CHAPTER I.

Produced by Tony Browne, Geetu Melwani, Greg Weeks, L.

Lynn Smith and the Online Distributed Proofreading Team

at https://www.pgdp.net

pg-i

SYMBOLIC LOGIC

By Lewis Carroll

pg-ii

pg-iii

pg-iv

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-v

SYMBOLIC 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-vi

RICHARD CLAY AND SONS, LIMITED,

LONDON AND BUNGAY

pg-vii

ADVERTISEMENT.

An envelope, containing two blank Diagrams (Biliteral and Triliteral)

and 9 counters (4 Red and 5 Grey), may be had, from Messrs. Macmillan,

for 3_d._, by post 4_d._

* * * * *

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-viii

PREFACE 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-ix

In 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 _which_)," 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."

pg-x

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

its 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-xi

INTRODUCTION.

_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. This, 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.

pg-xii

(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, _viva voce_, to _many_ children, and

have found 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!

pg-xiii

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

'=Things=' 1

'=Attributes=' "

'=Adjuncts=' "

CHAPTER II.

_CLASSIFICATION._

'=Classification=' 1 1/2

'=Class=' "

'=Peculiar=' Attributes "

'=Genus=' "

'=Species=' "

'=Differentia=' "

'=Real=' and '=Unreal=', or '=Imaginary=', Classes 2

'=Individual=' "

A Class regarded as a single Thing 2 1/2

pg-xvi

CHAPTER III.

_DIVISION._

Sec. 1.

_Introductory._

'=Division=' 3

'=Codivisional=' Classes "

Sec. 2.

_Dichotomy._

'=Dichotomy=' 3 1/2

Arbitrary limits of Classes "

Subdivision of Classes 4

CHAPTER IV.

_NAMES._

'=Name=' 4 1/2

'=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-xvii

=BOOK II.=

=PROPOSITIONS.=

CHAPTER I.

_PROPOSITIONS GENERALLY._

Sec. 1.

_Introductory._

Technical meaning of "some" 8

'=Proposition=' "

'=Normal form=' of a Proposition "

'=Subject=', '=Predicate=', and '=Terms=' 9

Sec. 2.

_Normal form of a Proposition._

Its four parts:--

(1) '=Sign of Quantity=' 9

(2) Name of Subject "

(3) '=Copula=' "

(4) Name of Predicate "

Sec. 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-xviii

A 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._

Sec. 1.

_Introductory._

'=Proposition of Relation=' 12

'=Universe of Discourse=,' or '=Univ.=' "

Sec. 2.

_Reduction of a Proposition of Relation

to Normal form._

Rules 13

Examples worked "

Sec. 3.

_A Proposition of Relation, beginning with "All",

is a Double Proposition._

Its equivalence to _two_ Propositions 17

pg-xix

Sec. 4.

_What is implied, in a Proposition of Relation,

as to the Reality of its Terms?_

Propositions beginning with "Some" 19

" " "No" "

" " "All" "

Sec. 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' 24

=Table I.= Attributes of Classes, and Compartments, or

Cells, assigned to them 25

pg-xx

CHAPTER II.

_COUNTERS._

Meaning of a Red Counter placed in a Cell 26

" " " " on a Partition "

American phrase "=sitting on the fence=" "

Meaning of a Grey Counter placed in a Cell "

CHAPTER III.

_REPRESENTATION OF PROPOSITIONS._

Sec. 1.

_Introductory._

The word "Things" to be henceforwards omitted 27

'=Uniliteral=' Proposition "

'=Biliteral=' do. "

Proposition '=in terms of=' certain Letters "

Sec. 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

Sec. 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

=Tables II, III.= Representation of Propositions of

Existence and Relation 34, 35

CHAPTER IV.

_INTERPRETATION OF BILITERAL DIAGRAM, WHEN MARKED WITH COUNTERS._

.-------.

|(.)| |

Interpretation of |---|---| 36

| | |

.-------.

And of three other similar arrangements "

pg-xxii

.-------.

|( )| |

Interpretation 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-xxiii

The 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._

Sec. 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 "

Sec. 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 "

=Tables V, VI, VII, VIII.= Representations of

Propositions in terms of x and m, or of

y and m 46 to 49

pg-xxiv

CHAPTER 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=' 57

'=Retinends=' "

'=Consequent=' "

The Symbol ".'." "

Specimen-Syllogisms "

pg-xxv

CHAPTER II.

_PROBLEMS IN SYLLOGISMS._

Sec. 1.

_Introductory._

'=Concrete=' and '=Abstract=' Propositions 59

Method of translating a Proposition from concrete into

abstract form "

Two forms of Problems "

Sec. 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

Sec. 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-xxvi

=BOOK VI.=

=THE METHOD OF SUBSCRIPTS.=

CHAPTER I.

_INTRODUCTORY._

Meaning of x_{1}, xy_{1}, &c. 70

'=Entity=' "

Meaning of x_{0}, xy_{0}, &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 versa_ "

pg-xxvii

CHAPTER III.

_SYLLOGISMS._

Sec. 1.

_Representation of Syllogisms._

Rules 73

Sec. 2.

_Formulae for Syllogisms._

Three Formulae worked out:--

Fig. I. xm_{0} + ym'_{0} > xy_{0} 75

its two Variants (a) and (b) "

Fig. II. xm_{0} + ym_{1} > x'y_{1} 76

Fig. III. xm_{0} + ym_{0} + m_{1} > x'y'_{1} 77

=Table IX.= Formulae and Rules 78

Examples worked briefly, as models "

Sec. 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 "

Sec. 4.

