM.

534.	Nature of the Inverse Problem	525
535.	A General Solution of the Inverse Problem	527
536.	Another Method of Solution of the Inverse Problem	530
537.	A Third Method of Solution of the Inverse Problem	531
538.	Mr Johnson’s Notation for the Solution of Logical Problems	533
539.	The Inverse Problem and Schröder’s Law of Reciprocal Equivalences	534
540 to 550.	Exercises	535

INDEX		539