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ELEMENTARY LOOIO
UNIVERSITY AND CERTIFICATE
STUDENTS.
(An Enlarged and Revised Edition of " Logic and Education.'^ )
AN ELEMENTARY TEXT-BOOK OF
DEDUCTIVE AND INDUCTIVE LOGIC.
KEY. JOHN LIGHTFOOT, M.A., D.So.
( Vicar of Cross Stone.)
AUIHOB OF " STUDIES IN PHILOSOPHY," " TEXT-BOOK OF THE XHIUTY-NINE
ARTICLES," ETC.
^ OF THE
UNIVERSITY
OF
IFORfiJ
Rauph. Holland & Co
Temple Chambers. E.G.
1905.
[all bights reserved.]
^WHTC
FiEST Edition, November, 1899.
New and Enlaeged Edition, May, 1902.
Repbinted, July, 1902.
Reprinted, September, 1904,
Reprinted, October, 1905.
^ PREFACE. -5^
This book is intended for students commencing the study
of Logic. It is an elementary exposition of the subject on
traditional lines. Modern logicians have added many retine-
ments to the analysis which has been generally received smce
the days of Aristotle. But the more subtle questions of
criticism can only be appreciated intelligently after the
elements of the subject have been mastered. A brief account
of the more important reforms in logical doctrine is given in
the Appendix now added. The treatment will be found
sufficiently full for the preliminary examinations at the
Universities, and for the purposes of students preparing for
any of the various examinations for a Teacher's Certificate.
In the present edition the exposition of Inductive Logic has
6een considerably extended, and will now be found ample for
all elementary purposes.
J. LIGHTFOOT.
Cross Stone Vicarage.
189155
AUTHOR'S NOTE.
students making their first acquaintance with the subject are
advised to adopt the following plan on their first reading of the book :—
Omit Chapter L Read Chapter II., omitting the section "Divisions
of the Subject." Omit Chapter III. Chapter IV. should be thoroughly
mastered. Read Chapter V., omitting the " Predicables." Omit
Chapter VI. Chapter VII. is very important, and the exercises on
page 50 should be carefully worked. Read Chapter VIII., but " Obver-
sion" and "Contraposition" maybe omitted on first reading. Chap-
ters IX. and X. are very important. Chapters XI. and XII. may be
neglected on first reading. Chapter XIII. is important (to bottom of
page 84). Chapter XTV. is easy and interesting. Pay special attention
to the "petitio principii" and "ignoratio elenchi" (pages 92-94).
Chapter XV. to end of book is very important. The Appendix may
be omitted on first reading.
On second reading no part of the book should be neglected. Notice
that the exercises contain only few questions involving mere repro-
duction of the text. It has been assumed that students can and will
construct such questions for themselves.
OF THE
IVERSITY
V OF
CONTENTS.
PAGB.
Chapter I. — The Eelation of Logic to other
Branches of Philosophy . . 7
Chapter II. — Definition of Logic and Divisions of the
Subject 12
Chapter III. — The Axioms of Logic .... 19
Chapter IV. — Terms : Their Definition and Classifica-
tion 22
Chapter V. — The Denotation and Connotation of
Terms 27
Chapter VI. — Definition and Division of Terms . . 34
Chapter VII. — Propositions 41
Chapter VIII. — Immediate Inference .... 51
Chapter IX. — Mediate Inference — The Syllogism . 61
Chapter X. — The Figures of the Syllogism . . 69
Chapter XI. — The Keduction of Syllogisms . . 74
Chapter XII. — Irregular and Compound Syllogisms . 79
Chapter XIII. — Conditional Syllogisms .... 82
Chapter XIV. — Fallacies of Deduction .... 89
Chapter XV. — Inductive Logic 97
Chapter XVI. — The Preliminaries to Induction . . 101
Chapter. XVIL— The Inductive Canons . . . .104
Chapter XVIII. — Arguments Similar to Induction . • 114
Appendix .....••••• 119
Index 124
CHAPTEB I.
The Relation of Logic to other
Branches of Philosophy.
Logic is usually considered the proper introduction to the
study of Philosophy. It is well, therefore, to get a preliminary
view of the subjects which are included under the general
term " Philosophy."