_Method of proceeding with a given Pair of Propositions._

Rules 84

pg-xxviii

=BOOK 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._

Sec. 1.

_Introductory._

Form of Problem 87

Two Methods of Solution "

Sec. 2.

_Solution by Method of Separate Syllogisms._

Rules 88

Example worked "

pg-xxix

Sec. 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._

Sec. 1.

_Propositions of Relation, to be reduced to normal form_ 97

Sec. 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

Sec. 3.

_Marked Triliteral Diagrams, to be interpreted in terms

of x and y_ 99

Sec. 4.

_Pairs of Abstract Propositions, proposed as Premisses:

Conclusions to be found_ 100

pg-xxx

Sec. 5.

_Pairs of Concrete Propositions, proposed as Premisses:

Conclusions to be found_ 101

Sec. 6.

_Trios of Abstract Propositions, proposed as Syllogisms:

to be examined_ 106

Sec. 7.

_Trios of Concrete Propositions, proposed as Syllogisms:

to be examined_ 107

Sec. 8.

_Sets of Abstract Propositions, proposed as Premisses for

Soriteses: Conclusions to be found_ 110

Sec. 9.

_Sets of Concrete Propositions, proposed as Premisses for

Soriteses: Conclusions to be found_ 112

CHAPTER II.

_ANSWERS._

Answers to

Sec. 1 125

Sec. 2 126

Secs. 3, 4 127

Sec. 5 128

Sec. 6 130

Sec. 7 131

Secs. 8, 9 132

pg-xxxi

CHAPTER III.

_SOLUTIONS._

Sec. 1.

_Propositions of Relation reduced to normal form._

Solutions for Sec. 1 134

Sec. 2.

_Method of Diagrams._

Solutions for

Sec. 4 Nos. 1 to 12 136

Sec. 5 " 1 to 12 138

Sec. 6 " 1 to 10 141

Sec. 7 " 1 to 6 144

Sec. 3.

_Method of Subscripts._

Solutions for

Sec. 4 146

Sec. 5 Nos. 13 to 24 147

Secs. 6, 7, 8, 9 148 to 157

=NOTES= 164

=APPENDIX, ADDRESSED TO TEACHERS= 165

=NOTES TO APPENDIX= 195

=INDEX.=

Sec. 1. Tables 197

Sec. 2. Words &c. explained "

pg-xxxii

pg001

BOOK I.

THINGS AND THEIR ATTRIBUTES.

CHAPTER I.

_INTRODUCTORY._

The Universe contains '=Things=.'

[For example, "I," "London," "roses," "redness," "old English

books," "the letter which I received yesterday."]

Things have '=Attributes=.'

[For example, "large," "red," "old," "which I received

yesterday."]

One Thing may have many Attributes; and one Attribute may belong to many

Things.

[Thus, the Thing "a rose" may have the Attributes "red,"

"scented," "full-blown," &c.; and the Attribute "red" may belong

to the Things "a rose," "a brick," "a ribbon," &c.]

Any Attribute, or any Set of Attributes, may be called an '=Adjunct=.'

[This word is introduced in order to avoid the constant

repetition of the phrase "Attribute or Set of Attributes."

Thus, we may say that a rose has the Attribute "red" (or the

Adjunct "red," whichever we prefer); or we may say that it has

the Adjunct "red, scented and full-blown."]

pg001 1/2

CHAPTER 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=.'

This Process may be performed in three different ways, as follows:--

(1) We may imagine that we have put together all Things. The Class so

formed (i.e. the Class "Things") contains the whole Universe.

(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='.

pg002

As 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=.'

[For example, we may imagine that we have picked out, from the

Class "Things," all the Things which possess the Adjunct

"material, artificial, consisting of houses and streets"; and we

may thus form the Real Class "towns." Here we may regard

"Things" as a _Genus_, "Towns" as a _Species_ of Things, and

"material, artificial, consisting of houses and streets" as its

_Differentia_.

Again, we may imagine that we have picked out all the Things

which possess the Adjunct "weighing a ton, easily lifted by a

baby"; and we may thus form the _Imaginary_ Class "Things that

weigh a ton and are easily lifted by a baby."]

(3) We may think of a certain Class, _not_ the Class "Things," and may

imagine that we have picked out from it all the Members of it which

possess a certain Adjunct _not_ possessed by the whole Class. This

Adjunct is said to be '=peculiar=' to the smaller Class so formed. In

this case, the Class thought of is called a '=Genus=' with regard to the

smaller Class picked out from it: the smaller Class is called a

'=Species=' of the larger: and its peculiar Adjunct is called its

'=Differentia='.

[For example, we may think of the Class "towns," and imagine

that we have picked out from it all the towns which possess the

Attribute "lit with gas"; and we may thus form the Real Class

"towns lit with gas." Here we may regard "Towns" as a _Genus_,

"Towns lit with gas" as a _Species_ of Towns, and "lit with gas"

as its _Differentia_.

If, in the above example, we were to alter "lit with gas" into

"paved with gold," we should get the _Imaginary_ Class "towns

paved with gold."]

A Class, containing only _one_ Member is called an '=Individual=.'

[For example, the Class "towns having four million inhabitants,"

which Class contains only _one_ Member, viz. "London."]

pg002 1/2

Hence, any single Thing, which we can name so as to distinguish it from

all other Things, may be regarded as a one-Member Class.

[Thus "London" may be regarded as the one-Member Class, picked

out from the Class "towns," which has, as its Differentia,

"having four million inhabitants."]

A Class, containing two or more Members, is sometimes regarded as _one

single Thing_. When so regarded, it may possess an Adjunct which is

_not_ possessed by any Member of it taken separately.

