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Title: Symbolic Logic

Author: Lewis Carroll

Release Date: May 5, 2009 [EBook #28696]

Language: English

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pg_iSYMBOLIC LOGIC

By Lewis Carroll

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A Syllogism worked out. That story of yours, about your once meeting the sea-serpent, always sets me off yawning; I never yawn, unless when I’m listening to something totally devoid of interest. The Premisses, separately. The Premisses, combined. The Conclusion. That story of yours, about your once meeting the sea-serpent, is totally devoid of interest.

pg_vSYMBOLIC LOGIC

PART I

ELEMENTARY

BY

LEWIS CARROLL

SECOND THOUSAND
FOURTH EDITION

PRICE TWO SHILLINGS

London
MACMILLAN AND CO., Limited
new york: the macmillan company
1897

All rights reserved

pg_viRichard Clay and Sons, Limited,
london and bungay

pg_viiADVERTISEMENT.

An envelope, containing two blank Diagrams (Biliteral and Triliteral) and 9 counters (4 Red and 5 Grey), may be had, from Messrs. Macmillan, for 3d., by post 4d.

I shall be grateful to any Reader of this book who will point out any mistakes or misprints he may happen to notice in it, or any passage which he thinks is not clearly expressed.

I have a quantity of MS. in hand for Parts II and III, and hope to be able——should life, and health, and opportunity, be granted to me, to publish them in the course of the next few years. Their contents will be as follows:—

PART II. ADVANCED.

Further investigations in the subjects of Part I. Propositions of other forms (such as “Not-all x are y”). Triliteral and Multiliteral Propositions (such as “All abc are de”). Hypotheticals. Dilemmas. &c. &c.

Part III. TRANSCENDENTAL.

Analysis of a Proposition into its Elements. Numerical and Geometrical Problems. The Theory of Inference. The Construction of Problems. And many other Curiosa Logica.

pg_viiiPREFACE TO THE FOURTH EDITION.

The chief alterations, since the First Edition, have been made in the Chapter on ‘Classification’ (pp. 2, 3) and the Book on ‘Propositions’ (pp. 10 to 19). The chief additions have been the questions on words and phrases, added to the Examination-Papers at p. 94, and the Notes inserted at pp. 164, 194.

In Book I, Chapter II, I have adopted a new definition of ‘Classification’, which enables me to regard the whole Universe as a ‘Class,’ and thus to dispense with the very awkward phrase ‘a Set of Things.’

In the Chapter on ‘Propositions of Existence’ I have adopted a new ‘normal form,’ in which the Class, whose existence is affirmed or denied, is regarded as the Predicate, instead of the Subject, of the Proposition, thus evading a very subtle difficulty which besets the other form. These subtle difficulties seem to lie at the root of every Tree of Knowledge, and they are far more hopeless to grapple with than any that occur in its higher branches. For example, the difficulties of the Forty-Seventh Proposition of Euclid are mere child’s play compared with the mental torture endured in the effort to think out the essential nature of a straight Line. And, in the present work, the difficulties of the “5 Liars” Problem, at p. 192, are “trifles, light as air,” compared with the bewildering question “What is a Thing?”

In the Chapter on ‘Propositions of Relation’ I have inserted a new Section, containing the proof that a Proposition, beginning with “All,” is a Double Proposition (a fact that is quite independent of the arbitrary rule, laid down in the next Section, that such a Proposition is to be understood as implying the actual existence of its Subject). This proof was given, in the earlier editions, incidentally, in the course of the discussion of the Biliteral Diagram: but its proper place, in this treatise, is where I have now introduced it.

pg_ixIn the Sorites-Examples, I have made a good many verbal alterations, in order to evade a difficulty, which I fear will have perplexed some of the Readers of the first three Editions. Some of the Premisses were so worded that their Terms were not Specieses of the Univ. named in the Dictionary, but of a larger Class, of which the Univ. was only a portion. In all such cases, it was intended that the Reader should perceive that what was asserted of the larger Class was thereby asserted of the Univ., and should ignore, as superfluous, all that it asserted of its other portion. Thus, in Ex. 15, the Univ. was stated to be “ducks in this village,” and the third Premiss was “Mrs. Bond has no gray ducks,” i.e. “No gray ducks are ducks belonging to Mrs. Bond.” Here the Terms are not Specieses of the Univ., but of the larger Class “ducks,” of which the Univ. is only a portion: and it was intended that the Reader should perceive that what is here asserted of “ ducks” is thereby asserted of “ ducks in this village.” and should treat this Premiss as if it were “Mrs. Bond has no gray ducks in this village,” and should ignore, as superfluous, what it asserts as to the other portion of the Class “ducks,” viz. “ Mrs. Bond has no gray ducks out of this village”.

In the Appendix I have given a new version of the Problem of the “Five Liars.” My object, in doing so, is to escape the subtle and mysterious difficulties which beset all attempts at regarding a Proposition as being its own Subject, or a Set of Propositions as being Subjects for one another. It is certainly, a most bewildering and unsatisfactory theory: one cannot help feeling that there is a great lack of substance in all this shadowy host——that, as the procession of phantoms glides before us, there is not one that we can pounce upon, and say “Here is a Proposition that must be either true or false!”——that it is but a Barmecide Feast, to which we have been bidden——and that its prototype is to be found in that mythical island, whose inhabitants “earned a precarious living by taking in each others’ washing”! By simply translating “telling 2 Truths” into “taking both of 2 condiments (salt and mustard),” “telling 2 Lies” into “taking neither of them” and “telling a Truth and a Lie (order not specified)” into “taking only one condiment (it is not specified pg_xwhich),” I have escaped all those metaphysical puzzles, and have produced a Problem which, when translated into a Set of symbolized Premisses, furnishes the very same Data as were furnished by the Problem of the “Five Liars.”

The coined words, introduced in previous editions, such as “Eliminands” and “Retinends”, perhaps hardly need any apology: they were indispensable to my system: but the new plural, here used for the first time, viz. “Soriteses”, will, I fear, be condemned as “bad English”, unless I say a word in its defence. We have three singular nouns, in English, of plural form, “series”, “species”, and “Sorites”: in all three, the awkwardness, of using the same word for both singular and plural, must often have been felt: this has been remedied, in the case of “series” by coining the plural “serieses”, which has already found it way into the dictionaries: so I am no rash innovator, but am merely “following suit”, in using the new plural “Soriteses”.

In conclusion, let me point out that even those, who are obliged to study Formal Logic, with a view to being able to answer Examination-Papers in that subject, will find the study of Symbolic Logic most helpful for this purpose, in throwing light upon many of the obscurities with which Formal Logic abounds, and in furnishing a delightfully easy method of testing the results arrived at by the cumbrous processes which Formal Logic enforces upon its votaries.

This is, I believe, the very first attempt (with the exception of my own little book, The Game of Logic, published in 1886, a very incomplete performance) that has been made to popularise this fascinating subject. It has cost me years of hard work: but if it should prove, as I hope it may, to be of real service to the young, and to be taken up, in High Schools and in private families, as a valuable addition to their stock of healthful mental recreations, such a result would more than repay ten times the labour that I have expended on it.

L. C.

29, Bedford Street, Strand.
Christmas, 1896.

pg_xiINTRODUCTION.

TO LEARNERS.

[N.B. Some remarks, addressed to Teachers, will be found in the Appendix, at p. 165.]

The Learner, who wishes to try the question fairly, whether this little book does, or does not, supply the materials for a most interesting mental recreation, is earnestly advised to adopt the following Rules:—

(1) Begin at the beginning, and do not allow yourself to gratify a mere idle curiosity by dipping into the book, here and there. This would very likely lead to your throwing it aside, with the remark “This is much too hard for me!”, and thus losing the chance of adding a very large item to your stock of mental delights. This Rule (of not dipping) is very desirable with other kinds of books——such as novels, for instance, where you may easily spoil much of the enjoyment you would otherwise get from the story, by dipping into it further on, so that what the author meant to be a pleasant surprise comes to you as a matter of course. Some people, I know, make a practice of looking into Vol. III first, just to see how the story ends: and perhaps it is as well just to know that all ends happily——that the much-persecuted lovers do marry after all, that he is proved to be quite innocent of the murder, that the wicked cousin is completely foiled in his plot and gets the punishment he deserves, and that the rich uncle in India (Qu. Why in India? Ans. Because, somehow, uncles never can get rich anywhere else) dies at exactly the right moment——before taking the trouble to read Vol. I. pg_xiiThis, I say, is just permissible with a novel, where Vol. III has a meaning, even for those who have not read the earlier part of the story; but, with a scientific book, it is sheer insanity: you will find the latter part hopelessly unintelligible, if you read it before reaching it in regular course.

(2) Don’t begin any fresh Chapter, or Section, until you are certain that you thoroughly understand the whole book up to that point, and that you have worked, correctly, most if not all of the examples which have been set. So long as you are conscious that all the land you have passed through is absolutely conquered, and that you are leaving no unsolved difficulties behind you, which will be sure to turn up again later on, your triumphal progress will be easy and delightful. Otherwise, you will find your state of puzzlement get worse and worse as you proceed, till you give up the whole thing in utter disgust.

(3) When you come to any passage you don’t understand, read it again: if you still don’t understand it, read it again: if you fail, even after three readings, very likely your brain is getting a little tired. In that case, put the book away, and take to other occupations, and next day, when you come to it fresh, you will very likely find that it is quite easy.

(4) If possible, find some genial friend, who will read the book along with you, and will talk over the difficulties with you. Talking is a wonderful smoother-over of difficulties. When I come upon anything——in Logic or in any other hard subject——that entirely puzzles me, I find it a capital plan to talk it over, aloud, even when I am all alone. One can explain things so clearly to one’s self! And then, you know, one is so patient with one’s self: one never gets irritated at one’s own stupidity!

If, dear Reader, you will faithfully observe these Rules, and so give my little book a really fair trial, I promise you, most confidently, that you will find Symbolic Logic to be one of the most, if not the most, fascinating of mental recreations! In this First Part, I have carefully avoided all difficulties which seemed to me to be beyond the grasp of an intelligent child of (say) twelve or fourteen years of age. I have myself taught most of its contents, vivâ voce, to many children, and have pg_xiiifound them take a real intelligent interest in the subject. For those, who succeed in mastering Part I, and who begin, like Oliver, “asking for more,” I hope to provide, in Part II, some tolerably hard nuts to crack——nuts that will require all the nut-crackers they happen to possess!

Mental recreation is a thing that we all of us need for our mental health; and you may get much healthy enjoyment, no doubt, from Games, such as Back-gammon, Chess, and the new Game “Halma”. But, after all, when you have made yourself a first-rate player at any one of these Games, you have nothing real to show for it, as a result! You enjoyed the Game, and the victory, no doubt, at the time: but you have no result that you can treasure up and get real good out of. And, all the while, you have been leaving unexplored a perfect mine of wealth. Once master the machinery of Symbolic Logic, and you have a mental occupation always at hand, of absorbing interest, and one that will be of real use to you in any subject you may take up. It will give you clearness of thought——the ability to see your way through a puzzle——the habit of arranging your ideas in an orderly and get-at-able form——and, more valuable than all, the power to detect fallacies, and to tear to pieces the flimsy illogical arguments, which you will so continually encounter in books, in newspapers, in speeches, and even in sermons, and which so easily delude those who have never taken the trouble to master this fascinating Art. Try it. That is all I ask of you!

