Symbolic Logic
by Lewis Carroll · English · Gutenberg #28696 · 456,023 chars · public domain (US)
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Title: Symbolic Logic
Author: Lewis Carroll
Release date: May 5, 2009 [eBook #28696]
Most recently updated: January 5, 2021
Language: English
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*** START OF THE PROJECT GUTENBERG EBOOK SYMBOLIC LOGIC ***
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, _vivâ 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=' 1½
'=Class=' "
'=Peculiar=' Attributes "
'=Genus=' "
'=Species=' "
'=Differentia=' "
'=Real=' and '=Unreal=', or '=Imaginary=', Classes 2
'=Individual=' "
A Class regarded as a single Thing 2½
pg-xvi
CHAPTER III.
_DIVISION._
§ 1.
_Introductory._
'=Division=' 3
'=Codivisional=' Classes "
§ 2.
_Dichotomy._
'=Dichotomy=' 3½
Arbitrary limits of Classes "
Subdivision of Classes 4
CHAPTER IV.
_NAMES._
'=Name=' 4½
'=Real=' and '=Unreal=' Names "
Three ways of expressing a Name "
Two senses in which a plural Name may be used 5
CHAPTER V.
_DEFINITIONS._
'=Definition=' 6
Examples worked as models "
pg-xvii
=BOOK II.=
=PROPOSITIONS.=
CHAPTER I.
_PROPOSITIONS GENERALLY._
§ 1.
_Introductory._
Technical meaning of "some" 8
'=Proposition=' "
'=Normal form=' of a Proposition "
'=Subject=', '=Predicate=', and '=Terms=' 9
§ 2.
_Normal form of a Proposition._
Its four parts:--
(1) '=Sign of Quantity=' 9
(2) Name of Subject "
(3) '=Copula=' "
(4) Name of Predicate "
§ 3.
_Various kinds of Propositions._
Three kinds of Propositions:--
(1) Begins with "Some". Called a '=Particular=' Proposition:
also a Proposition '=in I=' 10
(2) Begins with "No". Called a '=Universal Negative='
Proposition: also a Proposition '=in E=' "
(3) Begins with "All". Called a '=Universal Affirmative='
Proposition: also a Proposition '=in A=' "
pg-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._
§ 1.
_Introductory._
'=Proposition of Relation=' 12
'=Universe of Discourse=,' or '=Univ.=' "
§ 2.
_Reduction of a Proposition of Relation
to Normal form._
Rules 13
Examples worked "
§ 3.
_A Proposition of Relation, beginning with "All",
is a Double Proposition._
Its equivalence to _two_ Propositions 17
pg-xix
§ 4.
_What is implied, in a Proposition of Relation,
as to the Reality of its Terms?_
Propositions beginning with "Some" 19
" " "No" "
" " "All" "
§ 5.
_Translation of a Proposition of Relation into
one or more Propositions of Existence._
Rules 20
Examples worked "
=BOOK III.=
=THE BILITERAL DIAGRAM.=
CHAPTER I.
_SYMBOLS AND CELLS._
The Diagram assigned to a certain Set of Things, viz. our
Univ. 22
Univ. divided into 'the x-Class' and 'the x'-Class' 23
The North and South Halves assigned to these two Classes "
The x-Class subdivided into 'the xy-Class' and 'the xy'-Class' "
The North-West and North-East Cells assigned to these
two Classes "
The x'-Class similarly divided "
The South-West and South-East Cells similarly assigned "
The West and East Halves have thus been assigned to
'the y-Class' and 'the y'-Class' 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._
§ 1.
_Introductory._
The word "Things" to be henceforwards omitted 27
'=Uniliteral=' Proposition "
'=Biliteral=' do. "
Proposition '=in terms of=' certain Letters "
§ 2.
_Representation of Propositions of Existence._
The Proposition "Some x exist" 28
Three other similar Propositions "
The Proposition "No x exist" "
Three other similar Propositions 29
The Proposition "Some xy exist" "
Three other similar Propositions "
The Proposition "No xy exist" "
Three other similar Propositions "
The Proposition "No x exist" is _Double_, and is equivalent
to the two Propositions "No xy exist" and "No xy' exist" 30
pg-xxi
§ 3.
_Representation of Propositions of Relations._
The Proposition "Some x are y" "
Three other similar Propositions "
The Proposition "Some y are x" 31
Three other similar Propositions "
Trio of equivalent Propositions, viz.
"Some xy exist" = "Some x are y" = "Some y are x" "
'=Converse=' Propositions, and '=Conversion=' "
Three other similar Trios 32
The Proposition "No x are y" "
Three other similar Propositions "
The Proposition "No y are x" "
Three other similar Propositions "
Trio of equivalent Propositions, viz.
"No xy exist" = "No x are y" = "No y are x" 33
Three other similar Trios "
The Proposition "All x are y" is _Double_, and is equivalent
to the two Propositions "Some x are y" and "No x are _y'_" "
Seven other similar Propositions 34
=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._
§ 1.
_Representation of Propositions of Existence in terms
of x and m, or of y and m._
The Proposition "Some xm exist" 43
Seven other similar Propositions "
The Proposition "No xm exist" 44
Seven other similar Propositions "
§ 2.
_Representation of Propositions of Relation in terms
of x and m, or of y and m._
The Pair of Converse Propositions
"Some x are m" = "Some m are x" "
Seven other similar Pairs "
The Pair of Converse Propositions
"No x are m" = "No m are x" "
Seven other similar Pairs "
The Proposition "All x are m" 45
Fifteen other similar Propositions "
=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._
§ 1.
_Introductory._
'=Concrete=' and '=Abstract=' Propositions 59
Method of translating a Proposition from concrete into
abstract form "
Two forms of Problems "
§ 2.
_Given a Pair of Propositions of Relation, which contain
between them a Pair of codivisional Classes, and which are
proposed as Premisses: to ascertain what Conclusion, if any,
is consequent from them._
Rules 60
Examples worked fully "
The same worked briefly, as models 64
§ 3.
_Given a Trio of Propositions of Relation, of which every
two contain a Pair of codivisional Classes, and which are
proposed as a Syllogism: to ascertain whether the proposed
Conclusion is consequent from the proposed Premisses,
and, if so, whether it is complete._
Rules 66
Examples worked briefly, as models "
pg-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 versâ_ "
pg-xxvii
CHAPTER III.
_SYLLOGISMS._
§ 1.
_Representation of Syllogisms._
Rules 73
§ 2.
_Formulæ for Syllogisms._
Three Formulæ 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.= Formulæ and Rules 78
Examples worked briefly, as models "
§ 3.
_Fallacies._
'=Fallacy=' 81
Method of finding Forms of Fallacies 82
Forms best stated in _words_ "
Three Forms of Fallacies:--
(1) Fallacy of Like Eliminands not asserted to exist "
(2) Fallacy of Unlike Eliminands with an Entity-Premiss 83
(3) Fallacy of two Entity-Premisses "
§ 4.
_Method of proceeding with a given Pair of Propositions._
Rules 84
pg-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._
§ 1.
_Introductory._
Form of Problem 87
Two Methods of Solution "
§ 2.
_Solution by Method of Separate Syllogisms._
Rules 88
Example worked "
pg-xxix
§ 3.
_Solution by Method of Underscoring._
'=Underscoring=' 91
Subscripts to be omitted "
Example worked fully 92
Example worked briefly, as model 93
Seventeen Examination-Papers 94
=BOOK VIII.=
=EXAMPLES, WITH ANSWERS AND SOLUTIONS.=
CHAPTER I.
_EXAMPLES._
§ 1.
_Propositions of Relation, to be reduced to normal form_ 97
§ 2.
_Pairs of Abstract Propositions, one in terms of x and m,
and the other in terms of y and m, to be represented on
the same Triliteral Diagram_ 98
§ 3.
_Marked Triliteral Diagrams, to be interpreted in terms
of x and y_ 99
§ 4.
_Pairs of Abstract Propositions, proposed as Premisses:
Conclusions to be found_ 100
pg-xxx
§ 5.
_Pairs of Concrete Propositions, proposed as Premisses:
Conclusions to be found_ 101
§ 6.
_Trios of Abstract Propositions, proposed as Syllogisms:
to be examined_ 106
§ 7.
_Trios of Concrete Propositions, proposed as Syllogisms:
to be examined_ 107
§ 8.
_Sets of Abstract Propositions, proposed as Premisses for
Soriteses: Conclusions to be found_ 110
§ 9.
_Sets of Concrete Propositions, proposed as Premisses for
Soriteses: Conclusions to be found_ 112
CHAPTER II.
_ANSWERS._
Answers to
§ 1 125
§ 2 126
§§ 3, 4 127
§ 5 128
§ 6 130
§ 7 131
§§ 8, 9 132
pg-xxxi
CHAPTER III.
_SOLUTIONS._
§ 1.
_Propositions of Relation reduced to normal form._
Solutions for § 1 134
§ 2.
_Method of Diagrams._
Solutions for
§ 4 Nos. 1 to 12 136
§ 5 " 1 to 12 138
§ 6 " 1 to 10 141
§ 7 " 1 to 6 144
§ 3.
_Method of Subscripts._
Solutions for
§ 4 146
§ 5 Nos. 13 to 24 147
§§ 6, 7, 8, 9 148 to 157
=NOTES= 164
=APPENDIX, ADDRESSED TO TEACHERS= 165
=NOTES TO APPENDIX= 195
=INDEX.=
§ 1. Tables 197
§ 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."]
pg001½
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."]
pg002½
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._
§ 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.]
pg003½
§ 2.
_Dichotomy._
If we think of a certain Class, and imagine that we have picked out from
it a certain smaller Class, it is evident that the _Remainder_ of the
large Class does _not_ possess the Differentia of that smaller Class.
Hence it may be regarded as _another_ smaller Class, whose Differentia
may be formed, from that of the Class first picked out, by prefixing the
word "not"; and we may imagine that we have _divided_ the Class first
thought of into _two_ smaller Classes, whose Differentiæ are
_contradictory_. This kind of Division is called '=Dichotomy='.
[For example, we may divide "books" into the two Classes whose
Differentiæ 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 Differentiæ 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.]
pg004½
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._
§ 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.]
§ 2.
_Normal form of a Proposition._
A Proposition, in normal form, consists of four parts, viz.--
(1) The word "some," or "no," or "all." (This word,
which tells us _how many_ Members of the Subject
are also Members of the Predicate, is called the
'=Sign of Quantity=.')
(2) Name of Subject.
(3) The verb "are" (or "is"). (This is called the
'=Copula=.')
(4) Name of Predicate.
pg010
§ 3.
_Various kinds of Propositions._
A Proposition, that begins with "Some", is said to be '=Particular=.' It
is also called 'a Proposition =in I=.'