[Thus, the Class "The soldiers of the Tenth Regiment," when

regarded as _one single Thing_, may possess the Attribute

"formed in square," which is _not_ possessed by any Member of it

taken separately.]

pg003

CHAPTER III.

_DIVISION._

Sec. 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.

[Thus, we might think of the Class "books," and imagine that we

had divided it into the two smaller Classes "bound books" and

"unbound books," or into the three Classes, "books priced at

less than a shilling," "shilling-books," "books priced at more

than a shilling," or into the twenty-six Classes, "books whose

names begin with _A_," "books whose names begin with _B_," &c.]

A Class, that has been obtained by a certain Division, is said to be

'codivisional' with every Class obtained by that Division.

[Thus, the Class "bound books" is codivisional with each of the

two Classes, "bound books" and "unbound books."

Similarly, the Battle of Waterloo may be said to have been

"contemporary" with every event that happened in 1815.]

Hence a Class, obtained by Division, is codivisional with itself.

[Thus, the Class "bound books" is codivisional with itself.

Similarly, the Battle of Waterloo may be said to have been

"contemporary" with itself.]

pg003 1/2

Sec. 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 Differentiae are

_contradictory_. This kind of Division is called '=Dichotomy='.

[For example, we may divide "books" into the two Classes whose

Differentiae are "old" and "not-old."]

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.

[Thus, in dividing "books" into "old" and "not-old," we may say

"Let all books printed before A.D. 1801, be regarded as 'old,'

and all others as 'not-old'."]

Henceforwards let it be understood that, if a Class of Things be divided

into two Classes, whose Differentiae have contrary meanings, each

Differentia is to be regarded as equivalent to the other with the word

"not" prefixed.

[Thus, if "books" be divided into "old" and "new" the Attribute

"old" is to be regarded as equivalent to "not-new," and the

Attribute "new" as equivalent to "not-old."]

pg004

After 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.

[For example, we may divide "books" into "old" and "new" (i.e.

"_not_-old"): we may then sub-divide each of these into

"English" and "foreign" (i.e. "_not_-English"), thus getting

_four_ Classes, viz.

(1) old English;

(2) old foreign;

(3) new English;

(4) new foreign.

If we had begun by dividing into "English" and "foreign," and

had then sub-divided into "old" and "new," the four Classes

would have been

(1) English old;

(2) English new;

(3) foreign old;

(4) foreign new.

The Reader will easily see that these are the very same four

Classes which we had before.]

pg004 1/2

CHAPTER 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_.

Such a word (or phrase) is called a '=Name='; and, if there be an

existing Thing which it represents, it is said to be a Name of that

Thing.

[For example, the words "Thing," "Treasure," "Town," and the

phrases "valuable Thing," "material artificial Thing consisting

of houses and streets," "Town lit with gas," "Town paved with

gold," "old English Book."]

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.

[Thus, "Town lit with gas" is a _Real_ Name: "Town paved with

gold" is an _Unreal_ Name.]

Every Name is either a Substantive only, or else a phrase consisting of

a Substantive and one or more Adjectives (or phrases used as

Adjectives).

Every Name, except "Thing", may usually be expressed in three different

forms:--

(_a_) The Substantive "Thing", and one or more

Adjectives (or phrases used as Adjectives) conveying

the ideas of the Attributes;

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

(1) Members of a Class, _regarded as separate Things_;

or (2) a whole Class, _regarded as one single Thing_.

[Thus, when I say "Some soldiers of the Tenth Regiment are

tall," or "The soldiers of the Tenth Regiment are brave," I am

using the Name "soldiers of the Tenth Regiment" in the _first_

sense; and it is just the same as if I were to point to each of

them _separately_, and to say "_This_ soldier of the Tenth

Regiment is tall," "_That_ soldier of the Tenth Regiment is

tall," and so on.

But, when I say "The soldiers of the Tenth Regiment are formed

in square," I am using the phrase in the _second_ sense; and it

is just the same as if I were to say "The _Tenth Regiment_ is

formed in square."]

pg006

CHAPTER 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.

[Thus, we may define a "Treasure" as a "valuable Thing." In this

case we regard "Things" as the _Genus_, and "valuable" as the

_Differentia_.]

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

working others.

[Note that, in each Definition, the Substantive, representing a

Member (or Members) of the _Genus_, is printed in Capitals.]

1. Define "a Treasure."

_Ans._ "a valuable THING."

2. Define "Treasures."

_Ans._ "valuable THINGS."

3. Define "a Town."

_Ans._ "a material artificial THING, consisting of houses and streets."

pg007

4. 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.]

pg008

BOOK II.

PROPOSITIONS.

CHAPTER I.

_PROPOSITIONS GENERALLY._

Sec. 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.

[Thus the words "yes" and "no" are Propositions in the ordinary

sense of the word; and so are the phrases "you owe me five

farthings" and "I don't!"

Such words as "oh!" or "never!", and such phrases as "fetch me

that book!" "which book do you mean?" do not seem, at first

sight, to convey any _information_; but they can easily be

turned into equivalent forms which do so, viz. "I am surprised,"

"I will never consent to it," "I order you to fetch me that

book," "I want to know which book you mean."]

But a '=Proposition=,' as used in this First Part of "Symbolic Logic,"

has a peculiar form, which may be called its '=Normal form='; 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.

pg009

A '=Proposition=,' when in normal form, asserts, as to certain two

Classes, which are called its '=Subject=' and '=Predicate=,' either

(1) that _some_ Members of its Subject are Members

of its Predicate;

or (2) that _no_ Members of its Subject are Members

of its Predicate;

or (3) that _all_ Members of its Subject are Members

of its Predicate.