L. C.

29, Bedford Street, Strand.
February 21, 1896.

pg_xiv pg_xvCONTENTS BOOK I.   THINGS AND THEIR ATTRIBUTES.   CHAPTER I.   INTRODUCTORY.     page ‘Things’ 1  ‘Attributes’ 〃 ‘Adjuncts’ 〃 CHAPTER II.   CLASSIFICATION.   ‘Classification’ 1½ ‘Class’ 〃 ‘Peculiar’ Attributes 〃 ‘Genus’ 〃 ‘Species’ 〃 ‘Differentia’ 〃 ‘Real’ and ‘Unreal’, or ‘Imaginary’, Classes 2  ‘Individual’ 〃 A Class regarded as a single Thing 2½ pg_xviCHAPTER III.   DIVISION.   § 1.   Introductory.   ‘Division’ 3  ‘Codivisional’ Classes 〃 § 2.   Dichotomy.   ‘Dichotomy’ 3½ Arbitrary limits of Classes 〃 Subdivision of Classes 4  CHAPTER IV.   NAMES.   ‘Name’ 4½ ‘Real’ and ‘Unreal’ Names 〃 Three ways of expressing a Name 〃 Two senses in which a plural Name may be used 5  CHAPTER V.   DEFINITIONS.   ‘Definition’ 6  Examples worked as models 〃 pg_xviiBOOK II.   PROPOSITIONS.   CHAPTER I.   PROPOSITIONS GENERALLY.   § 1.   Introductory.   Technical meaning of “some” 8  ‘Proposition’ 〃 ‘Normal form’ of a Proposition 〃 ‘Subject’, ‘Predicate’, and ‘Terms’ 9  § 2.   Normal form of a Proposition.   Its four parts:—   (1) ‘Sign of Quantity’ 〃 (2) Name of Subject 〃 (3) ‘Copula’ 〃 (4) Name of Predicate 〃 § 3.   Various kinds of Propositions.   Three kinds of Propositions:—   (1) Begins with “Some”. Called a ‘Particular’ Proposition: also a Proposition ‘in I’ 10  (2) Begins with “No”. Called a ‘Universal Negative’ Proposition: also a Proposition ‘in E’ 〃  (3) Begins with “All”. Called a ‘Universal Affirmative’ Proposition: also a Proposition ‘in A’ 〃  pg_xviiiA Proposition, whose Subject is an Individual, is to be regarded as Universal 〃  Two kinds of Propositions, ‘Propositions of Existence’, and ‘Propositions of Relation’ 〃  CHAPTER II.   PROPOSITIONS OF EXISTENCE.   ‘Proposition of Existence ’ 11  CHAPTER III.   PROPOSITIONS OF RELATION.   § 1.   Introductory.   ‘Proposition of Relation’ 12  ‘Universe of Discourse,’ or ‘Univ.’ 〃  § 2.   Reduction of a Proposition of Relation to Normal form.   Rules 13  Examples worked 〃  § 3.   A Proposition of Relation, beginning with “All”, is a Double Proposition.   Its equivalence to two Propositions 17  pg_xix§ 4.   What is implied, in a Proposition of Relation, as to the Reality of its Terms?   Propositions beginning with “Some” 19  Propositions beginning with “No” 〃  Propositions beginning with “All” 〃  § 5.   Translation of a Proposition of Relation into one or more Propositions of Existence.   Rules 20  Examples worked 〃  BOOK III.   THE BILITERAL DIAGRAM.   CHAPTER I.   SYMBOLS AND CELLS.   The Diagram assigned to a certain Set of Things, viz. our Univ. 22  Univ. divided into ‘the x-Class’ and ‘the x′-Class’ 23  The North and South Halves assigned to these two Classes 〃  The x-Class subdivided into ‘the xy-Class’ and ‘the xy′-Class’ 〃  The North-West and North-East Cells assigned to these two Classes 〃  The x′-Class similarly divided 〃  The South-West and South-East Cells similarly assigned 〃  The West and East Halves have thus been assigned to ‘the y-Class’ and ‘the y′-Class’ 〃  Table I. Attributes of Classes, and Compartments, or Cells, assigned to them 25  pg_xxCHAPTER II.   COUNTERS.   Meaning of a Red Counter placed in a Cell 26  Meaning of a Red Counter placed on a Partition 〃  American phrase “sitting on the fence” 〃  Meaning of a Grey Counter placed in a Cell 〃  CHAPTER III.   REPRESENTATION OF PROPOSITIONS.   § 1.   Introductory.   The word “Things” to be henceforwards omitted 27  ‘Uniliteral’ Proposition 〃  ‘Biliteral’ do. 〃  Proposition ‘in terms of’ certain Letters 〃  § 2.   Representation of Propositions of Existence.   The Proposition “Some x exist” 28  Three other similar Propositions 〃  The Proposition “No x exist” 〃  Three other similar Propositions 29  The Proposition “Some xy exist” 〃  Three other similar Propositions 〃  The Proposition “No xy exist” 〃  Three other similar Propositions 〃  The Proposition “No x exist” is Double, and is equivalent to the two Propositions “No xy exist” and “No xy′ exist” 30  pg_xxi§ 3.   Representation of Propositions of Relations.   The Proposition “Some x are y” 〃  Three other similar Propositions 〃  The Proposition “Some y are x” 31  Three other similar Propositions 〃  Trio of equivalent Propositions, viz. “Some xy exist” = “Some x are y” = “Some y are x” 〃  ‘Converse’ Propositions, and ‘Conversion’ 〃  Three other similar Trios 32  The Proposition “No x are y” 〃  Three other similar Propositions 〃  The Proposition “No y are x” 〃  Three other similar Propositions 〃  Trio of equivalent Propositions, viz. “No xy exist” = “No x are y” = “No y are x” 33  Three other similar Trios 〃  The Proposition “All x are y” is Double, and is equivalent to the two Propositions “Some x are y” and “No x are y′” 〃  Seven other similar Propositions 34  Table II. Representation of Propositions of Existence 34  Table III. Representation of Propositions of Relation 35  CHAPTER IV.   INTERPRETATION OF BILITERAL DIAGRAM, WHEN MARKED WITH COUNTERS.   Interpretation of 36  And of three other similar arrangements 〃  pg_xxiiInterpretation of 〃  And of three other similar arrangements 〃  Interpretation of 37  And of three other similar arrangements 〃  Interpretation of 〃  And of three other similar arrangements 〃  Interpretation of 〃  And of three other similar arrangements 〃  Interpretation of 〃  And of seven other similar arrangements 38  BOOK IV.   THE TRILITERAL DIAGRAM.   CHAPTER I.   SYMBOLS AND CELLS.   Change of Biliteral into Triliteral Diagram 39  The xy-Class subdivided into ‘the xym-Class’ and ‘the xym′-Class’ 40  pg_xxiiiThe Inner and Outer Cells of the North-West Quarter assigned to these Classes 〃  The xy′-Class, the x′y-Class, and the x′y′-Class similarly subdivided 〃  The Inner and Outer Cells of the North-East, the South-West, and the South-East Quarter similarly assigned 〃  The Inner Square and the Outer Border have thus been assigned to ‘the m-Class’ and ‘the m′-Class’ 〃  Rules for finding readily the Compartment, or Cell, assigned to any given Attribute or Attributes 〃  Table IV. Attributes of Classes, and Compartments, or Cells, assigned to them 42  CHAPTER II.   REPRESENTATION OF PROPOSITIONS IN TERMS OF x AND m, OR OF y AND m.   § 1.   Representation of Propositions of Existence in terms of x and m, or of y and m.   The Proposition “Some xm exist” 43  Seven other similar Propositions 〃  The Proposition “No xm exist” 44  Seven other similar Propositions 〃  § 2.   Representation of Propositions of Relation in terms of x and m, or of y and m.   The Pair of Converse Propositions “Some x are m” = “Some m are x” 〃  Seven other similar Pairs 〃  The Pair of Converse Propositions “No x are m” = “No m are x” 〃  Seven other similar Pairs 〃  The Proposition “All x are m” 45  Fifteen other similar Propositions 〃  Table V. Representations of Propositions in terms of x and m 46  Table VI. Representations of Propositions in terms of y and m 47  Table VII. Representations of Propositions in terms of x and m 48  Table VIII. Representations of Propositions in terms of y and m 49  pg_xxivCHAPTER III.   REPRESENTATION OF TWO PROPOSITIONS OF RELATION, ONE IN TERMS OF x AND m, AND THE OTHER IN TERMS OF y AND m, ON THE SAME DIAGRAM.   The Digits “I” and “O” to be used instead of Red and Grey Counters 50  Rules 〃  Examples worked 〃  CHAPTER IV.   INTERPRETATION, IN TERMS OF x AND y, OF TRILITERAL DIAGRAM, WHEN MARKED WITH COUNTERS OR DIGITS.   Rules 53  Examples worked 54  BOOK V.   SYLLOGISMS.   CHAPTER I.   INTRODUCTORY.   ‘Syllogism’ 56  ‘Premisses’ 〃  ‘Conclusion’ 〃  ‘Eliminands’ 〃  ‘Retinends’ 〃  ‘Consequent’ 〃  The Symbol “∴” 〃  Specimen-Syllogisms 57  pg_xxvCHAPTER II.   PROBLEMS IN SYLLOGISMS.   § 1.   Introductory.   ‘Concrete’ and ‘Abstract’ Propositions 59  Method of translating a Proposition from concrete into abstract form 〃  Two forms of Problems 〃  § 2.   Given a Pair of Propositions of Relation, which contain between them a Pair of codivisional Classes, and which are proposed as Premisses: to ascertain what Conclusion, if any, is consequent from them.   Rules 60  Examples worked fully 〃  The same worked briefly, as models 64  § 3.   Given a Trio of Propositions of Relation, of which every two contain a Pair of codivisional Classes, and which are proposed as a Syllogism: to ascertain whether the proposed Conclusion is consequent from the proposed Premisses, and, if so, whether it is complete.   Rules 66  Examples worked briefly, as models 〃  pg_xxviBOOK VI.   THE METHOD OF SUBSCRIPTS.   CHAPTER I.   INTRODUCTORY.   Meaning of x1, xy1, &c. 70  ‘Entity’ 〃  Meaning of x0, xy0, &c. 〃  ‘Nullity’ 〃  The Symbols “†” and “¶” 〃  ‘Like’ and ‘unlike’ Signs 〃  CHAPTER II.   REPRESENTATION OF PROPOSITIONS OF RELATION.   The Pair of Converse Propositions “Some x are y” = “Some y are x” 71  Three other similar Pairs 〃  The Pair of Converse Propositions “No x are y” = “No y are x” 〃  Three other similar Pairs 〃  The Proposition “All x are y” 72  The Proposition “All x are y” is Double, and is equivalent to the two Propositions “Some x exist” and “No x and y′” 〃  Seven other similar Propositions 〃  Rule for translating “All x are y” from abstract into subscript form, and vice versâ 〃  pg_xxviiCHAPTER III.   SYLLOGISMS.   § 1.   Representation of Syllogisms.   Rules 73  § 2.   Formulæ for Syllogisms.   Three Formulæ worked out:—   Fig. I. xm0 † ym′0 ¶ xy0 75  its two Variants (α) and (β) 〃  Fig. II. xm0 † ym1 ¶ x′y1 76  Fig. III. xm0 † ym0 † m1 ¶ x′y′1 77  Table IX. Formulæ and Rules 78  Examples worked briefly, as models 〃  § 3.   Fallacies.   ‘Fallacy’ 81  Method of finding Forms of Fallacies 82  Forms best stated in words 〃  Three Forms of Fallacies:—   (1) Fallacy of Like Eliminands not asserted to exist 〃  (2) Fallacy of Unlike Eliminands with an Entity-Premiss 83  (3) Fallacy of two Entity-Premisses 〃  § 4.   Method of proceeding with a given Pair of Propositions.   Rules 84  pg_xxviiiBOOK VII.   SORITESES.   CHAPTER I.   INTRODUCTORY.   ‘Sorites’ 85  ‘Premisses’ 〃  ‘Partial Conclusion’ 〃  ‘Complete Conclusion’ (or ‘Conclusion’) 〃  ‘Eliminands’ 〃  ‘Retinends’ 〃  ‘consequent’ 〃  The Symbol “∴” 〃  Specimen-Soriteses 86  CHAPTER II.   PROBLEMS IN SORITESES.   § 1.   Introductory.   Form of Problem 87  Two Methods of Solution 〃  § 2.   Solution by Method of Separate Syllogisms.   Rules 88  Example worked 〃  pg_xxix§ 3.   Solution by Method of Underscoring.   ‘Underscoring’ 91  Subscripts to be omitted 〃  Example worked fully 92  Example worked briefly, as model 93  Seventeen Examination-Papers 94  BOOK VIII.   EXAMPLES, WITH ANSWERS AND SOLUTIONS.   CHAPTER I.   EXAMPLES.   § 1.   Propositions of Relation, to be reduced to normal form 97  § 2.   Pairs of Abstract Propositions, one in terms of x and m, and the other in terms of y and m, to be represented on the same Triliteral Diagram 98  § 3.   Marked Triliteral Diagrams, to be interpreted in terms of x and y 99  § 4.   Pairs of Abstract Propositions, proposed as Premisses: Conclusions to be found 100  pg_xxx§ 5.   Pairs of Concrete Propositions, proposed as Premisses: Conclusions to be found 101  § 6.   Trios of Abstract Propositions, proposed as Syllogisms: to be examined 106  § 7.   Trios of Concrete Propositions, proposed as Syllogisms: to be examined 107  § 8.   Sets of Abstract Propositions, proposed as Premisses for Soriteses: Conclusions to be found 110  § 9.   Sets of Concrete Propositions, proposed as Premisses for Soriteses: Conclusions to be found 112  CHAPTER II.   ANSWERS.   Answers to   § 1 125  § 2 126  § 3 127  § 4 〃  § 5 128  § 6 130  § 7 131  § 8 132  § 9 〃  pg_xxxiCHAPTER III.   SOLUTIONS.   § 1.   Propositions of Relation reduced to normal form.   Solutions for § 1 134  § 2.   Method of Diagrams.   Solutions for   § 4 Nos. 1 to 12 136  § 5  〃  1 to 12 138  § 6  〃  1 to 10 141  § 7  〃  1 to 6 144  § 3.   Method of Subscripts.   Solutions for   § 4 146  § 5 Nos. 13 to 24 147  § 6 148  § 7 150  § 8 155  § 9 157  NOTES 164  APPENDIX, ADDRESSED TO TEACHERS 165  NOTES TO APPENDIX 195  INDEX.   § 1. Tables 197  § 2. Words &c. explained 〃