[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._
§ 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
§ 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
§ 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
§ 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 § 3) a
similar Proposition beginning with "Some". Hence it is to be understood
as implying that _each_ Term, taken by itself, is _Real_.
[Thus, the Proposition "All hyænas are savage animals" contains
the Proposition "Some hyænas are savage animals". Hence it
implies that _each_ of the two Classes, "hyænas" and "savage
animals", taken by itself, is _Real_.]
pg020
§ 5.
_Translation of a Proposition of Relation into one or more Propositions
of Existence._
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 § 3)
equivalent to _two_ Propositions, one beginning with "Some" and the
other with "No", each of which we now know how to translate.
[Let us work a few Examples, to illustrate these Rules.
(1)
"Some apples are not ripe."
Here we arrange thus:--
"Some" _Sign of Quantity_.
"existing Things" _Subject_.
"are" _Copula_.
"not-ripe apples" _Predicate_.
or thus:--
"Some | existing Things | are | not-ripe apples."
pg021
(2)
"Some farmers always grumble at the weather, whatever
it may be."
Here we arrange thus:--
"Some | existing Things | are | farmers who always
grumble at the weather, whatever it may be."
(3)
"No lambs are accustomed to smoke cigars."
Here we arrange thus:--
"No | existing Things |are | lambs accustomed to
smoke cigars."
(4)
"None of my speculations have brought me as much
as 5 per cent."
Here we arrange thus:--
"No | existing Things | are | speculations of mine,
which have brought me as much as 5 per cent."
(5)
"None but the brave deserve the fair."
Here we note, to begin with, that the phrase "none but the
brave" is equivalent to "no not-brave men." We then arrange
thus:--
"No | existing Things | are | not-brave men deserving
of the fair."
(6)
"All bankers are rich men."
This is equivalent to the two Propositions "Some bankers are
rich men" and "No bankers are poor men."
Here we arrange thus:--
"Some | existing Things | are | rich bankers";
and
"No | existing Things | are | poor bankers."]
[Work Examples § =1=, 1-4 (p. 97).]
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
Differentiæ are "x" and "not-x" (which we may call "x'"), and that we
have assigned the _North_ Half of the Diagram to the one (which we may
call "the Class of x-Things," or "the x-Class"), and the _South_ Half to
the other (which we may call "the Class of x'-Things," or "the
x'-Class").
[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 Differentiæ 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
Differentiæ are "y" and "y'", and that we have assigned the North-_West_
Cell to the one (which we may call "the xy-Class"), and the North-_East_
Cell to the other (which we may call "the xy'-Class").
[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 Differentiæ
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._
§ 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
§ 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".]
§ 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 § =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 Differentiæ are m and m', and that we have assigned the N.W.
_Inner_ Cell to the one (which we may call "the Class of xym-Things", or
"the xym-Class"), and the N.W. _Outer_ Cell to the other (which we may
call "the Class of xym'-Things", or "the xym'-Class").
[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
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._
Let us take, first, the Proposition "Some xm exist".
[Note that the _full_ meaning of this Proposition is (as
explained at p. 12) "Some existing Things are xm-Things".]
This tells us that there is at least _one_ Thing in the Inner portion of
the North Half; that is, that this Compartment is _occupied_. And this
we can evidently represent by placing a _Red_ Counter on the partition
which divides it.
·---------------·
| | |
| ·---|---· |
| | (.) | |
|---|---|---|---|
| | | | |
| ·---|---· |
| | |
·---------------·
[In the "books" example, this Proposition would mean "Some old
bound books exist" (or "There are some old bound books").]
Similarly we may represent the seven similar Propositions, "Some xm'
exist", "Some x'm exist", "Some x'm' exist", "Some ym exist", "Some ym'
exist", "Some y'm exist", and "Some y'm' exist".
pg044
Let us take, next, the Proposition "No xm exist".
This tells us that there is _nothing_ in the Inner portion of the North
Half; that is, that this Compartment is _empty_. And this we can
represent by placing _two Grey_ Counters in it, one in each Cell.
·---------------·
| | |
| ·---|---· |
| |( )|( )| |
|---|---|---|---|
| | | | |
| ·---|---· |
| | |
·---------------·
Similarly we may represent the seven similar Propositions, in terms of x
and m, or of y and m, viz. "No xm' exist", "No x'm exist", &c.
* * * * *
These sixteen Propositions of Existence are the only ones that we shall
have to represent on this Diagram.
§ 2.
_Representation of Propositions of Relation in terms of x and m, or of y
and m._
Let us take, first, the Pair of Converse Propositions
"Some x are m" = "Some m are x."
We know that each of these is equivalent to the Proposition of Existence
"Some xm exist", which we already know how to represent.
·---------------·
| | |
| ·---|---· |
| | (.) | |
|---|---|---|---|
| | | | |
| ·---|---· |
| | |
·---------------·
Similarly for the seven similar Pairs, in terms of x and m, or of y and
m.
Let us take, next, the Pair of Converse Propositions
"No x are m" = "No m are x."
We know that each of these is equivalent to the Proposition of Existence
"No xm exist", which we already know how to represent.
·---------------·
| | |
| ·---|---· |
| |( )|( )| |
|---|---|---|---|
| | | | |
| ·---|---· |
| | |
·---------------·
Similarly for the seven similar Pairs, in terms of x and m, or of y and
m.
pg045
Let us take, next, the Proposition "All x are m."
·---------------·
|( ) | ( )|
| ·---|---· |
| | (.) | |
|---|---|---|---|
| | | | |
| ·---|---· |
| | |
·---------------·
We know (see p. 18) that this is a _Double_ Proposition, and equivalent
to the _two_ Propositions "Some x are m" and "No x are m' ", each of
which we already know how to represent.
Similarly for the fifteen similar Propositions, in terms of x and m, or
of y and m.
These thirty-two Propositions of Relation are the only ones that we
shall have to represent on this Diagram.
The Reader should now get his genial friend to question him on the
following four Tables.
The Victim should have nothing before him but a blank Triliteral
Diagram, a Red Counter, and 2 Grey ones, with which he is to represent
the various Propositions named by the Inquisitor, _e.g._ "No y' are m",
"Some xm' exist", &c., &c.
pg046
TABLE V.
·---------------------------------------------------------·
| ·---------------· Some xm exist | ·---------------· |
| | | | = Some x are m | | | | |
| | ·---|---· | = Some m are x | | ·---|---· | |
| | | (.) | | | | |( )|( )| | |
| |---|---|---|---| ·-----------------· |---|---|---|---| |
| | | | | | | No xm exist | | | | | |
| | ·---|---· | | = No x are m | ·---|---· | |
| | | | | = No m are x | | | |
| ·---------------· | ·---------------· |
|---------------------------------------------------------|
| ·---------------· Some xm' exist | ·---------------· |
| | (.) | = Some x are m' | |( ) | ( )| |
| | ·---|---· | = Some m' are x | | ·---|---· | |
| | | | | | | | | | | | |
| |---|---|---|---| ·-----------------· |---|---|---|---| |
| | | | | | | No xm' exist | | | | | |
| | ·---|---· | | = No x are m' | ·---|---· | |
| | | | | = No m' are x | | | |
| ·---------------· | ·---------------· |
|---------------------------------------------------------|
| ·---------------· Some x'm exist | ·---------------· |
| | | | = Some x' are m | | | | |
| | ·---|---· | = Some m are x' | | ·---|---· | |
| | | | | | | | | | | | |
| |---|---|---|---| ·-----------------· |---|---|---|---| |
| | | (.) | | | No x'm exist | |( )|( )| | |
| | ·---|---· | | = No x' are m | ·---|---· | |
| | | | | = No m are x' | | | |
| ·---------------· | ·---------------· |
|---------------------------------------------------------|
| ·---------------· Some x'm' exist | ·---------------· |
| | | | = Some x' are m'| | | | |
| | ·---|---· | = Some m' are x'| | ·---|---· | |
| | | | | | | | | | | | |
| |---|---|---|---| ·-----------------· |---|---|---|---| |
| | | | | | | No x'm' exist | | | | | |
| | ·---|---· | | = No x' are m' | ·---|---· | |
| | (.) | | = No m' are x' |( ) | ( )| |
| ·---------------· | ·---------------· |
·---------------------------------------------------------·
pg047
TABLE VI.
·---------------------------------------------------------·
| ·---------------· Some ym exist | ·---------------· |
| | | | = Some y are m | | | | |
| | ·---|---· | = Some m are y | | ·---|---· | |
| | | | | | | | |( )| | | |
| |---|(.)|---|---| ·-----------------· |---|---|---|---| |
| | | | | | | No ym exist | |( )| | | |
| | ·---|---· | | = No y are m | ·---|---· | |
| | | | | = No m are y | | | |
| ·---------------· | ·---------------· |
|---------------------------------------------------------|
| ·---------------· Some ym' exist | ·---------------· |
| | | | = Some y are m' | |( ) | | |
| | ·---|---· | = Some m' are y | | ·---|---· | |
| | | | | | | | | | | | |
| |(.)|---|---|---| ·-----------------· |---|---|---|---| |
| | | | | | | No ym' exist | | | | | |
| | ·---|---· | | = No y are m' | ·---|---· | |
| | | | | = No m' are y |( ) | | |
| ·---------------· | ·---------------· |
|---------------------------------------------------------|
| ·---------------· Some y'm exist | ·---------------· |
| | | | = Some y' are m | | | | |
| | ·---|---· | = Some m are y' | | ·---|---· | |
| | | | | | | | | |( )| | |
| |---|---|(.)|---| ·-----------------· |---|---|---|---| |
| | | | | | | No y'm exist | | |( )| | |
| | ·---|---· | | = No y' are m | ·---|---· | |
| | | | | = No m are y' | | | |
| ·---------------· | ·---------------· |
|---------------------------------------------------------|
| ·---------------· Some y'm' exist | ·---------------· |
| | | | = Some y' are m'| | | ( )| |
| | ·---|---· | = Some m' are y'| | ·---|---· | |
| | | | | | | | | | | | |
| |---|---|---|(.)| ·-----------------· |---|---|---|---| |
| | | | | | | No y'm' exist | | | | | |
| | ·---|---· | | = No y' are m' | ·---|---· | |
| | | | | = No m' are y' | | ( )| |
| ·---------------· | ·---------------· |
·---------------------------------------------------------·
pg048
TABLE VII.