The Subject and the Predicate of a Proposition are called its '=Terms=.'

Two Propositions, which convey the _same_ information, are said to be

'=equivalent='.

[Thus, the two Propositions, "I see John" and "John is seen by

me," are equivalent.]

Sec. 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

Sec. 3.

_Various kinds of Propositions._

A Proposition, that begins with "Some", is said to be '=Particular=.' It

is also called 'a Proposition =in I=.'

[Note, that it is called 'Particular,' because it refers to a

_part_ only of the Subject.]

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=.'

[Note, that they are called 'Universal', because they refer to

the _whole_ of the Subject.]

A Proposition, whose Subject is an _Individual_, is to be regarded as

_Universal_.

[Let us take, as an example, the Proposition "John is not well".

This of course implies that there is an _Individual_, to whom

the speaker refers when he mentions "John", and whom the

listener _knows_ to be referred to. Hence the Class "men

referred to by the speaker when he mentions 'John'" is a

one-Member Class, and the Proposition is equivalent to "_All_

the men, who are referred to by the speaker when he mentions

'John', are not well."]

Propositions are of two kinds, 'Propositions of Existence' and

'Propositions of Relation.'

These shall be discussed separately.

pg011

CHAPTER II.

_PROPOSITIONS OF EXISTENCE._

A '=Proposition of Existence=', when in normal form, has, for its

_Subject_, the Class "existing Things".

Its Sign of Quantity is "Some" or "No".

[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."]

It is called "a Proposition of Existence" because its effect is to

assert the _Reality_ (i.e. the real _existence_), or else the

_Imaginariness_, of its Predicate.

[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."]

pg012

CHAPTER III.

_PROPOSITIONS OF RELATION._

Sec. 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.='

The Sign of Quantity is "Some" or "No" or "All".

[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".]

It is called "a Proposition of Relation" because its effect is to assert

that a certain _relationship_ exists between its Terms.

pg013

Sec. 2.

_Reduction of a Proposition of Relation to Normal form._

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

(1) Ascertain what is the _Subject_ (i.e., ascertain what Class we are

_talking about_);

(2) If the verb, governed by the Subject, is _not_ the verb "are" (or

"is"), substitute for it a phrase beginning with "are" (or "is");

(3) Ascertain what is the _Predicate_ (i.e., ascertain what Class it is,

which is asserted to contain _some_, or _none_, or _all_, of the Members

of the Subject);

(4) If the Name of each Term is _completely expressed_ (i.e. if it

contains a Substantive), there is no need to determine the 'Univ.'; but,

if either Name is _incompletely expressed_, and contains _Attributes_

only, it is then necessary to determine a 'Univ.', in order to insert

its Name as the Substantive.

(5) Ascertain the _Sign of Quantity_;

(6) Arrange in the following order:--

Sign of Quantity,

Subject,

Copula,

Predicate.

[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

Sec. 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".

[Thus, the Proposition "_All_ bankers are rich men" evidently

contains the smaller Proposition "_Some_ bankers are rich men".]

The question now arises "What is the _rest_ of the information which

this Proposition gives us?"

In order to answer this question, let us begin with the smaller

Proposition, "_Some_ Members of the Subject are Members of the

Predicate," and suppose that this is _all_ we have been told; and let us

proceed to inquire what _else_ we need to be told, in order to know that

"_All_ Members of the Subject are Members of the Predicate".

[Thus, we may suppose that the Proposition "_Some_ bankers are

rich men" is all the information we possess; and we may proceed

to inquire what _other_ Proposition needs to be added to it, in

order to make up the entire Proposition "_All_ bankers are rich

men".]

Let us also suppose that the 'Univ.' (i.e. the Genus, of which both the

Subject and the Predicate are Specieses) has been divided (by the

Process of _Dichotomy_) into two smaller Classes, viz.

(1) the Predicate;

(2) the Class whose Differentia is _contradictory_ to that of

the Predicate.

[Thus, we may suppose that the Genus "men," (of which both

"bankers" and "rich men" are Specieses) has been divided into

the two smaller Classes, "rich men", "poor men".]

pg018

Now 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

(1) "_Some_ Members of the Subject are Members of the

Predicate";

(2) "_No_ Members of the Subject are Members of the

Class whose Differentia is _contradictory_ to that of

the Predicate".

[Thus, the Proposition "_All_ bankers are rich men" is a

_Double_ Proposition, and is equivalent to the _two_

Propositions

(1) "_Some_ bankers are rich men";

(2) "_No_ bankers are _poor_ men".]

pg019

Sec. 4.

_What is implied, in a Proposition of Relation, as to the Reality of its

Terms?_

Note that the rules, here laid down, are _arbitrary_, and only apply to

Part I of my "Symbolic Logic."

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_.

[Thus, the Proposition "Some rich men are invalids" is to be

understood as asserting that _some existing Things_ are "rich

invalids". Hence it implies that _each_ of the two Classes,

"rich men" and "invalids", 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.

[Thus, the Proposition "No mermaids are milliners" is to be

understood as asserting that _no existing Things_ are

"mermaid-milliners". But this implies nothing as to the

_Reality_, or the _Unreality_, of either of the two Classes,

"mermaids" and "milliners", taken by itself. In this case as it

happens, the Subject is _Imaginary_, and the Predicate _Real_.]