pg_xxxii
pg001BOOK I.

THINGS AND THEIR ATTRIBUTES.

CHAPTER I.

INTRODUCTORY.

The Universe contains ‘Things.’

Things have ‘Attributes.’

Any Attribute, or any Set of Attributes, may be called an ‘Adjunct.’

pg001½CHAPTER II.

CLASSIFICATION.

‘Classification,’ or the formation of Classes, is a Mental Process, in which we imagine that we have put together, in a group, certain Things. Such a group is called a ‘Class.’

(2) We may think of the Class “Things,” and may imagine that we have picked out from it all the Things which possess a certain Adjunct not possessed by the whole Class. This Adjunct is said to be ‘peculiar’ to the Class so formed. In this case, the Class “Things” is called a ‘Genus’ with regard to the Class so formed: the Class, so formed, is called a ‘Species’ of the Class “Things”: and its peculiar Adjunct is called its ‘Differentia’.

pg002As this Process is entirely Mental, we can perform it whether there is, or is not, an existing Thing which possesses that Adjunct. If there is, the Class is said to be ‘Real’; if not, it is said to be ‘Unreal’, or ‘Imaginary.’

A Class, containing only one Member is called an ‘Individual.’

pg002½Hence, any single Thing, which we can name so as to distinguish it from all other Things, may be regarded as a one-Member Class.

pg003CHAPTER III.

DIVISION.

§ 1.

Introductory.

‘Division’ is a Mental Process, in which we think of a certain Class of Things, and imagine that we have divided it into two or more smaller Classes.

A Class, that has been obtained by a certain Division, is said to be ‘codivisional’ with every Class obtained by that Division.

pg003½§ 2.

Dichotomy.

If we think of a certain Class, and imagine that we have picked out from it a certain smaller Class, it is evident that the Remainder of the large Class does not possess the Differentia of that smaller Class. Hence it may be regarded as another smaller Class, whose Differentia may be formed, from that of the Class first picked out, by prefixing the word “not”; and we may imagine that we have divided the Class first thought of into two smaller Classes, whose Differentiæ are contradictory. This kind of Division is called ‘Dichotomy’.

In performing this Process, we may sometimes find that the Attributes we have chosen are used so loosely, in ordinary conversation, that it is not easy to decide which of the Things belong to the one Class and which to the other. In such a case, it would be necessary to lay down some arbitrary rule, as to where the one Class should end and the other begin.

pg004After dividing a Class, by the Process of Dichotomy, into two smaller Classes, we may sub-divide each of these into two still smaller Classes; and this Process may be repeated over and over again, the number of Classes being doubled at each repetition.

pg004½CHAPTER IV.

NAMES.

The word “Thing”, which conveys the idea of a Thing, without any idea of an Adjunct, represents any single Thing. Any other word (or phrase), which conveys the idea of a Thing, with the idea of an Adjunct represents any Thing which possesses that Adjunct; i.e., it represents any Member of the Class to which that Adjunct is peculiar.

Just as a Class is said to be Real, or Unreal, according as there is, or is not, an existing Thing in it, so also a Name is said to be Real, or Unreal, according as there is, or is not, an existing Thing represented by it.

Every Name, except “Thing”, may usually be expressed in three different forms:—

pg005(b) A Substantive, conveying the idea of a Thing with the ideas of some of the Attributes, and one or more Adjectives (or phrases used as Adjectives) conveying the ideas of the other Attributes;

(c) A Substantive conveying the idea of a Thing with the ideas of all the Attributes.

[Thus, the phrase “material living Thing, belonging to the Animal Kingdom, having two hands and two feet” is a Name expressed in Form (a).

If we choose to roll up together the Substantive “Thing” and the Adjectives “material, living, belonging to the Animal Kingdom,” so as to make the new Substantive “Animal,” we get the phrase “Animal having two hands and two feet,” which is a Name (representing the same Thing as before) expressed in Form (b).

And, if we choose to roll up the whole phrase into one word, so as to make the new Substantive “Man,” we get a Name (still representing the very same Thing) expressed in Form (c).]

A Name, whose Substantive is in the plural number, may be used to represent either

pg006CHAPTER V.

DEFINITIONS.

It is evident that every Member of a Species is also a Member of the Genus out of which that Species has been picked, and that it possesses the Differentia of that Species. Hence it may be represented by a Name consisting of two parts, one being a Name representing any Member of the Genus, and the other being the Differentia of that Species. Such a Name is called a ‘Definition’ of any Member of that Species, and to give it such a Name is to ‘define’ it.

The following Examples, of this Process, may be taken as models for working others.

pg0074. Define “Men.”

Ans. “material, living Things, belonging to the Animal Kingdom, having two hands and two feet”;

or else

“Animals having two hands and two feet.”

5. Define “London.”

Ans. “the material artificial Thing, which consists of houses and streets, and has four million inhabitants”;

or else

“the Town which has four million inhabitants.”

[Note that we here use the article “the” instead of “a”, because we happen to know that there is only one such Thing.

The Reader can set himself any number of Examples of this Process, by simply choosing the Name of any common Thing (such as “house,” “tree,” “knife”), making a Definition for it, and then testing his answer by referring to any English Dictionary.]

pg008BOOK II.

PROPOSITIONS.

CHAPTER I.

PROPOSITIONS GENERALLY.

§ 1.

Introductory.

Note that the word “some” is to be regarded, henceforward, as meaning “one or more.”

The word ‘Proposition,’ as used in ordinary conversation, may be applied to any word, or phrase, which conveys any information whatever.

But a ‘Proposition,’ as used in this First Part of “Symbolic Logic,” has a peculiar form, which may be called its ‘Normal pg009form’; and if any Proposition, which we wish to use in an argument, is not in normal form, we must reduce it to such a form, before we can use it.

A ‘Proposition,’ when in normal form, asserts, as to certain two Classes, which are called its ‘Subject’ and ‘Predicate,’ either

The Subject and the Predicate of a Proposition are called its ‘Terms.’

§ 2.

Normal form of a Proposition.

A Proposition, in normal form, consists of four parts, viz.—

(1) The word “some,” or “no,” or “all.” (This word, which tells us how many Members of the Subject are also Members of the Predicate, is called the ‘Sign of Quantity.’)

(2) Name of Subject.

(3) The verb “are” (or “is”). (This is called the ‘Copula.’)

(4) Name of Predicate.

pg010§ 3.

Various kinds of Propositions.

A Proposition, that begins with “Some”, is said to be ‘Particular.’ It is also called ‘a Proposition in I.’

A Proposition, that begins with “No”, is said to be ‘Universal Negative.’ It is also called ‘a Proposition in E.’

A Proposition, that begins with “All”, is said to be ‘Universal Affirmative.’ It is also called ‘a Proposition in A.’

A Proposition, whose Subject is an Individual, is to be regarded as Universal.

Propositions are of two kinds, ‘Propositions of Existence’ and ‘Propositions of Relation.’

pg011CHAPTER II.

PROPOSITIONS OF EXISTENCE.