·---------------------------------------------------------·
| ·---------------· | ·---------------· |
| |( ) | ( )| All x are m | | (.) | |
| | ·---|---· | | | ·---|---· | |
| | | (.) | | | | |( )|( )| | |
| |---|---|---|---| ·-----------------· |---|---|---|---| |
| | | | | | | | | | | | |
| | ·---|---· | | All x are m' | ·---|---· | |
| | | | | | | | |
| ·---------------· | ·---------------· |
|---------------------------------------------------------|
| ·---------------· | ·---------------· |
| | | | All x' are m | | | | |
| | ·---|---· | | | ·---|---· | |
| | | | | | | | | | | | |
| |---|---|---|---| ·-----------------· |---|---|---|---| |
| | | (.) | | | | |( )|( )| | |
| | ·---|---· | | All x' are m' | ·---|---· | |
| |( ) | ( )| | | (.) | |
| ·---------------· | ·---------------· |
|---------------------------------------------------------|
| ·---------------· | ·---------------· |
| | | | All m are x | | | | |
| | ·---|---· | | | ·---|---· | |
| | | (.) | | | | |( )|( )| | |
| |---|---|---|---| ·-----------------· |---|---|---|---| |
| | |( )|( )| | | | | (.) | | |
| | ·---|---· | | All m are x' | ·---|---· | |
| | | | | | | | |
| ·---------------· | ·---------------· |
|---------------------------------------------------------|
| ·---------------· | ·---------------· |
| | (.) | All m' are x | |( ) | ( )| |
| | ·---|---· | | | ·---|---· | |
| | | | | | | | | | | | |
| |---|---|---|---| ·-----------------· |---|---|---|---| |
| | | | | | | | | | | | |
| | ·---|---· | | All m' are x' | ·---|---· | |
| |( ) | ( )| | | (.) | |
| ·---------------· | ·---------------· |
·---------------------------------------------------------·
pg049
TABLE VIII.
·---------------------------------------------------------·
| ·---------------· | ·---------------· |
| |( ) | | All y are m | | | | |
| | ·---|---· | | | ·---|---· | |
| | | | | | | | |( )| | | |
| |---|(.)|---|---| ·-----------------· |(.)|---|---|---| |
| | | | | | | | |( )| | | |
| | ·---|---· | | All y are m' | ·---|---· | |
| |( ) | | | | | | |
| ·---------------· | ·---------------· |
|---------------------------------------------------------|
| ·---------------· | ·---------------· |
| | | ( )| All y' are m | | | | |
| | ·---|---· | | | ·---|---· | |
| | | | | | | | | |( )| | |
| |---|---|(.)|---| ·-----------------· |---|---|---|(.)| |
| | | | | | | | | |( )| | |
| | ·---|---· | | All y' are m' | ·---|---· | |
| | | ( )| | | | | |
| ·---------------· | ·---------------· |
|---------------------------------------------------------|
| ·---------------· | ·---------------· |
| | | | All m are y | | | | |
| | ·---|---· | | | ·---|---· | |
| | | |( )| | | | |( )| | | |
| |---|(.)|---|---| ·-----------------· |---|---|(.)|---| |
| | | |( )| | | | |( )| | | |
| | ·---|---· | | All m are y' | ·---|---· | |
| | | | | | | | |
| ·---------------· | ·---------------· |
|---------------------------------------------------------|
| ·---------------· | ·---------------· |
| | | ( )| All m' are y | |( ) | | |
| | ·---|---· | | | ·---|---· | |
| | | | | | | | | | | | |
| |(.)|---|---|---| ·-----------------· |---|---|---|(.)| |
| | | | | | | | | | | | |
| | ·---|---· | | All m' are y' | ·---|---· | |
| | | ( )| | |( ) | | |
| ·---------------· | ·---------------· |
·---------------------------------------------------------·
pg050
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 Reader had better now begin to draw little Diagrams for himself, and
to mark them with the Digits "I" and "O", instead of using the Board and
Counters: he may put a "I" to represent a _Red_ Counter (this may be
interpreted to mean "There is at least _one_ Thing here"), and a "O" to
represent a _Grey_ Counter (this may be interpreted to mean "There is
_nothing_ here").
The Pair of Propositions, that we shall have to represent, will always
be, one in terms of x and m, and the other in terms of y and m.
When we have to represent a Proposition beginning with "All", we break
it up into the _two_ Propositions to which it is equivalent.
When we have to represent, on the same Diagram, Propositions, of which
some begin with "Some" and others with "No", we represent the _negative_
ones _first_. This will sometimes save us from having to put a "I" "on a
fence" and afterwards having to shift it into a Cell.
[Let us work a few examples.
(1)
"No x are m';
No y' are m".
Let us first represent "No x are m'". This gives us Diagram a.
Then, representing "No y' are m" on the same Diagram, we get
Diagram b.
pg051
a b
·---------------· ·---------------·
|(O) | (O)| |(O) | (O)|
| ·---|---· | | ·---|---· |
| | | | | | | |(O)| |
|---|---|---|---| |---|---|---|---|
| | | | | | | |(O)| |
| ·---|---· | | ·---|---· |
| | | | | |
·---------------· ·---------------·
(2)
"Some m are x;
No m are y".
If, neglecting the Rule, we were begin with "Some m are x", we
should get Diagram a.
And if we were then to take "No m are y", which tells us that
the Inner N.W. Cell is _empty_, we should be obliged to take the
"I" off the fence (as it no longer has the choice of _two_
Cells), and to put it into the Inner N.E. Cell, as in Diagram c.
This trouble may be saved by beginning with "No m are y", as in
Diagram b.
And _now_, when we take "Some m are x", there is no fence to sit
on! The "I" has to go, at once, into the N.E. Cell, as in
Diagram c.
a b c
·---------------· ·---------------· ·---------------·
| | | | | | | | |
| ·---|---· | | ·---|---· | | ·---|---· |
| | (I) | | | |(O)| | | | |(O)|(I)| |
|---|---|---|---| |---|---|---|---| |---|---|---|---|
| | | | | | |(O)| | | | |(O)| | |
| ·---|---· | | ·---|---· | | ·---|---· |
| | | | | | | | |
·---------------· ·---------------· ·---------------·
(3)
"No x' are m';
All m are y".
Here we begin by breaking up the Second into the two
Propositions to which it is equivalent. Thus we have _three_
Propositions to represent, viz.--
(1) "No x' are m';
(2) Some m are y;
(3) No m are y'".
These we will take in the order 1, 3, 2.
First we take No. (1), viz. "No x' are m'". This gives us
Diagram a.
pg052
Adding to this, No. (3), viz. "No m are y'", we get Diagram b.
This time the "I", representing No. (2), viz. "Some m are y,"
has to sit on the fence, as there is no "O" to order it off!
This gives us Diagram c.
a b c
·---------------· ·---------------· ·---------------·
| | | | | | | | |
| ·---|---· | | ·---|---· | | ·---|---· |
| | | | | | | |(O)| | | | |(O)| |
|---|---|---|---| |---|---|---|---| |---|(I)|---|---|
| | | | | | | |(O)| | | | |(O)| |
| ·---|---· | | ·---|---· | | ·---|---· |
|(O) | (O)| |(O) | (O)| |(O) | (O)|
·---------------· ·---------------· ·---------------·
(4)
"All m are x;
All y are m".
Here we break up _both_ Propositions, and thus get _four_ to
represent, viz.--
(1) "Some m are x;
(2) No m are x';
(3) Some y are m;
(4) No y are m'".
These we will take in the order 2, 4, 1, 3.
First we take No. (2), viz. "No m are x'". This gives us Diagram
a.
To this we add No. (4), viz. "No y are m'", and thus get Diagram
b.
If we were to add to this No. (1), viz. "Some m are x", we
should have to put the "I" on a fence: so let us try No. (3)
instead, viz. "Some y are m". This gives us Diagram c.
And now there is no need to trouble about No. (1), as it would
not add anything to our information to put a "I" on the fence.
The Diagram _already_ tells us that "Some m are x".]
a b c
·---------------· ·---------------· ·---------------·
| | | |(O) | | |(O) | |
| ·---|---· | | ·---|---· | | ·---|---· |
| | | | | | | | | | | |(I)| | |
|---|---|---|---| |---|---|---|---| |---|---|---|---|
| |(O)|(O)| | | |(O)|(O)| | | |(O)|(O)| |
| ·---|---· | | ·---|---· | | ·---|---· |
| | | |(O) | | |(O) | |
·---------------· ·---------------· ·---------------·
[Work Examples § =1=, 9-12 (p. 97); § =2=, 1-20 (p. 98).]
pg053
CHAPTER IV.
_INTERPRETATION, IN TERMS OF x AND y, OF TRILITERAL DIAGRAM, WHEN MARKED
WITH COUNTERS OR DIGITS._
The problem before us is, given a marked Triliteral Diagram, to
ascertain _what_ Propositions of Relation, in terms of x and y, are
represented on it.
The best plan, for a _beginner_, is to draw a _Biliteral_ Diagram
alongside of it, and to transfer, from the one to the other, all the
information he can. He can then read off, from the Biliteral Diagram,
the required Propositions. After a little practice, he will be able to
dispense with the Biliteral Diagram, and to read off the result from the
Triliteral Diagram itself.
To _transfer_ the information, observe the following Rules:--
(1) Examine the N.W. Quarter of the Triliteral Diagram.
(2) If it contains a "I", in _either_ Cell, it is certainly
_occupied_, and you may mark the N.W. Quarter of the
Biliteral Diagram with a "I".
(3) If it contains _two_ "O"s, one in _each_ Cell, it is
certainly _empty_, and you may mark the N.W. Quarter of the
Biliteral Diagram with a "O".
pg054
(4) Deal in the same way with the N.E., the S.W., and the
S.E. Quarter.
[Let us take, as examples, the results of the four Examples
worked in the previous Chapters.
(1)
·---------------·
|(O) | (O)|
| ·---|---· |
| | |(O)| |
|---|---|---|---|
| | |(O)| |
| ·---|---· |
| | |
·---------------·
In the N.W. Quarter, only _one_ of the two Cells is marked as
_empty_: so we do not know whether the N.W. Quarter of the
Biliteral Diagram is _occupied_ or _empty_: so we cannot mark
it.
·-------·
| |(O)|
|---|---|
| | |
·-------·
In the N.E. Quarter, we find _two_ "O"s: so _this_ Quarter is
certainly _empty_; and we mark it so on the Biliteral Diagram.
In the S.W. Quarter, we have no information _at all_.
In the S.E. Quarter, we have not enough to use.
We may read off the result as "No x are y'", or "No y' are x,"
whichever we prefer.
(2)
·---------------·
| | |
| ·---|---· |
| |(O)|(I)| |
|---|---|---|---|
| |(O)| | |
| ·---|---· |
| | |
·---------------·
In the N.W. Quarter, we have not enough information to use.
In the N.E. Quarter, we find a "I". This shows us that it is
_occupied_: so we may mark the N.E. Quarter on the Biliteral
Diagram with a "I".
