Symbolic Logic

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_xv
CONTENTS
BOOK I. 
THINGS AND THEIR ATTRIBUTES. 
CHAPTER I. 
INTRODUCTORY. 
 page
Things1 
Attributes
Adjuncts
CHAPTER II. 
CLASSIFICATION. 
Classification
Class
Peculiar’ Attributes
Genus
Species
Differentia
Real’ and ‘Unreal’, or ‘Imaginary’, Classes2 
Individual
A Class regarded as a single Thing
pg_xviCHAPTER III. 
DIVISION. 
§ 1. 
Introductory. 
Division3 
Codivisional’ Classes
§ 2. 
Dichotomy. 
Dichotomy
Arbitrary limits of Classes
Subdivision of Classes4 
CHAPTER IV. 
NAMES. 
Name
Real’ and ‘Unreal’ Names
Three ways of expressing a Name
Two senses in which a plural Name may be used5 
CHAPTER V. 
DEFINITIONS. 
Definition6 
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 ‘Terms9 
§ 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 I10 
(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 Existence11 
CHAPTER III. 
PROPOSITIONS OF RELATION. 
§ 1. 
Introductory. 
Proposition of Relation12 
Universe of Discourse,’ or ‘Univ. 
§ 2. 
Reduction of a Proposition of Relation to Normal form. 
Rules13 
Examples worked 
§ 3. 
A Proposition of Relation, beginning with “All”, is a Double Proposition. 
Its equivalence to two Propositions17 
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. 
Rules20 
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 them25 
pg_xxCHAPTER II. 
COUNTERS. 
Meaning of a Red Counter placed in a Cell26 
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 omitted27 
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 Propositions29 
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 x31 
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 Trios32 
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 x33 
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 Propositions34 
Table II. Representation of Propositions of Existence34 
Table III. Representation of Propositions of Relation35 
CHAPTER IV. 
INTERPRETATION OF BILITERAL DIAGRAM, WHEN MARKED WITH COUNTERS. 
Interpretation ofDiagram representing x y exists
36 
And of three other similar arrangements 
pg_xxiiInterpretation ofDiagram representing x y does not exist
 
And of three other similar arrangements 
Interpretation ofDiagram representing x exists
37 
And of three other similar arrangements 
Interpretation ofDiagram representing x exists with and without y
 
And of three other similar arrangements 
Interpretation ofDiagram representing x does not exist
 
And of three other similar arrangements 
Interpretation ofDiagram representing all x are y
 
And of seven other similar arrangements38 
BOOK IV. 
THE TRILITERAL DIAGRAM. 
CHAPTER I. 
SYMBOLS AND CELLS. 
Change of Biliteral into Triliteral Diagram39 
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 them42 
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 m45 
Fifteen other similar Propositions 
Table V. Representations of Propositions in terms of x and m46 
Table VI. Representations of Propositions in terms of y and m47 
Table VII. Representations of Propositions in terms of x and m48 
Table VIII. Representations of Propositions in terms of y and m49 
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 Counters50 
Rules 
Examples worked 
CHAPTER IV. 
INTERPRETATION, IN TERMS OF x AND y, OF TRILITERAL DIAGRAM, WHEN MARKED WITH COUNTERS OR DIGITS. 
Rules53 
Examples worked54 
BOOK V. 
SYLLOGISMS. 
CHAPTER I. 
INTRODUCTORY. 
Syllogism56 
Premisses 
Conclusion 
Eliminands 
Retinends 
Consequent 
The Symbol “∴” 
Specimen-Syllogisms57 
pg_xxvCHAPTER II. 
PROBLEMS IN SYLLOGISMS. 
§ 1. 
Introductory. 
Concrete’ and ‘Abstract’ Propositions59 
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. 
Rules60 
Examples worked fully 
The same worked briefly, as models64 
§ 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. 
Rules66 
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 x71 
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 y72 
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. 
Rules73 
§ 2. 
Formulæ for Syllogisms. 
Three Formulæ worked out:— 
Fig. I. xm0ym′0xy075 
its two Variants (α) and (β) 
Fig. II. xm0ym1x′y176 
Fig. III. xm0ym0m1x′y′177 
Table IX. Formulæ and Rules78 
Examples worked briefly, as models 
§ 3. 
Fallacies. 
Fallacy81 
Method of finding Forms of Fallacies82 
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-Premiss83 
(3) Fallacy of two Entity-Premisses 
§ 4. 
Method of proceeding with a given Pair of Propositions. 
Rules84 
pg_xxviiiBOOK VII. 
SORITESES. 
CHAPTER I. 
INTRODUCTORY. 
Sorites85 
Premisses 
Partial Conclusion 
Complete Conclusion’ (or ‘Conclusion’) 
Eliminands 
Retinends 
consequent 
The Symbol “∴” 
Specimen-Soriteses86 
CHAPTER II. 
PROBLEMS IN SORITESES. 
§ 1. 
Introductory. 
Form of Problem87 
Two Methods of Solution 
§ 2. 
Solution by Method of Separate Syllogisms. 
Rules88 
Example worked 
pg_xxix§ 3. 
Solution by Method of Underscoring. 
Underscoring91 
Subscripts to be omitted 
Example worked fully92 
Example worked briefly, as model93 
Seventeen Examination-Papers94 
BOOK VIII. 
EXAMPLES, WITH ANSWERS AND SOLUTIONS. 
pg_xxxii pg001BOOK I.
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