There are certain questions which must always be of great
importance to those whose profession it is to train the minds of
children. For instance, the question "What am J?" is
obviously as important as the question " What is the sun ? "
Now, there are many similar questions to " What am I ? "
that will suggest themselves as of great importance in this
respect ; e.g. : —
How did I become what I am ? What are the fixed
rules, or laws, which govern the development of mind in
man? To what laws must all my conscious thinking
conform, so that error and self-contradiction may be
avoided ?
Philosophy is the name given to that branch of study
which attempts to give an answer to these and similar
questions.
A course in Philosophy is usually divided into three
sections : — Logic, psychology, metaphysics ; and the object of
8 THE DIVISIONS OF PHILOSOPHY.
these three departments may be thus briefly stated : —
1. Logic. — Here we investigate the laws to which all
our " thinking " must conform, in order that we may avoid
error and self-contradiction in our thinking process.
2. Psychology. — Here is considered our power of think-
ing, its growth, and the laws by which "mind" in the
individual is governed.
3. Metaphysics. — Under this name we discuss a number
of very difficult and speculative questions about the
ultimate grounds of our beliefs and opinions.
This very brief statement must, for the sake of clearness,
be considered more in detail.
Philosophy, we have said, exists in three forms, and first
we have : —
I. PHILOSOPHY IN THE FORM OF LOGIC.
Knowledge in its simplest form and from our earliest
days, comes to us through sensation. But this elementary
knowledge is from the first extended by reflection {i.e., by
thinking) and by reasoning. The earliest efforts of a teacher
are devoted to making the scholar reason correctly. And
this effort is continued throughout the pupil's school-hfe.
Evidently, then, it is of first importance that all who teach
should themselves clearly recognise the laws to which
" correct reasoning " must conform.
Putting the matter in its briefest form, we may say that
" correct reasoning " must have two qualities, viz, :—
(a) Self-consistency ;
(b) Consistency with known facts.
If our "reasoning" is wanting in either of these qualities,
ib is incorrect, or fallacious.
In our efforts, then, to extend and to improve our know-
ledge. Logic comes to our aid by showing how fallacies of
THE DIVISIONS OP PHILOSOPHY. 9
thought and fallacies of expression may be avoided. Logic
is thus : —
A systematised body of tests and rules, by the aid
of which we may determine whether
(a) Our thinking is correct thinking (i.e., in accordance
with the laws of thought) ; and
(6) Whether our thinking is in agreement with factSf
and with the known laws of nature.
Logic and Rhetoric both aim at the formation of conclusions.
The Logician seeks to convince that the conclusion must be ;
The Rhetorician seeks to persuade that the conclusion ought to be.
II. PHILOSOPHY IN THE FORM OF PSYCHOLOGY.
The simplest reflection suggests to us that all our knowing,
reasoning and believing presupposes that we have a " mind "
which knows, reasons and believes. What the " mind " really
is we do not know. But we do know how the mind manifests
itself. If we do not know the mind itself, we know its
phenomena. Sensation, knowledge, memory, imagination,
reasoning, all these are phenomena of the mind. All these,
too, have their laws of growth a«d development, and Psycho-
logy is the orderly investigation of the phenomena of mind and
the laws by which they are governed.
Hereafter it will be seen that all man's conscious thinking
manifests itself in mental judgments. Logic investigates the laws
which these judgments, when formed, must obey. Thus we see the
relation between Logic and Psychology. Psychology investigates the
process by which the mind forms judgments. Logic studies the
result, i.e., the judgments when formed.
III. PHILOSOPHY IN THE FORM OF METAPHYSICS.
We have remarked that we do not know what '* mind "
really is. But all the same, men have felt themselves bound
to hold some theory about it. So, too, we do not know
precisely what " matter" really is, but chemists and others have
10 ETHICS, OR MORAIi PHILOSOPHY.
a theory abont it. So too in every department of human study
there are certain things which have to be taken for granted ;
things which cannot be proved, but which we are compelled
to assume. Now, Philosophy in the form of Metaphysics
attempts to give some account of these subjects. It attempts
to explain the ultimate nature of mind and matter, and to give
a reasoned account of those truths which ordinary science
takes for granted.
The word "Metaphysio" suggests the subjects with which its
study is concerned. The word means that which is " after," or
beyond physical or ordinary scientific investigation.