·-------·
| |(I)|
|---|---|
| | |
·-------·
In the S.W. Quarter, we have not enough information to use.
In the S.E. Quarter, we have none at all.
We may read off the result as "Some x are y'", or "Some y' are
x", whichever we prefer.
pg055
(3)
·---------------·
| | |
| ·---|---· |
| | |(O)| |
|---|(I)|---|---|
| | |(O)| |
| ·---|---· |
|(O) | (O)|
·---------------·
In the N.W. Quarter, we have _no_ information. (The "I", sitting
on the fence, is of no use to us until we know on _which_ side
he means to jump down!)
In the N.E. Quarter, we have not enough information to use.
Neither have we in the S.W. Quarter.
·-------·
| | |
|---|---|
| |(O)|
·-------·
The S.E. Quarter is the only one that yields enough information
to use. It is certainly _empty_: so we mark it as such on the
Biliteral Diagram.
We may read off the results as "No x' are y'", or "No y' are
x'", whichever we prefer.
(4)
·---------------·
|(O) | |
| ·---|---· |
| |(I)| | |
|---|---|---|---|
| |(O)|(O)| |
| ·---|---· |
|(O) | |
·---------------·
The N.W. Quarter is _occupied_, in spite of the "O" in the Outer
Cell. So we mark it with a "I" on the Biliteral Diagram.
The N.E. Quarter yields no information.
·-------·
|(I)| |
|---|---|
| | |
·-------·
The S.W. Quarter is certainly _empty_. So we mark it as such on
the Biliteral Diagram.
·-------·
|(I)| |
|---|---|
|(O)| |
·-------·
The S.E. Quarter does not yield enough information to use.
We read off the result as "All y are x."]
[Review Tables V, VI (pp. 46, 47). Work Examples § =1=, 13-16
(p. 97); § =2=, 21-32 (p. 98); § =3=, 1-20 (p. 99).]
pg056
BOOK V.
SYLLOGISMS.
CHAPTER I.
_INTRODUCTORY_
When a Trio of Biliteral Propositions of Relation is such that
(1) all their six Terms are Species of the same Genus,
(2) every two of them contain between them a Pair of
codivisional Classes,
(3) the three Propositions are so related that, if the first
two were true, the third would be true,
the Trio is called a '=Syllogism='; the Genus, of which each of the six
Terms is a Species, is called its ='Universe of Discourse=', or, more
briefly, its '=Univ.='; the first two Propositions are called its
'=Premisses=', and the third its '=Conclusion='; also the Pair of
codivisional Terms in the Premisses are called its '=Eliminands=', and
the other two its '=Retinends='.
The Conclusion of a Syllogism is said to be '=consequent=' from its
Premisses: hence it is usual to prefix to it the word "Therefore" (or
the Symbol ".'.").
pg057
[Note that the 'Eliminands' are so called because they are
_eliminated_, and do not appear in the Conclusion; and that the
'Retinends' are so called because they are _retained_, and _do_
appear in the Conclusion.
Note also that the question, whether the Conclusion is or is not
_consequent_ from the Premisses, is not affected by the _actual_
truth or falsity of any of the Trio, but depends entirely on
their _relationship to each other_.
As a specimen-Syllogism, let us take the Trio
"No x-Things are m-Things;
No y-Things are m'-Things.
No x-Things are y-Things."
which we may write, as explained at p. 26, thus:--
"No x are m;
No y are m'.
No x are y".
Here the first and second contain the Pair of codivisional
Classes m and m'; the first and third contain the Pair x and x;
and the second and third contain the Pair y and y.
Also the three Propositions are (as we shall see hereafter) so
related that, if the first two were true, the third would also
be true.
Hence the Trio is a _Syllogism_; the two Propositions, "No x are
m" and "No y are m'", are its _Premisses_; the Proposition "No x
are y" is its _Conclusion_; the Terms m and m' are its
_Eliminands_; and the Terms x and y are its _Retinends_.
Hence we may write it thus:--
"No x are m;
No y are m'.
.'. No x are y".
As a second specimen, let us take the Trio
"All cats understand French;
Some chickens are cats.
Some chickens understand French".
These, put into normal form, are
"All cats are creatures understanding French;
Some chickens are cats.
Some chickens are creatures understanding French".
Here all the six Terms are Species of the Genus "creatures."
Also the first and second Propositions contain the Pair of
codivisional Classes "cats" and "cats"; the first and third
contain the Pair "creatures understanding French" and "creatures
understanding French"; and the second and third contain the Pair
"chickens" and "chickens".
pg058
Also the three Propositions are (as we shall see at p. 64) so
related that, if the first two were true, the third would be
true. (The first two are, as it happens, _not_ strictly true in
_our_ planet. But there is nothing to hinder them from being
true in some _other_ planet, say _Mars_ or _Jupiter_--in which
case the third would _also_ be true in that planet, and its
inhabitants would probably engage chickens as
nursery-governesses. They would thus secure a singular
_contingent_ privilege, unknown in England, namely, that they
would be able, at any time when provisions ran short, to utilise
the nursery-governess for the nursery-dinner!)
Hence the Trio is a _Syllogism_; the Genus "creatures" is its
'Univ.'; the two Propositions, "All cats understand French" and
"Some chickens are cats", are its _Premisses_, the Proposition
"Some chickens understand French" is its _Conclusion_; the Terms
"cats" and "cats" are its _Eliminands_; and the Terms,
"creatures understanding French" and "chickens", are its
_Retinends_.
Hence we may write it thus:--
"All cats understand French;
Some chickens are cats;
.'. Some chickens understand French".]
pg059
CHAPTER II.
_PROBLEMS IN SYLLOGISMS._
§ 1.
_Introductory._
When the Terms of a Proposition are represented by _words_, it is said
to be '=concrete='; when by _letters_, '=abstract=.'
To translate a Proposition from concrete into abstract form, we fix on a
Univ., and regard each Term as a _Species_ of it, and we choose a letter
to represent its _Differentia_.
[For example, suppose we wish to translate "Some soldiers are
brave" into abstract form. We may take "men" as Univ., and
regard "soldiers" and "brave men" as _Species_ of the _Genus_
"men"; and we may choose x to represent the peculiar Attribute
(say "military") of "soldiers," and y to represent "brave." Then
the Proposition may be written "Some military men are brave
men"; _i.e._ "Some x-men are y-men"; _i.e._ (omitting "men," as
explained at p. 26) "Some x are y."
In practice, we should merely say "Let Univ. be "men",
x = soldiers, y = brave", and at once translate "Some soldiers
are brave" into "Some x are y."]
The Problems we shall have to solve are of two kinds, viz.
(1) "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."
(2) "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_."
These Problems we will discuss separately.
pg060
§ 2.
_Given a Pair of Propositions of Relation, which contain between them a
pair of codivisional Classes, and which are proposed as Premisses: to
ascertain what Conclusion, if any, is consequent from them._
The Rules, for doing this, are as follows:--
(1) Determine the 'Universe of Discourse'.
(2) Construct a Dictionary, making m and m (or m and m') represent the
pair of codivisional Classes, and x (or x') and y (or y') the other two.
(3) Translate the proposed Premisses into abstract form.
(4) Represent them, together, on a Triliteral Diagram.
(5) Ascertain what Proposition, if any, in terms of x and y, is _also_
represented on it.
(6) Translate this into concrete form.
It is evident that, if the proposed Premisses were true, this other
Proposition would _also_ be true. Hence it is a _Conclusion_ consequent
from the proposed Premisses.
[Let us work some examples.
(1)
"No son of mine is dishonest;
People always treat an honest man with respect".
Taking "men" as Univ., we may write these as follows:--
"No sons of mine are dishonest men;
All honest men are men treated with respect".
We can now construct our Dictionary, viz. m = honest; x = sons
of mine; y = treated with respect.
(Note that the expression "x = sons of mine" is an abbreviated
form of "x = the Differentia of 'sons of mine', when regarded as
a Species of 'men'".)
The next thing is to translate the proposed Premisses into
abstract form, as follows:--
"No x are m';
All m are y".
pg061
Next, by the process described at p. 50, we represent these on a
Triliteral Diagram, thus:--
·---------------·
|(O) | (O)|
| ·---|---· |
| | |(O)| |
|---|(I)|---|---|
| | |(O)| |
| ·---|---· |
| | |
·---------------·
Next, by the process described at p. 53, we transfer to a
Biliteral Diagram all the information we can.
·-------·
| |(O)|
|---|---|
| | |
·-------·
The result we read as "No x are y'" or as "No y' are x,"
whichever we prefer. So we refer to our Dictionary, to see which
will look best; and we choose
"No x are y'",
which, translated into concrete form, is
"No son of mine fails to be treated with respect".
(2)
"All cats understand French;
Some chickens are cats".
Taking "creatures" as Univ., we write these as follows:--
"All cats are creatures understanding French;
Some chickens are cats".
We can now construct our Dictionary, viz. m = cats;
x = understanding French; y = chickens.
The proposed Premisses, translated into abstract form, are
"All m are x;
Some y are m".
In order to represent these on a Triliteral Diagram, we break up
the first into the two Propositions to which it is equivalent,
and thus get the _three_ Propositions
(1) "Some m are x;
(2) No m are x';
(3) Some y are m".
The Rule, given at p. 50, would make us take these in the order
2, 1, 3.
This, however, would produce the result
·-----------------·
| | |
| ·----|----· |
| |(I)(I) | |
| |----|----| |
| |(O) | (O)| |
| ·----|----· |
| | |
·-----------------·
pg062
So it would be better to take them in the order 2, 3, 1. Nos.
(2) and (3) give us the result here shown; and now we need not
trouble about No. (1), as the Proposition "Some m are x" is
_already_ represented on the Diagram.
·---------------·
| | |
| ·---|---· |
| |(I)| | |
|---|---|---|---|
| |(O)|(O)| |
| ·---|---· |
| | |
·---------------·
Transferring our information to a Biliteral Diagram, we get
·-------·
|(I)| |
|---|---|
| | |
·-------·
This result we can read either as "Some x are y" or "Some y are
x".
After consulting our Dictionary, we choose
"Some y are x",
which, translated into concrete form, is
"Some chickens understand French."
(3)
"All diligent students are successful;
All ignorant students are unsuccessful".
Let Univ. be "students"; m = successful; x = diligent;
y = ignorant.
These Premisses, in abstract form, are
"All x are m;
All y are m'".
These, broken up, give us the four Propositions
(1) "Some x are m;
(2) No x are m';
(3) Some y are m';
(4) No y are m".
which we will take in the order 2, 4, 1, 3.