A Proposition of Relation, beginning with "All", contains (see Sec. 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 hyaenas are savage animals" contains

the Proposition "Some hyaenas are savage animals". Hence it

implies that _each_ of the two Classes, "hyaenas" and "savage

animals", taken by itself, is _Real_.]

pg020

Sec. 5.

_Translation of a Proposition of Relation into one or more Propositions

of Existence._

We have seen that a Proposition of Relation, beginning with "Some,"

asserts that _some existing Things_, being Members of its Subject, are

_also_ Members of its Predicate. Hence, it asserts that some existing

Things are Members of _both_; i.e. it asserts that some existing Things

are Members of the Class of Things which have _all_ the Attributes of

the Subject and the Predicate.

Hence, to translate it into a Proposition of Existence, we take

"existing Things" as the new _Subject_, and Things, which have _all_ the

Attributes of the Subject and the Predicate, as the new Predicate.

Similarly for a Proposition of Relation beginning with "No".

A Proposition of Relation, beginning with "All", is (as shown in Sec. 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 Sec. =1=, 1-4 (p. 97).]

pg022

BOOK III.

THE BILITERAL DIAGRAM.

.-------------.

| | |

| xy | xy' |

| | |

|------|------|

| | |

| x'y | x'y' |

| | |

.-------------.

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'.

[For example, we might say "Let Univ. be 'books'"; and we might

imagine the Diagram to be a large table, assigned to all

"books."]

[The Reader is strongly advised, in reading this Chapter, _not_

to refer to the above Diagram, but to draw a large one for

himself, _without any letters_, and to have it by him while he

reads, and keep his finger on that particular _part_ of it,

about which he is reading.]

pg023

Secondly, 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

Differentiae 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").

[For example, we might say "Let x mean 'old,' so that x' will

mean 'new'," and we might suppose that we had divided books into

the two Classes whose Differentiae are "old" and "new," and had

assigned the _North_ Half of the table to "_old_ books" and the

_South_ Half to "_new_ books."]

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

Differentiae 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").

[For example, we might say "Let y mean 'English,' so that y'

will mean 'foreign'", and we might suppose that we had

subdivided "old books" into the two Classes whose Differentiae

are "English" and "foreign", and had assigned the North-_West_

Cell to "old _English_ books", and the North-_East_ Cell to "old

_foreign_ books."]

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.

[For example, we might suppose that we had subdivided "new

books" into the two Classes "new _English_ books" and "new

_foreign_ books", and had assigned the South-_West_ Cell to the

one, and the South-_East_ Cell to the other.]

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 _same_ four

Classes. Hence we see that we have assigned the _West_ Half to the

y-Class, and the _East_ Half to the y'-Class.

pg024

[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."

.-------------------.

| old | old |

| English | foreign |

| books | books |

|---------|---------|

| new | new |

| English | foreign |

| books | 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:--

pg025

TABLE I.

.----------------------------------------.

| _Adjuncts_ | _Compartments, or Cells,_ |

| _of_ | _assigned to them._ |

| _Classes._ | |

|------------|---------------------------|

| x | North Half. |

| x' | South " |

| y | West " |

| y' | East " |

|------------|---------------------------|

| xy | North-West Cell. |

| xy' | " East " |

| x'y | South-West " |

| x'y' | " East " |

.----------------------------------------.

Q. "Adjunct for West Half?"

A. "y."

Q. "Compartment for xy'?"

A. "North-East Cell."

Q. "Adjunct for South-West Cell?"

A. "x'y."

&c., &c.

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.

pg026

CHAPTER 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").

[The Reader had better provide himself with 4 Red Counters and 5

Grey ones.]

pg027

CHAPTER III.

_REPRESENTATION OF PROPOSITIONS._

Sec. 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='.

[For example, "Some x exist", "No y' exist", &c.]

A Proposition, containing _two_ Letters, is said to be ='Biliteral'=.

[For example, "Some xy' exist", "No x' are y", &c.]

A Proposition is said to be '=in terms of=' the Letters it contains,

whether with or without accents.

[Thus, "Some xy' exist", "No x' are y", &c., are said to be _in

terms of_ x and y.]

pg028

Sec. 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".

[The Reader should make out all these for himself. In the

"books" example, these Propositions would be "Some new books

exist", &c.]

Let us take, next, the Proposition "No x exist".

This tells us that there is _nothing_ in the North Half; that is, that

the North Half is _empty_; that is, that the North-West Cell and the

North-East Cell are both of them _empty_. And this we can represent by

placing _two Grey_ Counters in the North Half, one in each Cell.

.-------.

|( )|( )|

|---|---|

| | |

.-------.

[The Reader may perhaps think that it would be enough to place a

_Grey_ Counter on the partition in the North Half, and that,

just as a _Red_ Counter, so placed, would mean "This Half is

_occupied_", so a _Grey_ one would mean "This Half is _empty_".

This, however, would be a mistake. We have seen that a _Red_

Counter, so placed, would mean "At least _one_ of these two

Cells is occupied: possibly _both_ are." Hence a _Grey_ one

would merely mean "At least _one_ of these two Cells is empty:

possibly _both_ are". But what we have to represent is, that

both Cells are _certainly_ empty: and this can only be done by

placing a _Grey_ Counter in _each_ of them.

In the "books" example, this Proposition would be "No old books

exist".]

pg029

Similarly we may represent the three similar Propositions "No x' exist",

"No y exist", and "No y' exist".

[The Reader should make out all these for himself. In the

"books" example, these three Propositions would be "No new books

exist", &c.]

Let us take, next, the Proposition "Some xy exist".

This tells us that there is at least _one_ Thing in the North-West Cell;

that is, that the North-West Cell is _occupied_. And this we can

represent by placing a _Red_ Counter in it.