It only remains in this general sketch of the province of
Philosophy that we should see precisely the place of Moral
Philosophy, or Ethics. Ethical study is the considera-
tion of a certain definite set of facts and opinions
which regulate the behaviour of men as individuals, and as
members of the community. It is because of the great
practical importance of these that they are reserved for
special and separate treatment. In Ethics, or Moral Philo-
sophy, we learn how the knowledge of moral distinctions
(i.e., of right and wrong) are obtained, and we investi-
gate the laws which govern the moral life. But it must be
observed that when we are studying the growth of the know-
ledge of moral distinctions we are really engaged in the study
of Ethical Psychology. So, too, when we further consider
questions like the ultimate nature and destiny of the soul, and
its relation to the Supreme Being, we are then in the province
of Ethical Metaphysics.
Having now got a general view of the whole province of
Philosophy, the student will appreciate the reason why Logic
is always prescribed as the proper introduction to this depart-
ment of study ; and, further, it will be clear that no course of
study in the theory and practice of education can be considered
at all complete which has not embraced the treatment of
EXERCISES. 11
Elementary Logic. At the Universities this subject always
finds a place in the general schemes of instruction, and the
tendency is to make it a compulsory one for all degree
examinations. The student who has mastered the treatment
in this work will be sufficiently well prepared for the pre-
liminary and intermediate examinations at the Universities,
as well as for the questions set by the Board of Education in
its certificate examinations.
EXERCISES ON CHAPTER I.
•2. Should Logic precede Psychology as a subject of study ? Give
reasons for your answer.
2. State briefly the fundamental relation in which Logic standi
to Psychology and Metaphysics,
CHAPTEE II.
Definition of Logic and Divisions of
the Subject.
LoQio is usually defined as " The Science of Reasoning or
Inference." This definition, though suf&cient for general
purposes, is not sufficiently precise. The object of Logic is to
unfold to us the ideal, or perfect conditions to which aU our
thinking must conform in order to be correct thinking. Its
object is to show us how we must arrange the matter about
which we think, in order that our thought shall be coherent
and non- contradictory. A better definition, therefore, will
be:—
Logic is the science of the laws which regulate valid
thought.
It is of the utmost importance to fix clearly the meaning attached
to each word in this definition.
(a) A Science is a systematised body of knowledge about some
particular subject-matter.
Q>) A Law is a statement of a general truth, i.e., a truth which
holds good generally in human experience,
(c) The word "Thought" is used both for the process of
thinking, and for the product of thinking.
N.B.— Knowledge of isolated facts is not Science ; nor ia a truth
which holds good only in certain instances a " Law."
In Logic we consider every simple complete thought as an
assertion or a denial.
DEFINITION OF IiOOIC. 18
Every assertion or denial we call a ** judgment."
Every judgment when expressed in words we call a
« proposition."
Every assertion or denial which it is possible to make is of
such a character that when it is made certain other assertions
or denials follow from it as a necessary consequence. These
latter are called " inferences."
An inference is thus a judgment which follows as a neces-
sary consequence from some previous assertion, and one
which the mind is obliged to make on pain of self-contradiction.
If I assert the general fact that " All men are mortal," I am
obliged, on pain of self-contradiction, to infer that this or that
particular man is mortal.
A child has had its attention called to the fact that a piece
of cork thrown into the water always floats, and is also taught
how to recognise a piece of cork. The child is now on any
occasion able to make two assertions : —
(a) Cork always floats ;
(6) This is a piece of cork ; •
and from these two assertions, the further one that we call an
" inference " follows, and must follow, vi^;., this piece of cork
thrown into the water wiU float.
From our earliest days, our conscious life is largely occupied
in drawing inferences. This will appear if we reflect how in
teaching and in ordinary conversation we are constantly using
such words as therefore, for, because, since, etc. Every
such word marks the drawing of some inference. Education
at school, and experience in after life is really little else than
the development of the inferential connections between pro-
positions.
Now this work of drawing inferences has certain definite
rules and laws which must be observed. And it is by these
laws that all our inferences may be tested. If the laws have
been broken then the inference drawn is an invahd one. An
14 LOGIC, A SCIBNCE OR ABT.
invalid inference we call a fallacy. No one knows better than a
teacher how prone children are to draw wrong inferences. And
indeed aU through life men are liable to draw inferences which
are fallacious, the main object of Logic being to show how these
wrong conclusions may be avoided. It does this by educating
man's power of distinguishing the consistent and the conclusive
from that which is inconsistent and inconclusive.