A ‘Proposition of Existence’, when in normal form, has, for its Subject, the Class “existing Things”.

[Note that, though its Sign of Quantity tells us how many existing Things are Members of its Predicate, it does not tell us the exact number: in fact, it only deals with two numbers, which are, in ascending order, “0” and “1 or more.”]

[Thus, the Proposition “Some existing Things are honest men” asserts that the Class “honest men” is Real.

This is the normal form; but it may also be expressed in any one of the following forms:—

(1) “Honest men exist”;
(2) “Some honest men exist”;
(3) “The Class ‘honest men’ exists”;
(4) “There are honest men”;
(5) “There are some honest men”.

Similarly, the Proposition “No existing Things are men fifty feet high” asserts that the Class “men 50 feet high” is Imaginary.

This is the normal form; but it may also be expressed in any one of the following forms:—

(1) “Men 50 feet high do not exist”;
(2) “No men 50 feet high exist”;
(3) “The Class ‘men 50 feet high’ does not exist”;
(4) “There are not any men 50 feet high”;
(5) “There are no men 50 feet high.”]

pg012CHAPTER III.

PROPOSITIONS OF RELATION.

§ 1.

Introductory.

A Proposition of Relation, of the kind to be here discussed, has, for its Terms, two Specieses of the same Genus, such that each of the two Names conveys the idea of some Attribute not conveyed by the other.

[Thus, the Proposition “Some merchants are misers” is of the right kind, since “merchants” and “misers” are Specieses of the same Genus “men”; and since the Name “merchants” conveys the idea of the Attribute “mercantile”, and the name “misers” the idea of the Attribute “miserly”, each of which ideas is not conveyed by the other Name.

But the Proposition “Some dogs are setters” is not of the right kind, since, although it is true that “dogs” and “setters” are Specieses of the same Genus “animals”, it is not true that the Name “dogs” conveys the idea of any Attribute not conveyed by the Name “setters”. Such Propositions will be discussed in Part II.]

The Genus, of which the two Terms are Specieses, is called the ‘Universe of Discourse,’ or (more briefly) the ‘Univ.’

[Note that, though its Sign of Quantity tells us how many Members of its Subject are also Members of its Predicate, it does not tell us the exact number: in fact, it only deals with three numbers, which are, in ascending order, “0”, “1 or more”, “the total number of Members of the Subject”.]

pg013§ 2.

Reduction of a Proposition of Relation to Normal form.

The Rules, for doing this, are as follows:—

[Let us work a few Examples, to illustrate these Rules.

(1)

“Some apples are not ripe.”

(1) The Subject is “apples.”

(2) The Verb is “are.”

(3) The Predicate is “not-ripe * * *.” (As no Substantive is expressed, and we have not yet settled what the Univ. is to be, we are forced to leave a blank.)

(4) Let Univ. be “fruit.”

(5) The Sign of Quantity is “some.”

(6) The Proposition now becomes

“Some | apples | are | not-ripe fruit.”

pg014(2)

“None of my speculations have brought me as much as 5 per cent.”

(1) The Subject is “my speculations.”

(2) The Verb is “have brought,” for which we substitute the phrase “are * * * that have brought”.

(3) The Predicate is “* * * that have brought &c.”

(4) Let Univ. be “transactions.”

(5) The Sign of Quantity is “none of.”

(6) The Proposition now becomes

“None of | my speculations | are | transactions that have brought me as much as 5 per cent.”

(3)

“None but the brave deserve the fair.”

To begin with, we note that the phrase “none but the brave” is equivalent to “no not-brave.”

(1) The Subject has for its Attribute “not-brave.” But no Substantive is supplied. So we express the Subject as “not-brave * * *.”

(2) The Verb is “deserve,” for which we substitute the phrase “are deserving of”.

(3) The Predicate is “* * * deserving of the fair.”

(4) Let Univ. be “persons.”

(5) The Sign of Quantity is “no.”

(6) The Proposition now becomes

“No | not-brave persons | are | persons deserving of the fair.”

(4)

“A lame puppy would not say “thank you” if you offered to lend it a skipping-rope.”

(1) The Subject is evidently “lame puppies,” and all the rest of the sentence must somehow be packed into the Predicate.

(2) The Verb is “would not say,” &c., for which we may substitute the phrase “are not grateful for.”

(3) The Predicate may be expressed as “* * * not grateful for the loan of a skipping-rope.”

(4) Let Univ. be “puppies.”

(5) The Sign of Quantity is “all.”

(6) The Proposition now becomes

“All | lame puppies | are | puppies not grateful for the loan of a skipping-rope.”

pg015(5)

“No one takes in the Times, unless he is well-educated.”

(1) The Subject is evidently persons who are not well-educated (“no one” evidently means “no person”).

(2) The Verb is “takes in,” for which we may substitute the phrase “are persons taking in.”

(3) The Predicate is “persons taking in the Times.”

(4) Let Univ. be “persons.”

(5) The Sign of Quantity is “no.”

(6) The Proposition now becomes

“No | persons who are not well-educated | are | persons taking in the Times.”

(6)

“My carriage will meet you at the station.”

(1) The Subject is “my carriage.” This, being an ‘Individual,’ is equivalent to the Class “my carriages.” (Note that this Class contains only one Member.)

(2) The Verb is “will meet”, for which we may substitute the phrase “are * * * that will meet.”

(3) The Predicate is “* * * that will meet you at the station.”

(4) Let Univ. be “things.”

(5) The Sign of Quantity is “all.”

(6) The Proposition now becomes

“All | my carriages | are | things that will meet you at the station.”

(7)

“Happy is the man who does not know what ‘toothache’ means!”

(1) The Subject is evidently “the man &c.” (Note that in this sentence, the Predicate comes first.) At first sight, the Subject seems to be an ‘Individual’; but on further consideration, we see that the article “the” does not imply that there is only one such man. Hence the phrase “the man who” is equivalent to “all men who”.

(2) The Verb is “are.”

(3) The Predicate is “happy * * *.”

(4) Let Univ. be “men.”

(5) The Sign of Quantity is “all.”

(6) The Proposition now becomes

“All | men who do not know what ‘toothache’ means | are | happy men.”

pg016(8)

“Some farmers always grumble at the weather, whatever it may be.”

(1) The Subject is “farmers.”

(2) The Verb is “grumble,” for which we substitute the phrase “are * * * who grumble.”

(3) The Predicate is “* * * who always grumble &c.”

(4) Let Univ. be “persons.”

(5) The Sign of Quantity is “some.”

(6) The Proposition now becomes

“Some | farmers | are | persons who always grumble at the weather, whatever it may be.”

(9)

“No lambs are accustomed to smoke cigars.”

(1) The Subject is “lambs.”

(2) The Verb is “are.”

(3) The Predicate is “* * * accustomed &c.”

(4) Let Univ. be “animals.”

(5) The Sign of Quantity is “no.”

(6) The Proposition now becomes

“No | lambs | are | animals accustomed to smoke cigars.”

(10)

“I ca’n’t understand examples that are not arranged in regular order, like those I am used to.”

(1) The Subject is “examples that,” &c.

(2) The Verb is “I ca’n’t understand,” which we must alter, so as to have “examples,” instead of “I,” as the nominative case. It may be expressed as “are not understood by me.”

(3) The Predicate is “* * * not understood by me.”

(4) Let Univ. be “examples.”

(5) The Sign of Quantity is “all.”

(6) The Proposition now becomes

“All | examples that are not arranged in regular order like those I am used to | are | examples not understood by me.”]

pg017§ 3.

A Proposition of Relation, beginning with “All”, is a Double Proposition.

A Proposition of Relation, beginning with “All”, asserts (as we already know) that “All Members of the Subject are Members of the Predicate”. This evidently contains, as a part of what it tells us, the smaller Proposition “Some Members of the Subject are Members of the Predicate”.

pg018Now we know that every Member of the Subject is (as shown at p. 6) a Member of the Univ. Hence every Member of the Subject is either in Class (1) or else in Class (2).

[Thus, we know that every banker is a Member of the Genus “men”. Hence, every banker is either in the Class “rich men”, or else in the Class “poor men”.]

Also we have been told that, in the case we are discussing, some Members of the Subject are in Class (1). What else do we need to be told, in order to know that all of them are there? Evidently we need to be told that none of them are in Class (2); i.e. that none of them are Members of the Class whose Differentia is contradictory to that of the Predicate.

[Thus, we may suppose we have been told that some bankers are in the Class “rich men”. What else do we need to be told, in order to know that all of them are there? Evidently we need to be told that none of them are in the Class “poor men”.]

Hence a Proposition of Relation, beginning with “All”, is a Double Proposition, and is ‘equivalent’ to (i.e. gives the same information as) the two Propositions

pg019§ 4.

What is implied, in a Proposition of Relation, as to the Reality of its Terms?

A Proposition of Relation, beginning with “Some”, is henceforward to be understood as asserting that there are some existing Things, which, being Members of the Subject, are also Members of the Predicate; i.e. that some existing Things are Members of both Terms at once. Hence it is to be understood as implying that each Term, taken by itself, is Real.

A Proposition of Relation, beginning with “No”, is henceforward to be understood as asserting that there are no existing Things which, being Members of the Subject, are also Members of the Predicate; i.e. that no existing Things are Members of both Terms at once. But this implies nothing as to the Reality of either Term taken by itself.

A Proposition of Relation, beginning with “All”, contains (see § 3) a similar Proposition beginning with “Some”. Hence it is to be understood as implying that each Term, taken by itself, is Real.

[Thus, the Proposition “All hyænas are savage animals” contains the Proposition “Some hyænas are savage animals”. Hence it implies that each of the two Classes, “hyænas” and “savage animals”, taken by itself, is Real.]

pg020§ 5.

Translation of a Proposition of Relation into one or more Propositions of Existence.

A Proposition of Relation, beginning with “All”, is (as shown in § 3) equivalent to two Propositions, one beginning with “Some” and the other with “No”, each of which we now know how to translate.

[Let us work a few Examples, to illustrate these Rules.

(1)

“Some apples are not ripe.”

Here we arrange thus:—

“Some”		Sign of Quantity.
“existing Things”		Subject.
“are”		Copula.
“not-ripe apples”		Predicate.

or thus:—

“Some | existing Things | are | not-ripe apples.”

pg021(2)

“Some farmers always grumble at the weather, whatever it may be.”

Here we arrange thus:—

“Some | existing Things | are | farmers who always grumble at the weather, whatever it may be.”

(3)

“No lambs are accustomed to smoke cigars.”

Here we arrange thus:—

“No | existing Things |are | lambs accustomed to smoke cigars.”

(4)

“None of my speculations have brought me as much as 5 per cent.”

Here we arrange thus:—

“No | existing Things | are | speculations of mine, which have brought me as much as 5 per cent.”

(5)

“None but the brave deserve the fair.”

Here we note, to begin with, that the phrase “none but the brave” is equivalent to “no not-brave men.” We then arrange thus:—

“No | existing Things | are | not-brave men deserving of the fair.”

(6)

“All bankers are rich men.”