Representing these on a Triliteral Diagram, we get
·---------------·
|(O) | (O)|
| ·---|---· |
| |(O)|(I)| |
|---|---|---|---|
| |(O)| | |
| ·---|---· |
|(I) | |
·---------------·
And this information, transferred to a Biliteral Diagram, is
·-------·
|(O)|(I)|
|---|---|
|(I)| |
·-------·
Here we get _two_ Conclusions, viz.
"All x are y';
All y are x'."
pg063
And these, translated into concrete form, are
"All diligent students are (not-ignorant, i.e.) learned;
All ignorant students are (not-diligent, i.e.) idle".
(See p. 4.)
(4)
"Of the prisoners who were put on their trial at the last
Assizes, all, against whom the verdict 'guilty' was
returned, were sentenced to imprisonment;
Some, who were sentenced to imprisonment, were also
sentenced to hard labour".
Let Univ. be "the prisoners who were put on their trial at the
last Assizes"; m = who were sentenced to imprisonment;
x = against whom the verdict 'guilty' was returned; y = who were
sentenced to hard labour.
The Premisses, translated into abstract form, are
"All x are m;
Some m are y".
Breaking up the first, we get the three
(1) "Some x are m;
(2) No x are m';
(3) Some m are y".
Representing these, in the order 2, 1, 3, on a Triliteral
Diagram, we get
·---------------·
|(O) | (O)|
| ·---|---· |
| | (I) | |
|---|(I)|---|---|
| | | | |
| ·---|---· |
| | |
·---------------·
Here we get no Conclusion at all.
You would very likely have guessed, if you had seen _only_ the
Premisses, that the Conclusion would be
"Some, against whom the verdict 'guilty' was returned,
were sentenced to hard labour".
But this Conclusion is not even _true_, with regard to the
Assizes I have here invented.
"Not _true!_" you exclaim. "Then who _were_ they, who were
sentenced to imprisonment and were also sentenced to hard
labour? They _must_ have had the verdict 'guilty' returned
against them, or how could they be sentenced?"
Well, it happened like _this_, you see. They were three
ruffians, who had committed highway-robbery. When they were put
on their trial, they _pleaded_ 'guilty'. So no _verdict_ was
returned at all; and they were sentenced at once.]
I will now work out, in their briefest form, as models for the Reader to
imitate in working examples, the above four concrete Problems.
pg064
(1) [see p. 60]
"No son of mine is dishonest;
People always treat an honest man with respect."
Univ. "men"; m = honest; x = my sons; y = treated with respect.
·---------------· ·-------·
"No x are m'; |(O) | (O)| | |(O)|
All m are y." | ·---|---· | |---|---|
| | |(O)| | | | |
|---|(I)|---|---| ·-------·
| | |(O)| |
| ·---|---· | .'. "No x are y'."
| | |
·---------------·
i.e. "No son of mine ever fails to be treated with respect."
(2) [see p. 61]
"All cats understand French;
Some chickens are cats".
Univ. "creatures"; m = cats; x = understanding French; y = chickens.
·---------------· ·-------·
"All m are x; | | | |(I)| |
Some y are m." | ·---|---· | |---|---|
| |(I)| | | | | |
|---|---|---|---| ·-------·
| |(O)|(O)| |
| ·---|---· | .'. "Some y are x."
| | |
·---------------·
i.e. "Some chickens understand French."
(3) [see p. 62]
"All diligent students are successful;
All ignorant students are unsuccessful".
Univ. "students"; m = successful; x = diligent; y = ignorant.
·---------------· ·-------·
"All x are m; |(O) | (O)| |(O)|(I)|
All y are m'." | ·---|---· | |---|---|
| |(O)|(I)| | |(I)| |
|---|---|---|---| ·-------·
| |(O)| | |
| ·---|---· | .'. "All x are y';
|(I) | | All y are x'."
·---------------·
i.e. "All diligent students are learned; and all ignorant students are
idle".
pg065
(4) [see p. 63]
"Of the prisoners who were put on their trial at the last
Assizes, all, against whom the verdict 'guilty' was
returned, were sentenced to imprisonment;
Some, who were sentenced to imprisonment, were also
sentenced to hard labour".
Univ. "prisoners who were put on their trial at the last Assizes",
m = sentenced to imprisonment; x = against whom the verdict 'guilty' was
returned; y = sentenced to hard labour.
·---------------·
"All x are m; |(O) | (O)|
Some m are y." | ·---|---· |
| | (I) | |
|---|(I)|---|---|
| | | | |
| ·---|---· | There is no
| | | Conclusion.
·---------------·
[Review Tables VII, VIII (pp. 48, 49). Work Examples § =1=,
17-21 (p. 97); § =4=, 1-6 (p. 100); § =5=, 1-6 (p. 101).]
pg066
§ 3.
_Given a Trio of Propositions of Relation, of which every two contain a
Pair of codivisional Classes, and which are proposed as a Syllogism; to
ascertain whether the proposed Conclusion is consequent from the
proposed Premisses, and, if so, whether it is complete._
The Rules, for doing this, are as follows:--
(1) Take the proposed Premisses, and ascertain, by the process
described at p. 60, what Conclusion, if any, is consequent
from them.
(2) If there be _no_ Conclusion, say so.
(3) If there _be_ a Conclusion, compare it with the proposed
Conclusion, and pronounce accordingly.
I will now work out, in their briefest form, as models for the Reader to
imitate in working examples, six Problems.
(1)
"All soldiers are strong;
All soldiers are brave.
Some strong men are brave."
Univ. "men"; m = soldiers; x = strong; y = brave.
pg067
·---------------· ·-------·
"All m are x; | | | |(I)| |
All m are y. | ·---|---· | |---|---|
Some x are y." | |(I)|(O)| | | | |
|---|---|---|---| ·-------·
| |(O)|(O)| |
| ·---|---· | .'. "Some x are y."
| | |
·---------------·
Hence proposed Conclusion is right.
(2)
"I admire these pictures;
When I admire anything I wish to examine it thoroughly.
I wish to examine some of these pictures thoroughly."
Univ. "things"; m = admired by me; x = these pictures; y = things which
I wish to examine thoroughly.
·---------------· ·-------·
"All x are m; |(O) | (O)| |(I)|(O)|
All m are y. | ·---|---· | |---|---|
Some x are y." | |(I)|(O)| | | | |
|---|---|---|---| ·-------·
| | |(O)| |
| ·---|---· | .'. "All x are y."
| | |
·---------------·
Hence proposed Conclusion is _incomplete_, the _complete_ one being "I
wish to examine _all_ these pictures thoroughly".
(3)
"None but the brave deserve the fair;
Some braggarts are cowards.
Some braggarts do not deserve the fair."
Univ. "persons"; m = brave; x = deserving of the fair; y = braggarts.
·---------------· ·-------·
"No m' are x; |(O) | (O)| | | |
Some y are m'. | ·---|---· | |---|---|
Some y are x'." | | | | | |(I)| |
|---|---|---|---| ·-------·
| | | | |
| ·---|---· | .'. "Some y are x'."
|(I) | |
·---------------·
Hence proposed Conclusion is right.
pg068
(4)
"All soldiers can march;
Some babies are not soldiers.
Some babies cannot march".
Univ. "persons"; m = soldiers; x = able to march; y = babies.
·---------------·
"All m are x; | | |
Some y are m'. | ·---|---· |
Some y are x'." | | (I) | |
|(I)|---|---|---|
| |(O)|(O)| |
| ·---|---· | There is no
| | | Conclusion.
·---------------·
(5)
"All selfish men are unpopular;
All obliging men are popular.
All obliging men are unselfish".
Univ. "men"; m = popular; x = selfish; y = obliging.
·---------------· ·-------·
"All x are m'; |(O) | (I)| |(O)|(I)|
All y are m. | ·---|---· | |---|---|
All y are x'." | |(O)|(O)| | |(I)| |
|---|---|---|---| ·-------·
| |(I)| | |
| ·---|---· | .'. "All x are y';
|(O) | | All y are x'."
·---------------·
Hence proposed Conclusion is _incomplete_, the _complete_ one
containing, in addition, "All selfish men are disobliging".
(6)
"No one, who means to go by the train and cannot get a
conveyance, and has not enough time to walk to the
station, can do without running;
This party of tourists mean to go by the train and cannot
get a conveyance, but they have plenty of time to walk
to the station.
This party of tourists need not run."
Univ. "persons meaning to go by the train, and unable to get a
conveyance"; m = having enough time to walk to the station; x = needing
to run; y = these tourists.
pg069
·---------------·
"No m' are x'; |(O) | |
All y are m. | ·---|---· |
All y are x'." | | | | |
|---|(I)|---|---|
| | | | |
| ·---|---· | There is no
|(O) | (O)| Conclusion.
·---------------·
[Here is _another_ opportunity, gentle Reader, for playing a
trick on your innocent friend. Put the proposed Syllogism before
him, and ask him what he thinks of the Conclusion.
He will reply "Why, it's perfectly correct, of course! And if
your precious Logic-book tells you it _isn't_, don't believe it!
You don't mean to tell me those tourists _need_ to run? If _I_
were one of them, and knew the _Premisses_ to be true, I should
be _quite_ clear that I _needn't_ run--and I _should walk!_"
And _you_ will reply "But suppose there was a mad bull behind
you?"
And then your innocent friend will say "Hum! Ha! I must think
that over a bit!"
You may then explain to him, as a convenient _test_ of the
soundness of a Syllogism, that, if circumstances can be invented
which, without interfering with the truth of the _Premisses_,
would make the _Conclusion_ false, the Syllogism _must_ be
unsound.]
[Review Tables V-VIII (pp. 46-49). Work Examples § =4=, 7-12 (p.
100); § =5=, 7-12 (p. 101); § =6=, 1-10 (p. 106); § =7=, 1-6
(pp. 107, 108).]
pg070
BOOK VI.
THE METHOD OF SUBSCRIPTS.
CHAPTER I.
_INTRODUCTORY._
Let us agree that "x_{1}" shall mean "Some existing Things have the
Attribute x", i.e. (more briefly) "Some x exist"; also that "xy_{1}"
shall mean "Some xy exist", and so on. Such a Proposition may be called
an '=Entity=.'
[Note that, when there are _two_ letters in the expression, it
does not in the least matter which stands _first_: "xy_{1}" and
"yx_{1}" mean exactly the same.]
Also that "x_{0}" shall mean "No existing Things have the Attribute x",
i.e. (more briefly) "No x exist"; also that "xy_{0}" shall mean "No xy
exist", and so on. Such a Proposition may be called a '=Nullity='.
Also that "+" shall mean "and".
[Thus "ab_{1} + cd_{0}" means "Some ab exist and no cd exist".]
Also that "¶" shall mean "would, if true, prove".