— The principles of Logic find their application in every walk
of life, and in every branch of science. This is recognised by
the names given to the various sciences. Thus in the name
*' Geology " the last four letters are only another form of
the word " logic," and the term " geology " means " Logic
applied to explain the crust of the earth." So, too,
theology means "Logic applied to explain Divine matters,"
and so on. Since then the rules of Logic find their application
in the processes of every special science, Logic has been very
properly called the Science of Sciences.
Some writers have considered it needful to discuss at great length
whether Logic should be called a Science or an Art. As a matter of
fact it may be considered as either or both. Science is sound know-
ledge, an Art is the instrument by which science works. In studying
a science we are gathering knowledge, in learning an art we are pre-
paring to do something.
Logic is a Science in so far as it unfolds the conditions of
valid thought.
Logic is an Art in so far as it devises rules for enabling men
to apply their thought to things consistently and coherently.
Of course it is not implied that a man is unable to think or
reason correctly unless he has learnt Logic. Plenty of people
speak correct English who have never learnt the rules of
Grammar. From our earliest childhood we have been
accustomed to draw conclusions, and no doubt we have
generally obeyed the laws of Logic in doing so, without being
in the least aware what those laws were. Li such cases we
were thinking logically without being conscious of the logical
DIVISIONS OF LOGIC AND PSYCHOLOGY.
il5
principles which our thinking exemplified. On the other hand,
a course of logical study must bring into prominence the laws
which constitute valid, consistent thought. The student who
has patiently worked through a course of Logic, is much more
likely hereafter to think consistently and coherently than one
who is ignorant of Logic. And this will be found to be
especially true in those cases where, owing to the complexity
of thought and the ambiguity of language, there is serious
danger of fallacy even to the cleverest intellects.
DIVISIONS OF THE SUBJECT.
There are two main divisions of Logic, viz. : —
(a) DeductiYe and (6) Inductive Logic.
We must get a preliminary view of the scope of these : —
(a) Deductive Logic (sometimes called pure or Formal
Logic). This division of the subject is the orderly, scientific
unfolding of those forms and conditions to which our " think-
ing " must conform in order to be valid thinking.
Our " thinking " manifests itself in what for convenience
may be called three stages, not that as a matter of fact they
are separate and distinct. Each of these ideal stages has its
peculiar product or result : —
STAGE 1.
PROCESS.
Formation of ideas of con-
cepts of things.
STAGE 2.
Forming judgments about
these ideas— i.e., making men-
tal assertions or denials about
them.
STAGE 3.
Drawing mental inferences Signifying this mental pro-
from these judgments. cess by the " Syllogism.
stage 1, is often described as " simple apprehension," and stands
for the action of the mind in being aware of anything, having an idea,
RESULT.
Signifying these concepts or
ideas by ^* Names."
Expressing these judgments
in •* Propositions."
16 INDUCTIVE LOGIC.
or concept of it. The psychological analysis of " simple apprehen-
sion," however, shows it to be a complex and not a simple process.
N.B.— Simple apprehension, Judgment and reasoning (inference)
are the psychological processes. Names (terms), propositionB and
syllogisms are the corresponding results which are the subject
matter of Logic.
Consequently Deductive Logic has three sub-divisions
answering to these three stages.
The laws of thought : —
1. Concerning our Concepts of things ;
2. Concerning our Judgment of things ;
3. Concerning oiur Reasoning about things.
And since our thinldng is liable to error, we shall require to
supplement the above by an account of the Fallacies to which
we are liable if the laws of pure logic are violated.
(b) Inductive Logic (sometimes called Applied or Mixed
Logic). The scope of this second main division is best seen
from an account of its sub-divisions.
1. Definition. — In all our thinking about nature we
are obliged to use language. But language, at the best, is
only an imperfect instrument for expressing all that is in
the thinker's mind. The Logical doctrine of definition
aims at improving the relations of language to thought,
and especially to man's thought about things.
2. Inductive and Analogical Proof. — This section
first investigates the meaning and value of the central
presupposition of all science, viz : " The uniformity of
nature." It reveals the fact that nature is not a chaos, but
an orderly coherent system of cause and effect. Next it
deals with the obvious fact that untrained minds are liable
to confuse mere accidental coincidence with true conse-
quence and real scientific connection. To guard against
this, Logic provides certain canons or rules, and by these
unfolds the standard of scientific proof. It thus enables
men to distinguish between evidence which is properly
LOGIC AND GBAMMAR. 17
{a) Inductive and therefore reliable, and evidence which is
only (b) Analogical or probable. This section is thus
really the logic of all the physical sciences.