This is equivalent to the two Propositions “Some bankers are rich men” and “No bankers are poor men.”

Here we arrange thus:—

“Some | existing Things | are | rich bankers”; and “No | existing Things | are | poor bankers.”]

[Work Examples § 1, 1–4 (p. 97).]

pg022BOOK III.

THE BILITERAL DIAGRAM.

CHAPTER I.

SYMBOLS AND CELLS.

First, let us suppose that the above Diagram is an enclosure assigned to a certain Class of Things, which we have selected as our ‘Universe of Discourse.’ or, more briefly, as our ‘Univ’.

pg023Secondly, let us suppose that we have selected a certain Adjunct, which we may call “x,” and have divided the large Class, to which we have assigned the whole Diagram, into the two smaller Classes whose Differentiæ are “x” and “not-x” (which we may call “x′”), and that we have assigned the North Half of the Diagram to the one (which we may call “the Class of x-Things,” or “the x-Class”), and the South Half to the other (which we may call “the Class of x′-Things,” or “the x′-Class”).

Thirdly, let us suppose that we have selected another Adjunct, which we may call “y”, and have subdivided the x-Class into the two Classes whose Differentiæ are “y” and “y′”, and that we have assigned the North-West Cell to the one (which we may call “the xy-Class”), and the North-East Cell to the other (which we may call “the xy′-Class”).

Fourthly, let us suppose that we have subdivided the x′-Class in the same manner, and have assigned the South-West Cell to the x′y-Class, and the South-East Cell to the x′y′-Class.

It is evident that, if we had begun by dividing for y and y′, and had then subdivided for x and x′, we should have got the pg024same four Classes. Hence we see that we have assigned the West Half to the y-Class, and the East Half to the y′-Class.

[Thus, in the above Example, we should find that we had assigned the West Half of the table to “English books” and the East Half to “foreign books.”
 

We have, in fact, assigned the four Quarters of the table to four different Classes of books, as here shown.]

The Reader should carefully remember that, in such a phrase as “the x-Things,” the word “Things” means that particular kind of Things, to which the whole Diagram has been assigned.

[Thus, if we say “Let Univ. be ‘books’,” we mean that we have assigned the whole Diagram to “books.” In that case, if we took “x” to mean “old”, the phrase “the x-Things” would mean “the old books.”]

The Reader should not go on to the next Chapter until he is quite familiar with the blank Diagram I have advised him to draw.

He ought to be able to name, instantly, the Adjunct assigned to any Compartment named in the right-hand column of the following Table.

Also he ought to be able to name, instantly, the Compartment assigned to any Adjunct named in the left-hand column.

To make sure of this, he had better put the book into the hands of some genial friend, while he himself has nothing but the blank Diagram, and get that genial friend to question him on this Table, dodging about as much as possible. The Questions and Answers should be something like this:—

After a little practice, he will find himself able to do without the blank Diagram, and will be able to see it mentally (“in my mind’s eye, Horatio!”) while answering the questions of his genial friend. When this result has been reached, he may safely go on to the next Chapter.

pg026CHAPTER II.

COUNTERS.

Let us agree that a Red Counter, placed within a Cell, shall mean “This Cell is occupied” (i.e. “There is at least one Thing in it”).

Let us also agree that a Red Counter, placed on the partition between two Cells, shall mean “The Compartment, made up of these two Cells, is occupied; but it is not known whereabouts, in it, its occupants are.” Hence it may be understood to mean “At least one of these two Cells is occupied: possibly both are.”

Our ingenious American cousins have invented a phrase to describe the condition of a man who has not yet made up his mind which of two political parties he will join: such a man is said to be “sitting on the fence.” This phrase exactly describes the condition of the Red Counter.

Let us also agree that a Grey Counter, placed within a Cell, shall mean “This Cell is empty” (i.e. “There is nothing in it”).

pg027CHAPTER III.

REPRESENTATION OF PROPOSITIONS.

§ 1.

Introductory.

Henceforwards, in stating such Propositions as “Some x-Things exist” or “No x-Things are y-Things”, I shall omit the word “Things”, which the Reader can supply for himself, and shall write them as “Some x exist” or “No x are y”.

[Note that the word “Things” is here used with a special meaning, as explained at p. 23.]

A Proposition, containing only one of the Letters used as Symbols for Attributes, is said to be ‘Uniliteral’.

A Proposition, containing two Letters, is said to be ‘Biliteral’.

A Proposition is said to be ‘in terms of’ the Letters it contains, whether with or without accents.

pg028§ 2.

Representation of Propositions of Existence.

Let us take, first, the Proposition “Some x exist”.

[Note that this Proposition is (as explained at p. 12) equivalent to “Some existing Things are x-Things.”]

This tells us that there is at least one Thing in the North Half; that is, that the North Half is occupied. And this we can evidently represent by placing a Red Counter (here represented by a dotted circle) on the partition which divides the North Half.

[In the “books” example, this Proposition would be “Some old books exist”.]

Similarly we may represent the three similar Propositions “Some x′ exist”, “Some y exist”, and “Some y′ exist”.

Let us take, next, the Proposition “No x exist”.

pg029Similarly we may represent the three similar Propositions “No x′ exist”, “No y exist”, and “No y′ exist”.

Let us take, next, the Proposition “Some xy exist”.

Similarly we may represent the three similar Propositions “Some xy′ exist”, “Some x′y exist”, and “Some x′y′ exist”.

Let us take, next, the Proposition “No xy exist”.

Similarly we may represent the three similar Propositions “No xy′ exist”, “No x′y exist”, and “No x′y′ exist”.

pg030

We have seen that the Proposition “No x exist” may be represented by placing two Grey Counters in the North Half, one in each Cell.

We have also seen that these two Grey Counters, taken separately, represent the two Propositions “No xy exist” and “No xy′ exist”.

Hence we see that the Proposition “No x exist” is a Double Proposition, and is equivalent to the two Propositions “No xy exist” and “No xy′ exist”.

§ 3.

Representation of Propositions of Relation.

Let us take, first, the Proposition “Some x are y”.

Similarly we may represent the three similar Propositions “Some x are y′”, “Some x′ are y”, and “Some x′ are y′”.

pg031Let us take, next, the Proposition “Some y are x”.

Similarly we may represent the three similar Propositions “Some y are x′”, “Some y′ are x”, and “Some y′ are x′”.

The two equivalent Propositions, “Some x are y” and “Some y are x”, are said to be ‘Converse’ to each other; and the Process, of changing one into the other, is called ‘Converting’, or ‘Conversion’.

pg032Similarly we may represent the three similar Trios of equivalent Propositions; the whole Set of four Trios being as follows:—

Let us take, next, the Proposition “No x are y”.

Similarly we may represent the three similar Propositions “No x are y′”, and “No x′ are y”, and “No x′ are y′”.

Let us take, next, the Proposition “No y are x”.

Similarly we may represent the three similar Propositions “No y are x′”, “No y′ are x”, and “No y′ are x′”.

pg033

Similarly we may represent the three similar Trios of equivalent Propositions; the whole Set of four Trios being as follows:—

We know (see p. 17) that this is a Double Proposition, and equivalent to the two Propositions “Some x are y” and “No x are y′”, each of which we already know how to represent.

[Note that the Subject of the given Proposition settles which Half we are to use; and that its Predicate settles in which portion of that Half we are to place the Red Counter.]

pg034TABLE II.   Some x exist     No x exist     Some x′ exist     No x′ exist     Some y exist     No y exist     Some y′ exist     No y′ exist

Similarly we may represent the seven similar Propositions “All x are y′”, “All x′ are y”, “All x′ are y′”, “All y are x”, “All y are x′”, “All y′ are x”, and “All y′ are x′”.

 
Let us take, lastly, the Double Proposition “Some x are y and some are y′”, each part of which we already know how to represent.

Similarly we may represent the three similar Propositions, “Some x′ are y and some are y′”, “Some y are x and some are x′”, “Some y′ are x and some are x′”.

The Reader should now get his genial friend to question him, severely, on these two Tables. The Inquisitor should have the Tables before him: but the Victim should have nothing but a blank Diagram, and the Counters with which he is to represent the various Propositions named by his friend, e.g. “Some y exist”, “No y′ are x”, “All x are y”, &c. &c.

pg035TABLE III.   Some xy exist    = Some x are y    = Some y are x     All x are y     Some xy′ exist    = Some x are y′    = Some y′ are x     All x are y′     Some x′y exist    = Some x′ are y    = Some y are x′     All x′ are y     Some x′y′ exist    = Some x′ are y′    = Some y′ are x′     All x′ are y′     No xy exist    = No x are y    = No y are x      All y are x     No xy′ exist    = No x are y′    = No y′ are x     All y are x′     No x′y exist    = No x′ are y    = No y are x′     All y′ are x     No x′y′ exist    = No x′ are y′    = No y′ are x′     All y′ are x′     Some x are y,   and some are y′     Some y are x   and some are x′     Some x′ are y,   and some are y′     Some y′ are x   and some are x′

pg036CHAPTER IV.

INTERPRETATION OF BILITERAL DIAGRAM WHEN MARKED WITH COUNTERS.

Similarly we may interpret a Red Counter, when placed in the North-East, or South-West, or South-East Cell.

Similarly we may interpret a Grey Counter, when placed in the North-East, or South-West, or South-East Cell.

pg037

Similarly we may interpret a Red Counter, when placed on the partition which divides the South, or West, or East Half.

Similarly we may interpret two Red Counters, when placed in the South, or West, or East Half.

Similarly we may interpret two Grey Counters, when placed in the South, or West, or East Half.

pg038Similarly we may interpret a Red and a Grey counter, when placed in any one of the seven similar positions

[Work Examples § 1, 5–8 (p. 97).]

pg039BOOK IV.

THE TRILITERAL DIAGRAM.

CHAPTER I.

SYMBOLS AND CELLS.

First, let us suppose that the above left-hand Diagram is the Biliteral Diagram that we have been using in Book III., and that we change it into a Triliteral Diagram by drawing an Inner Square, so as to divide each of its 4 Cells into 2 portions, thus making 8 Cells altogether. The right-hand Diagram shows the result.

pg040Secondly, let us suppose that we have selected a certain Adjunct, which we may call “m”, and have subdivided the xy-Class into the two Classes whose Differentiæ are m and m′, and that we have assigned the N.W. Inner Cell to the one (which we may call “the Class of xym-Things”, or “the xym-Class”), and the N.W. Outer Cell to the other (which we may call “the Class of xym′-Things”, or “the xym′-Class”).

Thirdly, let us suppose that we have subdivided the xy′-Class, the x′y-Class, and the x′y′-Class in the same manner, and have, in each case, assigned the Inner Cell to the Class possessing the Attribute m, and the Outer Cell to the Class possessing the Attribute m′.

It is evident that we have now assigned the Inner Square to the m-Class, and the Outer Border to the m′-Class.