[Thus, "x_{0} ¶ xy_{0}" means "The Proposition 'No x exist'
would, if true, prove the Proposition 'No xy exist'".]
When two Letters are both of them accented, or both _not_ accented, they
are said to have '=Like Signs=', or to be '=Like=': when one is
accented, and the other not, they are said to have '=Unlike Signs=', or
to be '=Unlike='.
pg071
CHAPTER II.
_REPRESENTATION OF PROPOSITIONS OF RELATION._
Let us take, first, the Proposition "Some x are y".
This, we know, is equivalent to the Proposition of Existence "Some xy
exist". (See p. 31.) Hence it may be represented by the expression
"xy_{1}".
The Converse Proposition "Some y are x" may of course be represented by
the _same_ expression, viz. "xy_{1}".
Similarly we may represent the three similar Pairs of Converse
Propositions, viz.--
"Some x are y'" = "Some y' are x",
"Some x' are y" = "Some y are x'",
"Some x' are y'" = "Some y' are x'".
Let us take, next, the Proposition "No x are y".
This, we know, is equivalent to the Proposition of Existence "No xy
exist". (See p. 33.) Hence it may be represented by the expression
"xy_{0}".
The Converse Proposition "No y are x" may of course be represented by
the _same_ expression, viz. "xy_{0}".
Similarly we may represent the three similar Pairs of Converse
Propositions, viz.--
"No x are y'" = "No y' are x",
"No x' are y" = "No y are x'",
"No x' are y'" = "No y' are x'".
pg072
Let us take, next, the Proposition "All x are y".
Now it is evident that the Double Proposition of Existence "Some x exist
and no xy' exist" tells us that _some_ x-Things exist, but that _none_
of them have the Attribute y': that is, it tells us that _all_ of them
have the Attribute y: that is, it tells us that "All x are y".
Also it is evident that the expression "x_{1} + xy'_{0}" represents this
Double Proposition.
Hence it also represents the Proposition "All x are y".
[The Reader will perhaps be puzzled by the statement that the
Proposition "All x are y" is equivalent to the Double
Proposition "Some x exist and no xy' exist," remembering that it
was stated, at p. 33, to be equivalent to the Double Proposition
"Some x are y and no x are y'" (i.e. "Some xy exist and no xy'
exist"). The explanation is that the Proposition "Some xy exist"
contains _superfluous information_. "Some x exist" is enough for
our purpose.]
This expression may be written in a shorter form, viz. "x_{1}y'_{0}",
since _each_ Subscript takes effect back to the _beginning_ of the
expression.
Similarly we may represent the seven similar Propositions "All x are
y'", "All x' are y", "All x' are y'", "All y are x", "All y are x'",
"All y' are x", and "All y' are x'".
[The Reader should make out all these for himself.]
It will be convenient to remember that, in translating a Proposition,
beginning with "All", from abstract form into subscript form, or _vice
versâ_, the Predicate _changes sign_ (that is, changes from positive to
negative, or else from negative to positive).
[Thus, the Proposition "All y are x'" becomes "y_{1}x_{0}",
where the Predicate changes from x' to x.
Again, the expression "x'_{1}y'_{0}" becomes "All x' are y",
where the Predicate changes for y' to y.]
pg073
CHAPTER III.
_SYLLOGISMS._
§ 1.
_Representation of Syllogisms._
We already know how to represent each of the three Propositions of a
Syllogism in subscript form. When that is done, all we need, besides, is
to write the three expressions in a row, with "+" between the Premisses,
and "¶" before the Conclusion.
[Thus the Syllogism
"No x are m';
All m are y.
.'. No x are y'."
may be represented thus:--
xm'_{0} + m_{1}y'_{0} ¶ xy'_{0}
When a Proposition has to be translated from concrete form into
subscript form, the Reader will find it convenient, just at
first, to translate it into _abstract_ form, and _thence_ into
subscript form. But, after a little practice, he will find it
quite easy to go straight from concrete form to subscript form.]
pg074
§ 2.
_Formulæ for solving Problems in Syllogisms._
When once we have found, by Diagrams, the Conclusion to a given Pair of
Premisses, and have represented the Syllogism in subscript form, we have
a _Formula_, by which we can at once find, without having to use
Diagrams again, the Conclusion to any _other_ Pair of Premisses having
the _same_ subscript forms.
[Thus, the expression
xm_{0} + ym'_{0} ¶ xy_{0}
is a Formula, by which we can find the Conclusion to any Pair of
Premisses whose subscript forms are
xm_{0} + ym'_{0}
For example, suppose we had the Pair of Propositions
"No gluttons are healthy;
No unhealthy men are strong".
proposed as Premisses. Taking "men" as our 'Universe', and
making m = healthy; x = gluttons; y = strong; we might translate
the Pair into abstract form, thus:--
"No x are m;
No m' are y".
These, in subscript form, would be
xm_{0} + m'y_{0}
which are identical with those in our _Formula_. Hence we at
once know the Conclusion to be
xy_{0}
that is, in abstract form,
"No x are y";
that is, in concrete form,
"No gluttons are strong".]
I shall now take three different forms of Pairs of Premisses, and work
out their Conclusions, once for all, by Diagrams; and thus obtain some
useful Formulæ. I shall call them "Fig. I", "Fig. II", and "Fig. III".
pg075
Fig. I.
This includes any Pair of Premisses which are both of them Nullities,
and which contain Unlike Eliminands.
The simplest case is
·---------------· ·-------·
xm_{0} + ym'_{0} |(O) | | |(O)| |
| ·---|---· | |---|---|
| |(O)|(O)| | | | |
|---|---|---|---| ·-------·
| | | | |
| ·---|---· | .'. xy_{0}
|(O) | |
·---------------·
In this case we see that the Conclusion is a Nullity, and that the
Retinends have kept their Signs.
And we should find this Rule to hold good with _any_ Pair of Premisses
which fulfil the given conditions.
[The Reader had better satisfy himself of this, by working out,
on Diagrams, several varieties, such as
m_{1}x_{0} + ym'_{0} (which ¶ xy_{0})
xm'_{0} + m_{1}y_{0} (which ¶ xy_{0})
x'm_{0} + ym'_{0} (which ¶ x'y_{0})
m'_{1}x'_{0} + m_{1}y'_{0} (which ¶ x'y'_{0}).]
If either Retinend is asserted in the _Premisses_ to exist, of course it
may be so asserted in the _Conclusion_.
Hence we get two _Variants_ of Fig. I, viz.
(a) where _one_ Retinend is so asserted;
(b) where _both_ are so asserted.
[The Reader had better work out, on Diagrams, examples of these
two Variants, such as
m_{1}x_{0} + y_{1}m'_{0} (which proves y_{1}x_{0})
x_{1}m'_{0} + m_{1}y_{0} (which proves x_{1}y_{0})
x'_{1}m_{0} + y_{1}m'_{0} (which proves x'_{1}y_{0} + y_{1}x'_{0}).]
The Formula, to be remembered, is
xm_{0} + ym'_{0} ¶ xy_{0}
with the following two Rules:--
(1) _Two Nullities, with Unlike Eliminands, yield a Nullity,
in which both Retinends keep their Signs._
pg076
(2) _A Retinend, asserted in the Premisses to exist, may be
so asserted in the Conclusion._
[Note that Rule (1) is merely the Formula expressed in words.]
Fig. II.
This includes any Pair of Premisses, of which one is a Nullity and the
other an Entity, and which contain Like Eliminands.
The simplest case is
xm_{0} + ym_{1}
·---------------· ·-------·
| | | | | |
| ·---|---· | |---|---|
| |(O)|(O)| | |(I)| |
|---|---|---|---| ·-------·
| |(I)| | |
| ·---|---· | .'. x'y_{1}
| | |
·---------------·
In this case we see that the Conclusion is an Entity, and that the
Nullity-Retinend has changed its Sign.
And we should find this Rule to hold good with _any_ Pair of Premisses
which fulfil the given conditions.
[The Reader had better satisfy himself of this, by working out,
on Diagrams, several varieties, such as
x'm_{0} + ym_{1} (which ¶ xy_{1})
x_{1}m'_{0} + y'm'_{1} (which ¶ x'y'_{1})
m_{1}x_{0} + y'm_{1} (which ¶ x'y'_{1}).]
The Formula, to be remembered, is,
xm_{0} + ym_{1} ¶ x'y_{1}
with the following Rule:--
_A Nullity and an Entity, with Like Eliminands, yield an
Entity, in which the Nullity-Retinend changes its Sign._
[Note that this Rule is merely the Formula expressed in words.]
pg077
Fig. III.
This includes any Pair of Premisses which are both of them Nullities,
and which contain Like Eliminands asserted to exist.
The simplest case is
xm_{0} + ym_{0} + m_{1}
[Note that "m_{1}" is here stated _separately_, because it does
not matter in which of the two Premisses it occurs: so that this
includes the _three_ forms "m_{1}x_{0} + ym_{0}", "xm_{0} +
m_{1}y_{0}", and "m_{1}x_{0} + m_{1}y_{0}".]
·---------------· ·-------·
| | | | | |
| ·---|---· | |---|---|
| |(O)|(O)| | | |(I)|
|---|---|---|---| ·-------·
| |(O)|(I)| |
| ·---|---· | .'. x'y'_{1}
| | |
·---------------·
In this case we see that the Conclusion is an Entity, and that _both_
Retinends have changed their Signs.
And we should find this Rule to hold good with _any_ Pair of Premisses
which fulfil the given conditions.
[The Reader had better satisfy himself of this, by working out,
on Diagrams, several varieties, such as
x'm_{0} + m_{1}y_{0} (which ¶ xy'_{1})
m'_{1}x_{0} + m'y'_{0} (which ¶ x'y_{1})
m_{1}x'_{0} + m_{1}y'_{0} (which ¶ xy_{1}).]
The Formula, to be remembered, is
xm_{0} + ym_{0} + m_{1} ¶ x'y'_{1}
with the following Rule (which is merely the Formula expressed in
words):--
_Two Nullities, with Like Eliminands asserted to exist, yield
an Entity, in which both Retinends change their Signs._
* * * * *
In order to help the Reader to remember the peculiarities and Formulæ of
these three Figures, I will put them all together in one Table.
pg078
TABLE IX.