3. Historical Proof. — If the language of nature is
liable to be misunderstood, much more so is the spoken
and written language of men. Therefore Logic has certain
rules to lay down respecting human testimony which form
the criteria of Historical Proof.
These divisions and sub-divisions of the whole domain of
Logic may be shown thus : —
Logic.
(The Science of the laws which regulate valid thought.)
Deductive Logic. Inductive Logic.
I
Thought as Thought as Thought as Definition Testimony
Concept. Judgment. Inference. and
Authority.
Proof :
(a) Inductive.
(6) Analogical.
Note that Logic is concerned primarily with Thought,
Grammar with Language. Logic considers Language only as
the instrument of Thought. The " Parts of Speech " which
are the main feature in the grammatical analysis of language
are not recognised by Logic. Only those words which can
properly express a concept are within its range. Words that
can express a concept are grouped together as Terms, and it is
quite immaterial whether, in grammatical language, they are
nouns, pronouns, adjectives, or verbs. In Grammar and Logic
the simplest expression of a complete thought is a Simple
Sentence. But the logical analysis of a sentence differs from
the grammatical. The predicate of a logical sentence is
always the complete assertion made of the subject.
18 EXEBCISBB.
EXEECISES ON CHAPTEE II.
1. Define " Science " and " Art." Discuss the question whether
Logic is a Science or an Art.
2. What is Logic ? What are the chief uses of its study ? Why
should teachers especially make a study of it ?
3. Many people think quite correctly who have never studied
Logic ; why, then, waste time in studying it ?
4. Logic has been defined as *' the science of the laws of thought,"
In this definition what is meant by the terms " Science,'' " Law,"
and " Thought " ?
6. Discuss the relation of Logic to Grammar and Rhetoric.
6. What practical valv,e may be attributed to Logic (a) in the
detection of error, (b) in tlie discovery of truth ?
7. Explain the logical words term, proposition and syllogism,
and give the psychological words for the corresponding mental act
of each.
CHAPTER III.
The Axioms of Logic.
Logic has thus far been shown to be the practical science
which unfolds to us the ideal of self-consistent thought. In its
later sections it supplies the student with certain canons or
rules for applying our thought to things. Now, there are
certain principles or axioms which form the essence of self-
consistency. "When these are drawn out they will appear to
the student as self-evident truths. All the same they require
consideration, and after we have examined them they must be
regarded as axioms.
Speaking quite generally we may say that these several
axioms imply one general truth, viz. : —
" Thought which is evidently self -contradictory is im-
possible.*'
Every description of fallacy is really a thought that is self-
contradictory. But we may so express ourselves that the
contradiction is not obvious to those to whom we are speaking,
nor even to ourselves. Thus, in such an argument as the
following : —
'* He who is most hungry eats most ;
He who eats least is most hungry :
Therefore, he who eats least eats most,"
we feel there is contradiction somewhere, but it would require
some consideration to point out where the contradiction
really lay.
It might be better to say that a fallacy is really the absence of
thought, i.e., of logical thought ; the absence usually being concealed
under a veil of words.
20 THE LAWS OF THOUGHT.
The general truth stated above — viz, " thought which is
evidently self- contradictory is impossible" — ^is expressed by
logicians in three different ways. These three different forms
are the three fundamental laws of which every valid thought
is the exemplification. In other words, they are the essence
of " self-consistent thought."
The three laws are known as : —
1. The Law of Identity.
2. The Law of Non-contradiction.
3. The Law of the Excluded Middle.
1. The Law of Identity. — This principle asserts that
if any proposition is true, then any other proposition which
is either {a) identical with it, or (6) logically included in it^
must also be true.
A is {i.e., must be) A.
Whatever is, is.
Everything is what it is.
If every A is B, then this A is B.
2. The Law of Non-contradiction. — This principle
asserts the necessary logical disagreement of assertions
with their contradictory denials. In other words, two
contradictory assertions cannot both be true.
A is not non-A.
Nothing can both be and not be.
The same attribute cannot be at the same time affirmed
and denied of the same subject.
3. The Law of the Excluded Middle. — This principle
asserts that of two downright contradictory statements,
either the one or the other must be true ; no third or inter-
mediate assertion is possible.
A either is or is not B.
Every assertion must be true or not true.