When the Reader has made himself familiar with this Diagram, he ought to be able to find, in a moment, the Compartment assigned to a particular pair of Attributes, or the Cell assigned to a particular trio of Attributes. The following Rules will help him in doing this:—

(1) Arrange the Attributes in the order x, y, m.
pg041 (2) Take the first of them and find the Compartment assigned to it.
(3) Then take the second, and find what portion of that compartment is assigned to it.
(4) Treat the third, if there is one, in the same way.

[For example, suppose we have to find the Compartment assigned to ym. We say to ourselves “y has the West Half; and m has the Inner portion of that West Half.”

Again, suppose we have to find the Cell assigned to x′ym′. We say to ourselves “x′ has the South Half; y has the West portion of that South Half, i.e. has the South-West Quarter; and m′ has the Outer portion of that South-West Quarter.”]

The Reader should now get his genial friend to question him on the Table given on the next page, in the style of the following specimen-Dialogue.

pg043CHAPTER II.

REPRESENTATION OF PROPOSITIONS IN TERMS OF x AND m, OR OF y AND m.

§ 1.

Representation of Propositions of Existence in terms of x and m, or of y and m.

Let us take, first, the Proposition “Some xm exist”.

[Note that the full meaning of this Proposition is (as explained at p. 12) “Some existing Things are xm-Things”.]

This tells us that there is at least one Thing in the Inner portion of the North Half; that is, that this Compartment is occupied. And this we can evidently represent by placing a Red Counter on the partition which divides it.

[In the “books” example, this Proposition would mean “Some old bound books exist” (or “There are some old bound books”).]

Similarly we may represent the seven similar Propositions, “Some xm′ exist”, “Some x′m exist”, “Some x′m′ exist”, “Some ym exist”, “Some ym′ exist”, “Some y′m exist”, and “Some y′m′ exist”.

pg044Let us take, next, the Proposition “No xm exist”.

Similarly we may represent the seven similar Propositions, in terms of x and m, or of y and m, viz. “No xm′ exist”, “No x′m exist”, &c.

§ 2.

Representation of Propositions of Relation in terms of x and m, or of y and m.

Let us take, first, the Pair of Converse Propositions

Similarly for the seven similar Pairs, in terms of x and m, or of y and m.

Let us take, next, the Pair of Converse Propositions

Similarly for the seven similar Pairs, in terms of x and m, or of y and m.

pg045Let us take, next, the Proposition “All x are m.”

We know (see p. 18) that this is a Double Proposition, and equivalent to the two Propositions “Some x are m” and “No x are m′ ”, each of which we already know how to represent.

Similarly for the fifteen similar Propositions, in terms of x and m, or of y and m.

These thirty-two Propositions of Relation are the only ones that we shall have to represent on this Diagram.

The Reader should now get his genial friend to question him on the following four Tables.

The Victim should have nothing before him but a blank Triliteral Diagram, a Red Counter, and 2 Grey ones, with which he is to represent the various Propositions named by the Inquisitor, e.g. “No y′ are m”, “Some xm′ exist”, &c., &c.

pg046TABLE V.    Some xm exist     = Some x are m     = Some m are x       No xm exist     = No x are m     = No m are x       Some xm′ exist     = Some x are m′     = Some m′ are x       No xm′ exist     = No x are m′     = No m′ are x       Some x′m exist     = Some x′ are m     = Some m are x′       No x′m exist     = No x′ are m     = No m are x′       Some x′m′ exist     = Some x′ are m′     = Some m′ are x′       No x′m′ exist     = No x′ are m′     = No m′ are x′

pg047TABLE VI.    Some ym exist     = Some y are m     = Some m are y       No ym exist     = No y are m     = No m are y       Some ym′ exist     = Some y are m′     = Some m′ are y       No ym′ exist     = No y are m′     = No m′ are y       Some y′m exist     = Some y′ are m     = Some m are y′       No y′m exist     = No y′ are m     = No m are y′       Some y′m′ exist     = Some y′ are m′     = Some m′ are y′       No y′m′ exist     = No y′ are m′     = No m′ are y′

pg048TABLE VII.    All x are m       All x are m′       All x′ are m       All x′ are m′       All m are x       All m are x′       All m′ are x       All m′ are x′

pg049TABLE VIII.    All y are m       All y are m′       All y′ are m       All y′ are m′       All m are y       All m are y′       All m′ are y       All m′ are y′

pg050CHAPTER III.

REPRESENTATION OF TWO PROPOSITIONS OF RELATION, ONE IN TERMS OF x AND m, AND THE OTHER IN TERMS OF y AND m, ON THE SAME DIAGRAM.

The Reader had better now begin to draw little Diagrams for himself, and to mark them with the Digits “I” and “O”, instead of using the Board and Counters: he may put a “I” to represent a Red Counter (this may be interpreted to mean “There is at least one Thing here”), and a “O” to represent a Grey Counter (this may be interpreted to mean “There is nothing here”).

When we have to represent a Proposition beginning with “All”, we break it up into the two Propositions to which it is equivalent.

[Let us work a few examples.

(1)

“No x are m′;
  No y′ are m”.

Let us first represent “No x are m′”. This gives us Diagram a.

Then, representing “No y′ are m” on the same Diagram, we get Diagram b.

pg051a     b

(2)

“Some m are x;
  No m are y”.

If, neglecting the Rule, we were begin with “Some m are x”, we should get Diagram a.

And if we were then to take “No m are y”, which tells us that the Inner N.W. Cell is empty, we should be obliged to take the “I” off the fence (as it no longer has the choice of two Cells), and to put it into the Inner N.E. Cell, as in Diagram c.

This trouble may be saved by beginning with “No m are y”, as in Diagram b.

And now, when we take “Some m are x”, there is no fence to sit on! The “I” has to go, at once, into the N.E. Cell, as in Diagram c.

a     b     c

(3)

“No x′ are m′;
  All m are y”.

Here we begin by breaking up the Second into the two Propositions to which it is equivalent. Thus we have three Propositions to represent, viz.—

(1) “No x′ are m′;
(2)   Some m are y;
(3)   No m are y′”.

These we will take in the order 1, 3, 2.

First we take No. (1), viz. “No x′ are m′”. This gives us Diagram a.

pg052Adding to this, No. (3), viz. “No m are y′”, we get Diagram b.

This time the “I”, representing No. (2), viz. “Some m are y,” has to sit on the fence, as there is no “O” to order it off! This gives us Diagram c.

a     b     c

(4)

“All m are x;
  All y are m”.

Here we break up both Propositions, and thus get four to represent, viz.—

(1) “Some m are x;
(2)   No m are x′;
(3)   Some y are m;
(4)   No y are m′”.

These we will take in the order 2, 4, 1, 3.

First we take No. (2), viz. “No m are x′”. This gives us Diagram a.

To this we add No. (4), viz. “No y are m′”, and thus get Diagram b.

If we were to add to this No. (1), viz. “Some m are x”, we should have to put the “I” on a fence: so let us try No. (3) instead, viz. “Some y are m”. This gives us Diagram c.

And now there is no need to trouble about No. (1), as it would not add anything to our information to put a “I” on the fence. The Diagram already tells us that “Some m are x”.]

a     b     c

[Work Examples § 1, 9–12 (p. 97); § 2, 1–20 (p. 98).]

pg053CHAPTER IV.

INTERPRETATION, IN TERMS OF x AND y, OF TRILITERAL DIAGRAM, WHEN MARKED WITH COUNTERS OR DIGITS.

To transfer the information, observe the following Rules:—

(1) Examine the N.W. Quarter of the Triliteral Diagram.
(2) If it contains a “I”, in either Cell, it is certainly occupied, and you may mark the N.W. Quarter of the Biliteral Diagram with a “I”.
(3) If it contains two “O”s, one in each Cell, it is certainly empty, and you may mark the N.W. Quarter of the Biliteral Diagram with a “O”.
pg054(4) Deal in the same way with the N.E., the S.W., and the S.E. Quarter.

[Let us take, as examples, the results of the four Examples worked in the previous Chapters.

(1)

In the N.W. Quarter, only one of the two Cells is marked as empty: so we do not know whether the N.W. Quarter of the Biliteral Diagram is occupied or empty: so we cannot mark it.

In the N.E. Quarter, we find two “O”s: so this Quarter is certainly empty; and we mark it so on the Biliteral Diagram.

In the S.W. Quarter, we have no information at all.

In the S.E. Quarter, we have not enough to use.

We may read off the result as “No x are y′”, or “No y′ are x,” whichever we prefer.

(2)

In the N.W. Quarter, we have not enough information to use.

In the N.E. Quarter, we find a “I”. This shows us that it is occupied: so we may mark the N.E. Quarter on the Biliteral Diagram with a “I”.

In the S.W. Quarter, we have not enough information to use.

In the S.E. Quarter, we have none at all.

We may read off the result as “Some x are y′”, or “Some y′ are x”, whichever we prefer.

pg055(3)

In the N.W. Quarter, we have no information. (The “I”, sitting on the fence, is of no use to us until we know on which side he means to jump down!)

In the N.E. Quarter, we have not enough information to use.

Neither have we in the S.W. Quarter.

The S.E. Quarter is the only one that yields enough information to use. It is certainly empty: so we mark it as such on the Biliteral Diagram.

We may read off the results as “No x′ are y′”, or “No y′ are x′”, whichever we prefer.

(4)

The N.W. Quarter is occupied, in spite of the “O” in the Outer Cell. So we mark it with a “I” on the Biliteral Diagram.

The N.E. Quarter yields no information.

The S.W. Quarter is certainly empty. So we mark it as such on the Biliteral Diagram.

The S.E. Quarter does not yield enough information to use.

We read off the result as “All y are x.”]

[Review Tables V, VI (pp. 46, 47). Work Examples § 1, 13–16 (p. 97); § 2, 21–32 (p. 98); § 3, 1–20 (p. 99).]

pg056BOOK V.

SYLLOGISMS.

CHAPTER I.

INTRODUCTORY

When a Trio of Biliteral Propositions of Relation is such that

the Trio is called a ‘Syllogism’; the Genus, of which each of the six Terms is a Species, is called its ‘Universe of Discourse’, or, more briefly, its ‘Univ.’; the first two Propositions are called its ‘Premisses’, and the third its ‘Conclusion’; also the Pair of codivisional Terms in the Premisses are called its ‘Eliminands’, and the other two its ‘Retinends’.

The Conclusion of a Syllogism is said to be ‘consequent’ from its Premisses: hence it is usual to prefix to it the word “Therefore” (or the Symbol “∴”).

pg057[Note that the ‘Eliminands’ are so called because they are eliminated, and do not appear in the Conclusion; and that the ‘Retinends’ are so called because they are retained, and do appear in the Conclusion.

Note also that the question, whether the Conclusion is or is not consequent from the Premisses, is not affected by the actual truth or falsity of any of the Trio, but depends entirely on their relationship to each other.

As a specimen-Syllogism, let us take the Trio

“No x-Things are m-Things;
  No y-Things are m′-Things.
          No x-Things are y-Things.”

which we may write, as explained at p. 26, thus:—

“No x are m;
  No y are m′.
          No x are y”.