.-------.

|(.)| |

|---|---|

| | |

.-------.

[In the "books" example, this Proposition would be "Some old

English books exist".]

Similarly we may represent the three similar Propositions "Some xy'

exist", "Some x'y exist", and "Some x'y' exist".

[The Reader should make out all these for himself. In the

"books" example, these three Propositions would be "Some old

foreign books exist", &c.]

Let us take, next, the Proposition "No xy exist".

This tells us that there is _nothing_ in the North-West Cell; that is,

that the North-West Cell is _empty_. And this we can represent by

placing a _Grey_ Counter in it.

.-------.

|( )| |

|---|---|

| | |

.-------.

[In the "books" example, this Proposition would be "No old

English books exist".]

Similarly we may represent the three similar Propositions "No xy'

exist", "No x'y exist", and "No x'y' exist".

[The Reader should make out all these for himself. In the

"books" example, these three Propositions would be "No old

foreign books exist", &c.]

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".

[In the "books" example, this Proposition would be "No old books

exist".

Hence this is a _Double_ Proposition, and is equivalent to the

_two_ Propositions "No old _English_ books exist" and "No old

_foreign_ books exist".]

Sec. 3.

_Representation of Propositions of Relation._

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

This tells us that at least _one_ Thing, in the _North_ Half, is also in

the _West_ Half. Hence it must be in the space _common_ to them, that

is, in the _North-West Cell_. Hence the North-West Cell is _occupied_.

And this we can represent by placing a _Red_ Counter in it.

.-------.

|(.)| |

|---|---|

| | |

.-------.

[Note that the _Subject_ of the Proposition settles which _Half_

we are to use; and that the _Predicate_ settles in which

_portion_ of it we are to place the Red Counter.

In the "books" example, this Proposition would be "Some old

books are English".]

Similarly we may represent the three similar Propositions "Some x are

y'", "Some x' are y", and "Some x' are y'".

[The Reader should make out all these for himself. In the

"books" example, these three Propositions would be "Some old

books are foreign", &c.]

pg031

Let us take, next, the Proposition "Some y are x".

This tells us that at least _one_ Thing, in the _West_ Half, is also in

the _North_ Half. Hence it must be in the space _common_ to them, that

is, in the _North-West Cell_. Hence the North-West Cell is _occupied_.

And this we can represent by placing a _Red_ Counter in it.

.-------.

|(.)| |

|---|---|

| | |

.-------.

[In the "books" example, this Proposition would be "Some English

books are old".]

Similarly we may represent the three similar Propositions "Some y are

x'", "Some y' are x", and "Some y' are x'".

[The Reader should make out all these for himself. In the

"books" example, these three Propositions would be "Some English

books are new", &c.]

We see that this _one_ Diagram has now served to represent no less than

_three_ Propositions, viz.

(1) "Some xy exist;

(2) Some x are y;

(3) Some y are x".

.-------.

|(.)| |

|---|---|

| | |

.-------.

Hence these three Propositions are equivalent.

[In the "books" example, these Propositions would be

(1) "Some old English books exist;

(2) Some old books are English;

(3) Some English books are old".]

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='.

[For example, if we were told to convert the Proposition

"Some apples are not ripe,"

we should first choose our Univ. (say "fruit"), and then

complete the Proposition, by supplying the Substantive "fruit"

in the Predicate, so that it would be

"Some apples are not-ripe fruit";

and we should then convert it by interchanging its Terms, so

that it would be

"Some not-ripe fruit are apples".]

pg032

Similarly we may represent the three similar Trios of equivalent

Propositions; the whole Set of _four_ Trios being as follows:--

(1) "Some xy exist" = "Some x are y" = "Some y are x".

(2) "Some xy' exist" = "Some x are y'" = "Some y' are x".

(3) "Some x'y exist" = "Some x' are y" = "Some y are x'".

(4) "Some x'y' exist" = "Some x' are y'" = "Some y' are x'".

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

This tell us that no Thing, in the _North_ Half, is also in the _West_

Half. Hence there is _nothing_ in the space _common_ to them, that is,

in the _North-West Cell_. Hence the North-West Cell is _empty_. And this

we can represent by placing a _Grey_ Counter in it.

.-------.

|( )| |

|---|---|

| | |

.-------.

[In the "books" example, this Proposition would be "No old books

are English".]

Similarly we may represent the three similar Propositions "No x are y'",

and "No x' are y", and "No x' are y'".

[The Reader should make out all these for himself. In the

"books" example, these three Propositions would be "No old books

are foreign", &c.]

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

This tells us that no Thing, in the _West_ Half, is also in the _North_

Half. Hence there is _nothing_ in the space _common_ to them, that is,

in the _North-West Cell_. That is, the North-West Cell is _empty_. And

this we can represent by placing a _Grey_ Counter in it.

.-------.

|( )| |

|---|---|

| | |

.-------.

[In the "books" example, this Proposition would be "No English

books are old".]

Similarly we may represent the three similar Propositions "No y are x'",

"No y' are x", and "No y' are x'".

[The Reader should make out all these for himself. In the

"books" example, these three Propositions would be "No English

books are new", &c.]

pg033

.-------.

|( )| |

|---|---|

| | |

.-------.

We see that this _one_ Diagram has now served to present no less than

_three_ Propositions, viz.

(1) "No xy exist;

(2) No x are y;

(3) No y are x."

Hence these three Propositions are equivalent.

[In the "books" example, these Propositions would be

(1) "No old English books exist;

(2) No old books are English;

(3) No English books are old".]