_______________________________________________________
| |
| Fig. I. |
| |
| xm_{0} + ym'_{0} ¶ xy_{0} |
| |
| Two Nullities, with Unlike Eliminands, yield a |
| Nullity, in which both Retinends keep their Signs. |
| |
| A Retinend, asserted in the Premisses to exist, may |
| be so asserted in the Conclusion. |
|_______________________________________________________|
| |
| Fig. II. |
| |
| xm_{0} + ym_{1} ¶ x'y_{1} |
| |
| A Nullity and an Entity, with Like Eliminands, |
| yield an Entity, in which the Nullity-Retinend |
| changes its Sign. |
|_______________________________________________________|
| |
| Fig. III. |
| |
| xm_{0} + ym_{0} + m_{1} ¶ x'y'_{1} |
| |
| Two Nullities, with Like Eliminands asserted |
| to exist, yield an Entity, in which both Retinends |
| change their Signs. |
|_______________________________________________________|
I will now work out, by these Formulæ, as models for the Reader to
imitate, some Problems in Syllogisms which have been already worked, by
Diagrams, in Book V., Chap. II.
(1) [see p. 64]
"No son of mine is dishonest;
People always treat an honest man with respect."
Univ. "men"; m = honest; x = my sons; y = treated with respect.
xm'_{0} + m_{1}y'_{0} ¶ xy'_{0} [Fig. I.
_i.e._ "No son of mine ever fails to be treated with respect."
pg079
(2) [see p. 64]
"All cats understand French;
Some chickens are cats."
Univ. "creatures"; m = cats; x = understanding French; y = chickens.
m_{1}x'_{0} + ym_{1} ¶ xy_{1} [Fig. II.
_i.e._ "Some chickens understand French."
(3) [see p. 64]
"All diligent students are successful;
All ignorant students are unsuccessful."
Univ. "students"; m = successful; x = diligent; y = ignorant.
x_{1}m'_{0} + y_{1}m_{0} ¶ x_{1}y_{0} + y_{1}x_{0} [Fig. I (b).
_i.e._ "All diligent students are learned; and all ignorant students are
idle."
(4) [see p. 66]
"All soldiers are strong;
All soldiers are brave.
Some strong men are brave."
Univ. "men"; m = soldiers; x = strong; y = brave.
m_{1}x'_{0} + m_{1}y'_{0} ¶ xy_{1} [Fig. III.
Hence proposed Conclusion is right.
(5) [see p. 67]
"I admire these pictures;
When I admire anything, I wish to examine it thoroughly.
I wish to examine some of these pictures thoroughly."
Univ. "things"; m = admired by me; x = these; y = things which I wish to
examine thoroughly.
x_{1}m'_{0} + m_{1}y'_{0} ¶ x_{1}y'_{0} [Fig. I (a).
Hence proposed Conclusion, xy_{1}, is _incomplete_, the _complete_ one
being "I wish to examine _all_ these pictures thoroughly."
pg080
(6) [see p. 67]
"None but the brave deserve the fair;
Some braggarts are cowards.
Some braggarts do not deserve the fair."
Univ. "persons"; m = brave; x = deserving of the fair; y = braggarts.
m'x_{0} + ym'_{1} ¶ x'y_{1} [Fig. II.
Hence proposed Conclusion is right.
(7) [see p. 69]
"No one, who means to go by the train and cannot get a
conveyance, and has not enough time to walk to the
station, can do without running;
This party of tourists mean to go by the train and cannot
get a conveyance, but they have plenty of time to
walk to the station.
This party of tourists need not run."
Univ. "persons meaning to go by the train, and unable to get a
conveyance"; m = having enough time to walk to the station; x = needing
to run; y = these tourists.
m'x'_{0} + y_{1}m'_{0} do not come under any of the three Figures. Hence
it is necessary to return to the Method of Diagrams, as shown at p. 69.
Hence there is no Conclusion.
[Work Examples § =4=, 12-20 (p. 100); § =5=, 13-24 (pp. 101,
102); § =6=, 1-6 (p. 106); § =7=, 1-3 (pp. 107, 108). Also read
Note (A), at p. 164.]
pg081
§ 3.
_Fallacies._
Any argument which _deceives_ us, by seeming to prove what it does not
really prove, may be called a '=Fallacy=' (derived from the Latin verb
_fallo_ "I deceive"): but the particular kind, to be now discussed,
consists of a Pair of Propositions, which are proposed as the Premisses
of a Syllogism, but yield no Conclusion.
When each of the proposed Premisses is a Proposition in _I_, or _E_, or
_A_, (the only kinds with which we are now concerned,) the Fallacy may
be detected by the 'Method of Diagrams,' by simply setting them out on a
Triliteral Diagram, and observing that they yield no information which
can be transferred to the Biliteral Diagram.
But suppose we were working by the 'Method of _Subscripts_,' and had to
deal with a Pair of proposed Premisses, which happened to be a
'Fallacy,' how could we be certain that they would not yield any
Conclusion?
Our best plan is, I think, to deal with _Fallacies_ in the same was as
we have already dealt with _Syllogisms_: that is, to take certain forms
of Pairs of Propositions, and to work them out, once for all, on the
Triliteral Diagram, and ascertain that they yield _no_ Conclusion; and
then to record them, for future use, as _Formulæ for Fallacies_, just as
we have already recorded our three _Formulæ for Syllogisms_.
pg082
Now, if we were to record the two Sets of Formulæ in the _same_ shape,
viz. by the Method of Subscripts, there would be considerable risk of
confusing the two kinds. Hence, in order to keep them distinct, I
propose to record the Formulæ for _Fallacies_ in _words_, and to call
them "Forms" instead of "Formulæ."
Let us now proceed to find, by the Method of Diagrams, three "Forms of
Fallacies," which we will then put on record for future use. They are as
follows:--
(1) Fallacy of Like Eliminands not asserted to exist.
(2) Fallacy of Unlike Eliminands with an Entity-Premiss.
(3) Fallacy of two Entity-Premisses.
These shall be discussed separately, and it will be seen that each fails
to yield a Conclusion.
(1) _Fallacy of Like Eliminands not asserted to exist._
It is evident that neither of the given Propositions can be an _Entity_,
since that kind asserts the _existence_ of both of its Terms (see p.
20). Hence they must both be _Nullities_.
Hence the given Pair may be represented by (xm_{0} + ym_{0}), with or
without x_{1}, y_{1}.
These, set out on Triliteral Diagrams, are
xm_{0} + ym_{0} x_{1}m_{0} + ym_{0}
·---------------· ·---------------·
| | | | (I) |
| ·---|---· | | ·---|---· |
| |(O)|(O)| | | |(O)|(O)| |
|---|---|---|---| |---|---|---|---|
| |(O)| | | | |(O)| | |
| ·---|---· | | ·---|---· |
| | | | | |
·---------------· ·---------------·
xm_{0} + y_{1}m_{0} x_{1}m_{0} + y_{1}m_{0}
·---------------· ·---------------·
| | | | (I) |
| ·---|---· | | ·---|---· |
| |(O)|(O)| | | |(O)|(O)| |
|(I)|---|---|---| |(I)|---|---|---|
| |(O)| | | | |(O)| | |
| ·---|---· | | ·---|---· |
| | | | | |
·---------------· ·---------------·
pg083
(2) _Fallacy of Unlike Eliminands with an Entity-Premiss._
Here the given Pair may be represented by (xm_{0} + ym'_{1}) with or
without x_{1} or m_{1}.
These, set out on Triliteral Diagrams, are
xm_{0} + ym'_{1} x_{1}m_{0} + ym'_{1} m_{1}x_{0} + ym'_{1}
·---------------· ·---------------· ·---------------·
| | | | (I) | | | |
| ·---|---· | | ·---|---· | | ·---|---· |
| |(O)|(O)| | | |(O)|(O)| | | |(O)|(O)| |
|(I)|---|---|---| |(I)|---|---|---| |(I)|---|---|---|
| | | | | | | | | | | | (I) | |
| ·---|---· | | ·---|---· | | ·---|---· |
| | | | | | | | |
·---------------· ·---------------· ·---------------·
(3) _Fallacy of two Entity-Premisses._
Here the given Pair may be represented by either (xm_{1} + ym_{1}) or
(xm_{1} + ym'_{1}).
These, set out on Triliteral Diagrams, are
xm_{1} + ym_{1} xm_{1} + ym'_{1}
·---------------· ·---------------·
| | | | | |
| ·---|---· | | ·---|---· |
| | (I) | | | | (I) | |
|---|(I)|---|---| |(I)|---|---|---|
| | | | | | | | | |
| ·---|---· | | ·---|---· |
| | | | | |
·---------------· ·---------------·
pg084
§ 4.
_Method of proceeding with a given Pair of Propositions._
Let us suppose that we have before us a Pair of Propositions of
Relation, which contain between them a Pair of codivisional Classes, and
that we wish to ascertain what Conclusion, if any, is consequent from
them. We translate them, if necessary, into subscript-form, and then
proceed as follows:--
(1) We examine their Subscripts, in order to see whether they are
(a) a Pair of Nullities;
or (b) a Nullity and an Entity;
or (c) a Pair of Entities.
(2) If they are a Pair of Nullities, we examine their Eliminands, in
order to see whether they are Unlike or Like.
If their Eliminands are _Unlike_, it is a case of Fig. I. We then
examine their Retinends, to see whether one or both of them are asserted
to _exist_. If one Retinend is so asserted, it is a case of Fig. I (a);
if both, it is a case of Fig. I (b).
If their Eliminands are Like, we examine them, in order to see whether
either of them is asserted to exist. If so, it is a case of Fig. III.;
if not, it is a case of "Fallacy of Like Eliminands not asserted to
exist."
(3) If they are a Nullity and an Entity, we examine their Eliminands, in
order to see whether they are Like or Unlike.
If their Eliminands are Like, it is a case of Fig. II.; if _Unlike_, it
is a case of "Fallacy of Unlike Eliminands with an Entity-Premiss."
(4) If they are a Pair of Entities, it is a case of "Fallacy of two
Entity-Premisses."
[Work Examples § =4=, 1-11 (p. 100); § =5=, 1-12 (p. 101);
§ =6=, 7-12 (p. 106); § =7=, 7-12 (p. 108).]
pg085
BOOK VII.
SORITESES.
CHAPTER I.
_INTRODUCTORY._
When a Set of three or more Biliteral Propositions are such that all
their Terms are Species of the same Genus, and are also so related that
two of them, taken together, yield a Conclusion, which, taken with
another of them, yields another Conclusion, and so on, until all have
been taken, it is evident that, if the original Set were true, the last
Conclusion would _also_ be true.
Such a Set, with the last Conclusion tacked on, is called a '=Sorites=';
the original Set of Propositions is called its '=Premisses='; each of
the intermediate Conclusions is called a '=Partial Conclusion=' of the
Sorites; the last Conclusion is called its '=Complete Conclusion=,' or,
more briefly, its '=Conclusion='; the Genus, of which all the Terms are
Species, is called its '=Universe of Discourse=', or, more briefly, its
'=Univ.='; the Terms, used as Eliminands in the Syllogisms, are called
its '=Eliminands='; and the two Terms, which are retained, and therefore
appear in the Conclusion, are called its '=Retinends='.