Here the first and second contain the Pair of codivisional Classes m and m′; the first and third contain the Pair x and x; and the second and third contain the Pair y and y.

Also the three Propositions are (as we shall see hereafter) so related that, if the first two were true, the third would also be true.

Hence the Trio is a Syllogism; the two Propositions, “No x are m” and “No y are m′”, are its Premisses; the Proposition “No x are y” is its Conclusion; the Terms m and m′ are its Eliminands; and the Terms x and y are its Retinends.

Hence we may write it thus:—

“No x are m;
  No y are m′.
  ∴ No x are y”.

As a second specimen, let us take the Trio

“All cats understand French;
  Some chickens are cats.
          Some chickens understand French”.

These, put into normal form, are

“All cats are creatures understanding French;
  Some chickens are cats.
          Some chickens are creatures understanding French”.

Here all the six Terms are Species of the Genus “creatures.”

Also the first and second Propositions contain the Pair of codivisional Classes “cats” and “cats”; the first and third contain the Pair “creatures understanding French” and “creatures understanding French”; and the second and third contain the Pair “chickens” and “chickens”.

pg058Also the three Propositions are (as we shall see at p. 64) so related that, if the first two were true, the third would be true. (The first two are, as it happens, not strictly true in our planet. But there is nothing to hinder them from being true in some other planet, say Mars or Jupiter—in which case the third would also be true in that planet, and its inhabitants would probably engage chickens as nursery-governesses. They would thus secure a singular contingent privilege, unknown in England, namely, that they would be able, at any time when provisions ran short, to utilise the nursery-governess for the nursery-dinner!)

Hence the Trio is a Syllogism; the Genus “creatures” is its ‘Univ.’; the two Propositions, “All cats understand French“ and ”Some chickens are cats”, are its Premisses, the Proposition “Some chickens understand French” is its Conclusion; the Terms “cats” and “cats” are its Eliminands; and the Terms, “creatures understanding French” and “chickens”, are its Retinends.

Hence we may write it thus:—

“All cats understand French;
  Some chickens are cats;
  ∴ Some chickens understand French”.]

pg059CHAPTER II.

PROBLEMS IN SYLLOGISMS.

§ 1.

Introductory.

When the Terms of a Proposition are represented by words, it is said to be ‘concrete’; when by letters, ‘abstract.’

To translate a Proposition from concrete into abstract form, we fix on a Univ., and regard each Term as a Species of it, and we choose a letter to represent its Differentia.

[For example, suppose we wish to translate “Some soldiers are brave” into abstract form. We may take “men” as Univ., and regard “soldiers” and “brave men” as Species of the Genus “men”; and we may choose x to represent the peculiar Attribute (say “military”) of “soldiers,” and y to represent “brave.” Then the Proposition may be written “Some military men are brave men”; i.e. “Some x-men are y-men”; i.e. (omitting “men,” as explained at p. 26) “Some x are y.”

In practice, we should merely say “Let Univ. be “men”, x = soldiers, y = brave”, and at once translate “Some soldiers are brave” into “Some x are y.”]

The Problems we shall have to solve are of two kinds, viz.

pg060§ 2.

Given a Pair of Propositions of Relation, which contain between them a pair of codivisional Classes, and which are proposed as Premisses: to ascertain what Conclusion, if any, is consequent from them.

The Rules, for doing this, are as follows:—

[Let us work some examples.

(1)

“No son of mine is dishonest;
  People always treat an honest man with respect”.

Taking “men” as Univ., we may write these as follows:—

“No sons of mine are dishonest men;
  All honest men are men treated with respect”.

We can now construct our Dictionary, viz. m = honest; x = sons of mine; y = treated with respect.

(Note that the expression “x = sons of mine” is an abbreviated form of “x = the Differentia of ‘sons of mine’, when regarded as a Species of ‘men’”.)

The next thing is to translate the proposed Premisses into abstract form, as follows:—

“No x are m′;
  All m are y”.

pg061Next, by the process described at p. 50, we represent these on a Triliteral Diagram, thus:—

Next, by the process described at p. 53, we transfer to a Biliteral Diagram all the information we can.

The result we read as “No x are y′” or as “No y′ are x,” whichever we prefer. So we refer to our Dictionary, to see which will look best; and we choose

“No x are y′”,

which, translated into concrete form, is

“No son of mine fails to be treated with respect”.

(2)

“All cats understand French;
  Some chickens are cats”.

Taking “creatures” as Univ., we write these as follows:—

“All cats are creatures understanding French;
  Some chickens are cats”.

We can now construct our Dictionary, viz. m = cats; x = understanding French; y = chickens.

The proposed Premisses, translated into abstract form, are

“All m are x;
  Some y are m”.

In order to represent these on a Triliteral Diagram, we break up the first into the two Propositions to which it is equivalent, and thus get the three Propositions

(1) “Some m are x;
(2)   No m are x′;
(3)   Some y are m”.

The Rule, given at p. 50, would make us take these in the order 2, 1, 3.

This, however, would produce the result

pg062So it would be better to take them in the order 2, 3, 1. Nos. (2) and (3) give us the result here shown; and now we need not trouble about No. (1), as the Proposition “Some m are x” is already represented on the Diagram.

Transferring our information to a Biliteral Diagram, we get

This result we can read either as “Some x are y” or “Some y are x”.

After consulting our Dictionary, we choose

“Some y are x”,

which, translated into concrete form, is

“Some chickens understand French.”

(3)

“All diligent students are successful;
  All ignorant students are unsuccessful”.

Let Univ. be “students”; m = successful; x = diligent; y = ignorant.

These Premisses, in abstract form, are

“All x are m;
  All y are m′”.

These, broken up, give us the four Propositions

(1) “Some x are m;
(2)   No x are m′;
(3)   Some y are m′;
(4)   No y are m”.

which we will take in the order 2, 4, 1, 3.

Representing these on a Triliteral Diagram, we get

And this information, transferred to a Biliteral Diagram, is

Here we get two Conclusions, viz.

“All x are y′;
  All y are x′.”

pg063And these, translated into concrete form, are

“All diligent students are (not-ignorant, i.e.) learned;
  All ignorant students are (not-diligent, i.e.) idle”. (See p. 4.)

(4)

“Of the prisoners who were put on their trial at the last
      Assizes, all, against whom the verdict ‘guilty’ was
      returned, were sentenced to imprisonment;
  Some, who were sentenced to imprisonment, were also
      sentenced to hard labour”.

Let Univ. be “the prisoners who were put on their trial at the last Assizes”; m = who were sentenced to imprisonment; x = against whom the verdict ‘guilty’ was returned; y = who were sentenced to hard labour.

The Premisses, translated into abstract form, are

“All x are m;
  Some m are y”.

Breaking up the first, we get the three

(1) “Some x are m;
(2)   No x are m′;
(3)   Some m are y”.

Representing these, in the order 2, 1, 3, on a Triliteral Diagram, we get

Here we get no Conclusion at all.

You would very likely have guessed, if you had seen only the Premisses, that the Conclusion would be

“Some, against whom the verdict ‘guilty’ was returned,
  were sentenced to hard labour”.

But this Conclusion is not even true, with regard to the Assizes I have here invented.

“Not true!” you exclaim. “Then who were they, who were sentenced to imprisonment and were also sentenced to hard labour? They must have had the verdict ‘guilty’ returned against them, or how could they be sentenced?”

Well, it happened like this, you see. They were three ruffians, who had committed highway-robbery. When they were put on their trial, they pleaded ‘guilty’. So no verdict was returned at all; and they were sentenced at once.]

I will now work out, in their briefest form, as models for the Reader to imitate in working examples, the above four concrete Problems.

pg064(1) [see p. 60]

“No son of mine is dishonest;
  People always treat an honest man with respect.”

Univ. “men”; m = honest; x = my sons; y = treated with respect.

“No x are m′;  All m are y.”

∴ “No x are y′.”

i.e. “No son of mine ever fails to be treated with respect.”

(2) [see p. 61]

“All cats understand French;
  Some chickens are cats”.

Univ. “creatures”; m = cats; x = understanding French; y = chickens.

“All m are x;  Some y are m.”

∴ “Some y are x.”

i.e. “Some chickens understand French.”

(3) [see p. 62]

“All diligent students are successful;
  All ignorant students are unsuccessful”.

Univ. “students”; m = successful; x = diligent; y = ignorant.

“All x are m;  All y are m′.”

∴ “All x are y′;     All y are x′.”

i.e. “All diligent students are learned; and all ignorant students are idle”.

pg065(4) [see p. 63]

“Of the prisoners who were put on their trial at the last Assizes, all, against whom the verdict ‘guilty’ was returned, were sentenced to imprisonment;

Some, who were sentenced to imprisonment, were also sentenced to hard labour”.

Univ. “prisoners who were put on their trial at the last Assizes”, m = sentenced to imprisonment; x = against whom the verdict ‘guilty’ was returned; y = sentenced to hard labour.

“All x are m;  Some m are y.”

There is no Conclusion.

[Review Tables VII, VIII (pp. 48, 49). Work Examples § 1, 17–21 (p. 97); § 4, 1–6 (p. 100); § 5, 1–6 (p. 101).]

pg066§ 3.

Given a Trio of Propositions of Relation, of which every two contain a Pair of codivisional Classes, and which are proposed as a Syllogism; to ascertain whether the proposed Conclusion is consequent from the proposed Premisses, and, if so, whether it is complete.

The Rules, for doing this, are as follows:—

(1) Take the proposed Premisses, and ascertain, by the process described at p. 60, what Conclusion, if any, is consequent from them.

(2) If there be no Conclusion, say so.

(3) If there be a Conclusion, compare it with the proposed Conclusion, and pronounce accordingly.

I will now work out, in their briefest form, as models for the Reader to imitate in working examples, six Problems.

(1)

“All soldiers are strong;
  All soldiers are brave.
          Some strong men are brave.”

Univ. “men”; m = soldiers; x = strong; y = brave.

pg067 “All m are x;  All m are y.    Some x are y.”

∴ “Some x are y.”

Hence proposed Conclusion is right.

(2)

“I admire these pictures;
  When I admire anything I wish to examine it thoroughly.
      I wish to examine some of these pictures thoroughly.”

Univ. “things”; m = admired by me; x = these pictures; y = things which I wish to examine thoroughly.

“All x are m;  All m are y.    Some x are y.”

∴ “All x are y.”

Hence proposed Conclusion is incomplete, the complete one being “I wish to examine all these pictures thoroughly”.

(3)

“None but the brave deserve the fair;
  Some braggarts are cowards.
      Some braggarts do not deserve the fair.”

Univ. “persons”; m = brave; x = deserving of the fair; y = braggarts.

“No m′ are x;  Some y are m′.    Some y are x′.”

∴ “Some y are x′.”

Hence proposed Conclusion is right.

pg068(4)

“All soldiers can march;
  Some babies are not soldiers.
      Some babies cannot march”.

Univ. “persons”; m = soldiers; x = able to march; y = babies.

“All m are x;  Some y are m′.    Some y are x′.”

There is no Conclusion.