The two equivalent Propositions, "No x are y" and "No y are x", are said

to be 'Converse' to each other.

[For example, if we were told to convert the Proposition

"No porcupines are talkative",

we should first choose our Univ. (say "animals"), and then

complete the Proposition, by supplying the Substantive "animals"

in the Predicate, so that it would be

"No porcupines are talkative animals", and we should then

convert it, by interchanging its Terms, so that it would be

"No talkative animals are porcupines".]

Similarly we may represent the three similar Trios of equivalent

Propositions; the whole Set of _four_ Trios being as follows:--

(1) "No xy exist" = "No x are y" = "No y are x".

(2) "No xy' exist" = "No x are y'" = "No y' are x".

(3) "No x'y exist" = "No x' are y" = "No y are x'".

(4) "No x'y' exist" = "No x' are y'" = "No y' are x'".

Let us take, next, the Proposition "All x are y".

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.]

pg034

TABLE 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.

pg035

TABLE III.

.-------------------------------------------------------------.

| | .-------. | | .-------. |

| Some xy exist | |(.)| | | | |(.)|( )| |

| = Some x are y | |---|---| | All x are y | |---|---| |

| = Some y are x | | | | | | | | | |

| | .-------. | | .-------. |

|------------------|-----------|------------------|-----------|

| | .-------. | | .-------. |

| Some xy' exist | | |(.)| | | |( )|(.)| |

| = Some x are y' | |---|---| | All x are y' | |---|---| |

| = Some y' are x | | | | | | | | | |

| | .-------. | | .-------. |

|------------------|-----------|------------------|-----------|

| | .-------. | | .-------. |

| Some x'y exist | | | | | | | | | |

| = Some x' are y | |---|---| | All x' are y | |---|---| |

| = Some y are x' | |(.)| | | | |(.)|( )| |

| | .-------. | | .-------. |

|------------------|-----------|------------------|-----------|

| | .-------. | | .-------. |

| Some x'y' exist | | | | | | | | | |

| = Some x' are y'| |---|---| | All x' are y' | |---|---| |

| = Some y' are x'| | |(.)| | | |( )|(.)| |

| | .-------. | | .-------. |

.-------------------------------------------------------------.

.-------------------------------------------------------------.

| | .-------. | | .-------. |

| No xy exist | |( )| | | | |(.)| | |

| = No x are y | |---|---| | All y are x | |---|---| |

| = No y are x | | | | | | |( )| | |

| | .-------. | | .-------. |

|------------------|-----------|------------------|-----------|

| | .-------. | | .-------. |

| No xy' exist | | |( )| | | |( )| | |

| = No x are y' | |---|---| | All y are x' | |---|---| |

| = No y' are x | | | | | | |(.)| | |

| | .-------. | | .-------. |

|------------------|-----------|------------------|-----------|

| | .-------. | | .-------. |

| No x'y exist | | | | | | | |(.)| |

| = No x' are y | |---|---| | All y' are x | |---|---| |

| = No y are x' | |( )| | | | | |( )| |

| | .-------. | | .-------. |

|------------------|-----------|------------------|-----------|

| | .-------. | | .-------. |

| No x'y' exist | | | | | | | |( )| |

| = No x' are y' | |---|---| | All y' are x' | |---|---| |

| = No y' are x' | | |( )| | | | |(.)| |

| | .-------. | | .-------. |

.-------------------------------------------------------------.

.-------------------------------------------------------------.

| | .-------. | | .-------. |

| | |(.)|(.)| | | |(.)| | |

| Some x are y, | |---|---| | Some y are x | |---|---| |

| and some are y' | | | | | and some are x' | |(.)| | |

| | .-------. | | .-------. |

|------------------|-----------|------------------|-----------|

| | .-------. | | .-------. |

| | | | | | | | |(.)| |

| Some x' are y, | |---|---| | Some y' are x | |---|---| |

| and some are y' | |(.)|(.)| | and some are x' | | |(.)| |

| | .-------. | | .-------. |

.-------------------------------------------------------------.

pg036

CHAPTER IV.

_INTERPRETATION OF BILITERAL DIAGRAM WHEN MARKED WITH COUNTERS._

The Diagram is supposed to be set before us, with certain Counters

placed upon it; and the problem is to find out what Proposition, or

Propositions, the Counters represent.

As the process is simply the reverse of that discussed in the previous

Chapter, we can avail ourselves of the results there obtained, as far as

they go.

First, let us suppose that we find a _Red_ Counter placed in the

North-West Cell.

.-------.

|(.)| |

|---|---|

| | |

.-------.

We know that this represents each of the Trio of equivalent Propositions

"Some xy exist" = "Some x are y" = "Some y are x".

Similarly we may interpret a _Red_ Counter, when placed in the

North-East, or South-West, or South-East Cell.

Next, let us suppose that we find a _Grey_ Counter placed in the

North-West Cell.

.-------.

|( )| |

|---|---|

| | |

.-------.

We know that this represents each of the Trio of equivalent Propositions

"No xy exist" = "No x are y" = "No y are x".

Similarly we may interpret a _Grey_ Counter, when placed in the

North-East, or South-West, or South-East Cell.

pg037

Next, let us suppose that we find a _Red_ Counter placed on the

partition which divides the North Half.

.-------.

| (.) |

|---|---|

| | |

.-------.

We know that this represents the Proposition "Some x exist."

Similarly we may interpret a _Red_ Counter, when placed on the partition

which divides the South, or West, or East Half.

* * * * *

Next, let us suppose that we find _two Red_ Counters placed in the North

Half, one in each Cell.