[Note that each _Partial_ Conclusion contains one or two
_Eliminands_; but that the _Complete_ Conclusion contains
_Retinends_ only.]
The Conclusion is said to be '=consequent=' from the Premisses; for
which reason it is usual to prefix to it the word "Therefore" (or the
symbol ".'.").
[Note that the question, whether the Conclusion is or is not
_consequent_ from the Premisses, is not affected by the _actual_
truth or falsity of any one of the Propositions which make up
the Sorites, by depends entirely on their _relationship to one
another_.
pg086
As a specimen-Sorites, let us take the following Set of 5
Propositions:--
(1) "No a are b';
(2) All b are c;
(3) All c are d;
(4) No e' are a';
(5) All h are e'".
Here the first and second, taken together, yield "No a are c'".
This, taken along with the third, yields "No a are d'".
This, taken along with the fourth, yields "No d' are e'".
And this, taken along with the fifth, yields "All h are d".
Hence, if the original Set were true, this would _also_ be true.
Hence the original Set, with this tacked on, is a _Sorites_; the
original Set is its _Premisses_; the Proposition "All h are d"
is its _Conclusion_; the Terms a, b, c, e are its _Eliminands_;
and the Terms d and h are its _Retinends_.
Hence we may write the whole Sorites thus:--
"No a are b';
All b are c;
All c are d;
No e' are a';
All h are e'.
.'. All h are d".
In the above Sorites, the 3 Partial Conclusions are the
Positions "No a are e'", "No a are d'", "No d' are e'"; but, if
the Premisses were arranged in other ways, other Partial
Conclusions might be obtained. Thus, the order 41523 yields the
Partial Conclusions "No c' are b'", "All h are b", "All h are
c". There are altogether _nine_ Partial Conclusions to this
Sorites, which the Reader will find it an interesting task to
make out for himself.]
pg087
CHAPTER II.
_PROBLEMS IN SORITESES._
§ 1.
_Introductory._
The Problems we shall have to solve are of the following form:--
"Given three or more Propositions of Relation, which are proposed as
Premisses: to ascertain what Conclusion, if any, is consequent from
them."
We will limit ourselves, at present, to Problems which can be worked by
the Formulæ of Fig. I. (See p. 75.) Those, that require _other_ Formulæ,
are rather too hard for beginners.
Such Problems may be solved by either of two Methods, viz.
(1) The Method of Separate Syllogisms;
(2) The Method of Underscoring.
These shall be discussed separately.
pg088
§ 2.
_Solution by Method of Separate Syllogisms._
The Rules, for doing this, are as follows:--
(1) Name the 'Universe of Discourse'.
(2) Construct a Dictionary, making a, b, c, &c. represent
the Terms.
(3) Put the Proposed Premisses into subscript form.
(4) Select two which, containing between them a pair of
codivisional Classes, can be used as the Premisses of a
Syllogism.
(5) Find their Conclusion by Formula.
(6) Find a third Premiss which, along with this Conclusion,
can be used as the Premisses of a second Syllogism.
(7) Find a second Conclusion by Formula.
(8) Proceed thus, until all the proposed Premisses have
been used.
(9) Put the last Conclusion, which is the Complete
Conclusion of the Sorites, into concrete form.
[As an example of this process, let us take, as the proposed Set
of Premisses,
(1) "All the policemen on this beat sup with our cook;
(2) No man with long hair can fail to be a poet;
(3) Amos Judd has never been in prison;
(4) Our cook's 'cousins' all love cold mutton;
(5) None but policemen on this beat are poets;
(6) None but her 'cousins' ever sup with our cook;
(7) Men with short hair have all been in prison."
Univ. "men"; a = Amos Judd; b = cousins of our cook; c = having
been in prison; d = long-haired; e = loving cold mutton;
h = poets; k = policemen on this beat; l = supping with our cook
pg089
We now have to put the proposed Premisses into _subscript_ form.
Let us begin by putting them into _abstract_ form. The result is
(1) "All k are l;
(2) No d are h';
(3) All a are c';
(4) All b are e;
(5) No k' are h;
(6) No b' are l;
(7) All d' are c."
And it is now easy to put them into _subscript_ form, as
follows:--
(1) k_{1}l'_{0}
(2) dh'_{0}
(3) a_{1}c_{0}
(4) b_{1}e'_{0}
(5) k'h_{0}
(6) b'l_{0}
(7) d'_{1}c'_{0}
We now have to find a pair of Premisses which will yield a
Conclusion. Let us begin with No. (1), and look down the list,
till we come to one which we can take along with it, so as to
form Premisses belonging to Fig. I. We find that No. (5) will
do, since we can take k as our Eliminand. So our first syllogism
is
(1) k_{1}l'_{0}
(5) k'h_{0}
.'. l'h_{0} ... (8)
We must now begin again with l'h_{0} and find a Premiss to go
along with it. We find that No. (2) will do, h being our
Eliminand. So our next Syllogism is
(8) l'h_{0}
(2) dh'_{0}
.'. l'd_{0} ... (9)
We have now used up Nos. (1), (5), and (2), and must search
among the others for a partner for l'd_{0}. We find that No. (6)
will do. So we write
(9) l'd_{0}
(6) b'l_{0}
.'. db'_{0} ... (10)
Now what can we take along with db'_{0}? No. (4) will do.
(10) db'_{0}
(4) b_{1}e'_{0}
.'. de'_{0} ... (11)
pg090
Along with this we may take No. (7).
(11) de'_{0}
(7) d'_{1}c'_{0}
.'. c'e'_{0} ... (12)
And along with this we may take No. (3).
(12) c'e'_{0}
(3) a_{1}c_{0}
.'. a_{1}e'_{0}
This Complete Conclusion, translated into _abstract_ form, is
"All a are e";
and this, translated into _concrete_ form, is
"Amos Judd loves cold mutton."
In actually _working_ this Problem, the above explanations
would, of course, be omitted, and all, that would appear on
paper, would be as follows:--
(1) k_{1}l'_{0}
(2) dh'_{0}
(3) a_{1}c_{0}
(4) b_{1}e'_{0}
(5) k'h_{0}
(6) b'l_{0}
(7) d'_{1}c'_{0}
(1) k_{1}l'_{0}
(5) k'h_{0}
.'. l'h_{0} ... (8)
(8) l'h_{0}
(2) dh'_{0}
.'. l'd_{0} ... (9)
(9) l'd_{0}
(6) b'l_{0}
.'. db'_{0} ... (10)
(10) db'_{0}
(4) b_{1}e'_{0}
.'. de'_{0} ... (11)
(11) de'_{0}
(7) d'_{1}c'_{0}
.'. c'e'_{0} ... (12)
(12) c'e'_{0}
(3) a_{1}c_{0}
.'. a_{1}e'_{0}
Note that, in working a Sorites by this Process, we may begin
with _any_ Premiss we choose.]
pg091
§ 3.
_Solution by Method of Underscoring._
Consider the Pair of Premisses
xm_{0} + ym'_{0}
which yield the Conclusion xy_{0}
We see that, in order to get this Conclusion, we must eliminate m and
m', and write x and y together in one expression.
Now, if we agree to _mark_ m and m' as eliminated, and to read the two
expressions together, as if they were written in one, the two Premisses
will then exactly represent the _Conclusion_, and we need not write it
out separately.
Let us agree to mark the eliminated letters by _underscoring_ them,
putting a _single_ score under the _first_, and a _double_ one under the
_second_.
The two Premisses now become
xm_{0} + ym'_{0}
- =
which we read as "xy_{0}".
In copying out the Premisses for underscoring, it will be convenient to
_omit all subscripts_. As to the "0's" we may always _suppose_ them
written, and, as to the "1's", we are not concerned to know _which_
Terms are asserted to _exist_, except those which appear in the
_Complete_ Conclusion; and for _them_ it will be easy enough to refer to
the original list.
pg092
[I will now go through the process of solving, by this method,
the example worked in § 2.
The Data are
1 2 3 4
k_{1}l'_{0} + dh'_{0} + a_{1}c_{0} + b_{1}e'_{0} +
5 6 7
k'h_{0} + b'l_{0} + d'_{1}c'_{0}
The Reader should take a piece of paper, and write out this
solution for himself. The first line will consist of the above
Data; the second must be composed, bit by bit, according to the
following directions.
We begin by writing down the first Premiss, with its numeral
over it, but omitting the subscripts.
We have now to find a Premiss which can be combined with this,
_i.e._, a Premiss containing either k' or l. The first we find
is No. 5; and this we tack on, with a +.
To get the _Conclusion_ from these, k and k' must be eliminated,
and what remains must be taken as one expression. So we
_underscore_ them, putting a _single_ score under k, and a
_double_ one under k'. The result we read as l'h.
We must now find a Premiss containing either l or h'. Looking
along the row, we fix on No. 2, and tack it on.
Now these 3 Nullities are really equivalent to (l'h + dh'), in
which h and h' must be eliminated, and what remains taken as one
expression. So we _underscore_ them. The result reads as l'd.
We now want a Premiss containing l or d'. No. 6 will do.
These 4 Nullities are really equivalent to (l'd + b'l). So we
underscore l' and l. The result reads as db'.
We now want a Premiss containing d' or b. No. 4 will do.
Here we underscore b' and b. The result reads as de'.
We now want a Premiss containing d' or e. No. 7 will do.
Here we underscore d and d'. The result reads as c'e'.
We now want a Premiss containing c or e. No. 3 will do--in fact
_must_ do, as it is the only one left.
Here we underscore c' and c; and, as the whole thing now reads
as e'a, we tack on e'a_{0} as the _Conclusion_, with a ¶.
We now look along the row of Data, to see whether e' or a has
been given as _existent_. We find that a has been so given in
No. 3. So we add this fact to the Conclusion, which now stands
as ¶ e'a_{0} + a_{1}, _i.e._ ¶ a_{1}e'_{0}; i.e. "All a are e."
If the Reader has faithfully obeyed the above directions, his
written solution will now stand as follows:--
1 2 3 4
k_{1}l'_{0} + dh'_{0} + a_{1}c_{0} + b_{1}e'_{0} +
5 6 7
k'h_{0} + b'l_{0} + d'_{1}c'_{0}
1 5 2 6 4 7 3
kl' + k'h + dh' + b'l + be' + d'c' + ac
-- = - -= - = = = - =
¶ e'a_{0} + a_{1}
_i.e._ ¶ a_{1}e'_{0};
_i.e._ "All a are e."
pg093
The Reader should now take a second piece of paper, and copy the
Data only, and try to work out the solution for himself,
b
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