(5)

“All selfish men are unpopular;
  All obliging men are popular.
      All obliging men are unselfish”.

Univ. “men”; m = popular; x = selfish; y = obliging.

“All x are m′;  All y are m.    All y are x′.”

∴ “All x are y′;    All y are x′.”

Hence proposed Conclusion is incomplete, the complete one containing, in addition, “All selfish men are disobliging”.

(6)

”No one, who means to go by the train and cannot get a conveyance, and has not enough time to walk to the station, can do without running;

  This party of tourists mean to go by the train and cannot get a conveyance, but they have plenty of time to walk to the station.

              This party of tourists need not run.”

Univ. “persons meaning to go by the train, and unable to get a conveyance”; m = having enough time to walk to the station; x = needing to run; y = these tourists.

pg069 “No m′ are x′;  All y are m.    All y are x′.”

There is no Conclusion.

[Here is another opportunity, gentle Reader, for playing a trick on your innocent friend. Put the proposed Syllogism before him, and ask him what he thinks of the Conclusion.

He will reply “Why, it’s perfectly correct, of course! And if your precious Logic-book tells you it isn’t, don’t believe it! You don’t mean to tell me those tourists need to run? If I were one of them, and knew the Premisses to be true, I should be quite clear that I needn’t run—and I should walk!”

And you will reply “But suppose there was a mad bull behind you?”

And then your innocent friend will say “Hum! Ha! I must think that over a bit!”

You may then explain to him, as a convenient test of the soundness of a Syllogism, that, if circumstances can be invented which, without interfering with the truth of the Premisses, would make the Conclusion false, the Syllogism must be unsound.]

[Review Tables V–VIII (pp. 46–49). Work Examples § 4, 7–12 (p. 100); § 5, 7–12 (p. 101); § 6, 1–10 (p. 106); § 7, 1–6 (pp. 107, 108).]

pg070BOOK VI.

THE METHOD OF SUBSCRIPTS.

CHAPTER I.

INTRODUCTORY.

Let us agree that “x₁” shall mean “Some existing Things have the Attribute x”, i.e. (more briefly) “Some x exist”; also that “xy₁” shall mean “Some xy exist”, and so on. Such a Proposition may be called an ‘Entity.’

Also that “x₀” shall mean “No existing Things have the Attribute x”, i.e. (more briefly) “No x exist”; also that “xy₀” shall mean “No xy exist”, and so on. Such a Proposition may be called a ‘Nullity’.

Also that “†” shall mean “and”.

When two Letters are both of them accented, or both not accented, they are said to have ‘Like Signs’, or to be ‘Like’: when one is accented, and the other not, they are said to have ‘Unlike Signs’, or to be ‘Unlike’.

pg071CHAPTER II.

REPRESENTATION OF PROPOSITIONS OF RELATION.

Let us take, first, the Proposition “Some x are y”.

This, we know, is equivalent to the Proposition of Existence “Some xy exist”. (See p. 31.) Hence it may be represented by the expression “xy₁”.

The Converse Proposition “Some y are x” may of course be represented by the same expression, viz. “xy₁”.

Similarly we may represent the three similar Pairs of Converse Propositions, viz.—

Let us take, next, the Proposition “No x are y”.

This, we know, is equivalent to the Proposition of Existence “No xy exist”. (See p. 33.) Hence it may be represented by the expression “xy₀”.

The Converse Proposition “No y are x” may of course be represented by the same expression, viz. “xy₀”.

Similarly we may represent the three similar Pairs of Converse Propositions, viz.—

pg072Let us take, next, the Proposition “All x are y”.

Now it is evident that the Double Proposition of Existence “Some x exist and no xy′ exist” tells us that some x-Things exist, but that none of them have the Attribute y′: that is, it tells us that all of them have the Attribute y: that is, it tells us that “All x are y”.

[The Reader will perhaps be puzzled by the statement that the Proposition “All x are y” is equivalent to the Double Proposition “Some x exist and no xy′ exist,” remembering that it was stated, at p. 33, to be equivalent to the Double Proposition “Some x are y and no x are y′” (i.e. “Some xy exist and no xy′ exist”). The explanation is that the Proposition “Some xy exist” contains superfluous information. “Some x exist” is enough for our purpose.]

This expression may be written in a shorter form, viz. “x₁y′₀”, since each Subscript takes effect back to the beginning of the expression.

Similarly we may represent the seven similar Propositions “All x are y′”, “All x′ are y”, “All x′ are y′”, “All y are x”, “All y are x′”, “All y′ are x”, and “All y′ are x′”.

It will be convenient to remember that, in translating a Proposition, beginning with “All”, from abstract form into subscript form, or vice versâ, the Predicate changes sign (that is, changes from positive to negative, or else from negative to positive).

pg073CHAPTER III.

SYLLOGISMS.

§ 1.

Representation of Syllogisms.

We already know how to represent each of the three Propositions of a Syllogism in subscript form. When that is done, all we need, besides, is to write the three expressions in a row, with “†” between the Premisses, and “¶” before the Conclusion.

pg074§ 2.

Formulæ for solving Problems in Syllogisms.

I shall now take three different forms of Pairs of Premisses, and work out their Conclusions, once for all, by Diagrams; and thus obtain some useful Formulæ. I shall call them “Fig. I”, “Fig. II”, and “Fig. III”.

pg075Fig. I.

This includes any Pair of Premisses which are both of them Nullities, and which contain Unlike Eliminands.

The simplest case is

xm0 † ym′0
	∴ xy0

In this case we see that the Conclusion is a Nullity, and that the Retinends have kept their Signs.

And we should find this Rule to hold good with any Pair of Premisses which fulfil the given conditions.

[The Reader had better satisfy himself of this, by working out, on Diagrams, several varieties, such as

m₁x₀ † ym′₀ (which ¶ xy₀)
xm′₀ † m₁y₀ (which ¶ xy₀)
x′m₀ † ym′₀ (which ¶ x′y₀)
m′₁x′₀ † m₁y′₀ (which ¶ x′y′₀).]

If either Retinend is asserted in the Premisses to exist, of course it may be so asserted in the Conclusion.

Hence we get two Variants of Fig. I, viz.

pg076(2) A Retinend, asserted in the Premisses to exist, may be so asserted in the Conclusion.

[Note that Rule (1) is merely the Formula expressed in words.]

Fig. II.

This includes any Pair of Premisses, of which one is a Nullity and the other an Entity, and which contain Like Eliminands.

The simplest case is

xm0 † ym1
	∴ x′y1

In this case we see that the Conclusion is an Entity, and that the Nullity-Retinend has changed its Sign.

And we should find this Rule to hold good with any Pair of Premisses which fulfil the given conditions.

[The Reader had better satisfy himself of this, by working out, on Diagrams, several varieties, such as

x′m₀ † ym₁ (which ¶ xy₁)
x₁m′₀ † y′m′₁ (which ¶ x′y′₁)
m₁x₀ † y′m₁ (which ¶ x′y′₁).]

The Formula, to be remembered, is,

xm₀ † ym₁ ¶ x′y₁

with the following Rule:—

A Nullity and an Entity, with Like Eliminands, yield an Entity, in which the Nullity-Retinend changes its Sign.

[Note that this Rule is merely the Formula expressed in words.]

pg077Fig. III.

This includes any Pair of Premisses which are both of them Nullities, and which contain Like Eliminands asserted to exist.

The simplest case is

xm₀ † ym₀ † m₁

[Note that “m₁” is here stated separately, because it does not matter in which of the two Premisses it occurs: so that this includes the three forms “m₁x₀ † ym₀”, “xm₀ † m₁y₀”, and “m₁x₀ † m₁y₀”.]

∴ x′y′1

In this case we see that the Conclusion is an Entity, and that both Retinends have changed their Signs.

And we should find this Rule to hold good with any Pair of Premisses which fulfil the given conditions.

[The Reader had better satisfy himself of this, by working out, on Diagrams, several varieties, such as

x′m₀ † m₁y₀ (which ¶ xy′₁)
m′₁x₀ † m′y′₀ (which ¶ x′y₁)
m₁x′₀ † m₁y′₀ (which ¶ xy₁).]

The Formula, to be remembered, is

xm₀ † ym₀ † m₁ ¶ x′y′₁

with the following Rule (which is merely the Formula expressed in words):—

Two Nullities, with Like Eliminands asserted to exist, yield an Entity, in which both Retinends change their Signs.

---

In order to help the Reader to remember the peculiarities and Formulæ of these three Figures, I will put them all together in one Table.

pg078TABLE IX. Fig. I. xm0 † ym′0 ¶ xy0 Two Nullities, with Unlike Eliminands, yield a Nullity, in which both Retinends keep their Signs. A Retinend, asserted in the Premisses to exist, may be so asserted in the Conclusion. Fig. II. xm0 † ym1 ¶ x′y1 A Nullity and an Entity, with Like Eliminands, yield an Entity, in which the Nullity-Retinend changes its Sign. Fig. III. xm0 † ym0 † m1 ¶ x′y′1 Two Nullities, with Like Eliminands asserted to exist, yield an Entity, in which both Retinends change their Signs.

I will now work out, by these Formulæ, as models for the Reader to imitate, some Problems in Syllogisms which have been already worked, by Diagrams, in Book V., Chap. II.

(1) [see p. 64]

“No son of mine is dishonest;
  People always treat an honest man with respect.”

Univ. “men”; m = honest; x = my sons; y = treated with respect.

xm′₀ † m₁y′₀ ¶ xy′₀         [Fig. I.

i.e. “No son of mine ever fails to be treated with respect.”

pg079(2) [see p. 64]

“All cats understand French;
  Some chickens are cats.”

Univ. “creatures”; m = cats; x = understanding French; y = chickens.

m₁x′₀ † ym₁ ¶ xy₁         [Fig. II.

i.e. “Some chickens understand French.”

(3) [see p. 64]

“All diligent students are successful;
  All ignorant students are unsuccessful.”

Univ. “students”; m = successful; x = diligent; y = ignorant.

x₁m′₀ † y₁m₀ ¶ x₁y₀ † y₁x₀         [Fig. I (β).

i.e. “All diligent students are learned; and all ignorant students are idle.”

(4) [see p. 66]

“All soldiers are strong;
  All soldiers are brave.
          Some strong men are brave.”

Univ. “men”; m = soldiers; x = strong; y = brave.

m₁x′₀ † m₁y′₀ ¶ xy₁         [Fig. III.

Hence proposed Conclusion is right.

(5) [see p. 67]

“I admire these pictures;
  When I admire anything, I wish to examine it thoroughly.
          I wish to examine some of these pictures thoroughly.”

Univ. “things”; m = admired by me; x = these; y = things which I wish to examine thoroughly.

x₁m′₀ † m₁y′₀ ¶ x₁y′₀         [Fig. I (α).

Hence proposed Conclusion, xy₁, is incomplete, the complete one being “I wish to examine all these pictures thoroughly.”

pg080(6) [see p. 67]

“None but the brave deserve the fair;
  Some braggarts are cowards.
          Some braggarts do not deserve the fair.”