.-------.

|(.)|(.)|

|---|---|

| | |

.-------.

We know that this represents the _Double_ Proposition "Some x are y and

some are y'".

Similarly we may interpret _two Red_ Counters, when placed in the South,

or West, or East Half.

* * * * *

Next, let us suppose that we find _two Grey_ Counters placed in the

North Half, one in each Cell.

.-------.

|( )|( )|

|---|---|

| | |

.-------.

We know that this represents the Proposition "No x exist".

Similarly we may interpret _two Grey_ Counters, when placed in the

South, or West, or East Half.

* * * * *

Lastly, let us suppose that we find a _Red_ and a _Grey_ Counter placed

in the North Half, the _Red_ in the North-_West_ Cell, and the _Grey_ in

the North-_East_ Cell.

.-------.

|(.)|( )|

|---|---|

| | |

.-------.

We know that this represents the Proposition, "All x are y".

[Note that the _Half_, occupied by the two Counters, settles

what is to be the _Subject_ of the Proposition, and that the

_Cell_, occupied by the _Red_ Counter, settles what is to be its

_Predicate_.]

pg038

Similarly we may interpret a _Red_ and a _Grey_ counter, when placed in

any one of the seven similar positions

Red in North-East, Grey in North-West;

Red in South-West, Grey in South-East;

Red in South-East, Grey in South-West;

Red in North-West, Grey in South-West;

Red in South-West, Grey in North-West;

Red in North-East, Grey in South-East;

Red in South-East, Grey in North-East.

Once more the genial friend must be appealed to, and requested to

examine the Reader on Tables II and III, and to make him not only

_represent_ Propositions, but also _interpret_ Diagrams when marked with

Counters.

The Questions and Answers should be like this:--

Q. Represent "No x' are y'."

A. Grey Counter in S.E. Cell.

Q. Interpret Red Counter on E. partition.

A. "Some y' exist."

Q. Represent "All y' are x."

A. Red in N.E. Cell; Grey in S.E.

Q. Interpret Grey Counter in S.W. Cell.

A. "No x'y exist" = "No x' are y" = "No y are x'".

&c., &c.

At first the Examinee will need to have the Board and Counters before

him; but he will soon learn to dispense with these, and to answer with

his eyes shut or gazing into vacancy.

[Work Examples Sec. =1=, 5-8 (p. 97).]

pg039

BOOK IV.

THE TRILITERAL DIAGRAM.

.-----------------. .-----------------.

| | | | xy | xy' |

| | | | m' | m' |

| xy | xy' | | .----|----. |

| | | | | xy | xy'| |

| | | | | m | m | |

|--------|--------| |---|----|----|---|

| | | | |x'y |x'y'| |

| | | | | m | m | |

| x'y | x'y' | | .----|----. |

| | | | x'y | x'y' |

| | | | m' | m' |

.-----------------. .-----------------.

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.

[The Reader is strongly advised, in reading this Chapter, _not_

to refer to the above Diagrams, but to make a large copy of the

right-hand one for himself, _without any letters_, and to have

it by him while he reads, and keep his finger on that particular

_part_ of it, about which he is reading.]

pg040

Secondly, 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 Differentiae 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").

[Thus, in the "books" example, we might say "Let m mean 'bound',

so that m' will mean 'unbound'", and we might suppose that we

had subdivided the Class "old English books" into the two

Classes, "old English bound books" and "old English unbound

books", and had assigned the N.W. _Inner_ Cell to the one, and

the N.W. _Outer_ Cell to the other.]

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'.

[Thus, in the "books" example, we might suppose that we had

subdivided the "new English books" into the two Classes, "new

English bound books" and "new English unbound books", and had

assigned the S.W. _Inner_ Cell to the one, and the S.W. _Outer_

Cell to the other.]

It is evident that we have now assigned the _Inner Square_ to the

m-Class, and the _Outer Border_ to the m'-Class.

[Thus, in the "books" example, we have assigned the _Inner

Square_ to "bound books" and the _Outer Border_ to "unbound

books".]

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.

Q. Adjunct for South Half, Inner Portion?

A. x'm.

Q. Compartment for m'?

A. The Outer Border.

Q. Adjunct for North-East Quarter, Outer Portion?

A. xy'm'.

Q. Compartment for ym?

A. West Half, Inner Portion.

Q. Adjunct for South Half?

A. x'.

Q. Compartment for x'y'm?

A. South-East Quarter, Inner Portion.

&c. &c.

pg042

TABLE IV.

.-----------------------------------------------.

| Adjunct | |

| of | Compartments, or Cells, assigned |

| Classes. | to them. |

|----------|------------------------------------|

| x | North Half. |

| x' | South " |

| y | West " |

| y' | East " |

| m | Inner Square. |

| m' | Outer Border. |

|----------|------------------------------------|

| xy | North-West Quarter. |

| xy' | " East " |

| x'y | South-West " |

| x'y' | " East " |

| xm | North Half, Inner Portion. |

| xm' | " " Outer " |

| x'm | South " Inner " |

| x'm' | " " Outer " |

| ym | West " Inner " |

| ym' | " " Outer " |

| y'm | East " Inner " |

| y'm' | " " Outer " |

|----------|------------------------------------|

| xym | North-West Quarter, Inner Portion. |

| xym' | " " " Outer " |

| xy'm | " East " Inner " |

| xy'm' | " " " Outer " |

| x'ym | South-West " Inner " |

| x'ym' | " " " Outer " |

| x'y'm | " East " Inner " |

| x'y'm' | " " " Outer " |

.-----------------------------------------------.

pg043