Besides these three generally accepted principles, there is
another one which is less universally adopted. It is known as
EXBRCISES. 31
The Law of Sufficient Eeason, and the law may be expressed
thus : — Nothing happens without a reason why it should he so,
rather than otherwise. For instance: — If two forces are in
exact equilibrium, there is no reason why the body on which
they act should move in the direction of either force. If a body
is acted on by two unequal forces in exactly opposite directions,
it will move in the direction of the greater force. If the
"reason why" is granted you must allow the consequence.
The student will notice that these laws are self-evident
truths, they cannot be proved by reference to any thing simpler.
They neither require proof nor are' capable of it.
Until the student has made some progress in the study of
Logic, he is liable to confuse opposite terms with contradictory
ones. Thus in considering the law of the excluded middle, he
might fancy it possible to make intermediate assertions which
the law says are impossible. Thus, if we say " every substance
is either hard or not hard," the reply might be made that some
substances are neither hard nor soft, but of medium quality.
But Logic has nothing to do with degrees of hardness. It
assumes that the word " hard " has a definite meaning, and all
things which do not exactly agree with this meaning are " not
hard." Even concerning things of which "hardness " could not
properly be predicted, e.g., heat, colour, taste, etc., it is still
possible to say " heat is either hard or not hard."
EXERCISES ON CHAPTER III.
i. " Things luhich are equal to the same thing are equal to each
ether.'* Show that this is only another form of the Law of Identity.
2. What are the laws of thought ? State clearly what you
understand by a law of tJwught.
3. Euclid in comparing things shows that they are either greater,
equal to, or less than each other. How would these three alternatives
be expressed in Logic t
CHAPTER IV.
Terms : their Definition and
Classification.
The simplest and most elementary manifestation of thought
is an assertion or denial. Every assertion or denial that we
can make is called in Logic a judgment. Now in every
assertion or denial we affirm or deny something of something
else.
When the judgment is expressed in words we call it a
proposition. Every proposition must therefore contain two
names : —
(a) The name of the thing about which the assertion is
made ; (6) The assertion itself, e.g., —
(a) Gold (b) is a metal.
These two names are called the " terms " of the proposition^
because they are the boundaries (terminals) of the proposition.
Definition of a Term. — A Term is a word or a combina-
tion of words, which can properly stand as the subject or
predicate of a proposition.
Every proposition must of course have a subject and a predicate
either expressed or implied. Even in an exclamation such as "Fire 1 "
the word is the predicate of a proposition the subject of which is
implied, thus: — " This house (etc.) is on fire."
From the definition given it follows that all words may
be divided into two classes : —
{a) Those which can be used as Terms
(6) Those which cannot.
CLASSIFICATION OP TERMS. 23
Words belonging to the former class are called categorematic.
Those belonging to the latter class are called syncategorematic.
A categorematic word is one which can by itself be used
as a term, i.e., which can stand alone as the subject op
predicate of a proposition. (In grammar such words are
distinguished as nouns, pronouns, adjective, participles,
but in Logic they form one class.)
A syncategorematic word is one which cannot by itself
be used as a term, but only in combination with one op
more other words. (Adverbs, prepositions, conjunctions
and interjections.)
The student must carefully avoid speaking of synoategorematio
terms. The contradiction is obvious.
Terms (categorematic words) are classified in five groups as
follows : —
1. The Common term (or, as it is often called, the
General term), as contrasted with the Singular (or
Proper) ; and the Collective term.
2. Concrete and Abstract terms.
3. Positive and Negative terms.
4. Connotative and Non-connotative terms.
5. Absolute and Relative terms.
1. (a) The Common or General Term. — To logicians
this is by far the most important of all. It is a term
which can be affirmed or denied in the same sense of
more things than one : as book, dog, man.
(6) The Singular or Proper Term is one which can
be affirmed, in the same sense, of only one single thing.
N.B. — A common term may of course be transformed
into a singular term by means of some individualising
prefix. Thus *' man " is a common term, but " the first
man " is a singular term.
24 CliASSIFICATION OF TERMS.
(c) The Collective Term is one which can be afi&rmed
or denied of two or more things taken together, but which
cannot, hke a common term, be affirmed or denied of each
one of these when taken separately : as armyj flock,
library, etc.
Notice that in words like " library " the sense in which the word
is being used must be taken into account. Library, a collection of
books, is a collective term. Library {i.e., any library) is a common term.
2. (a) A Concrete Term is the name of an object; it
stands for some individual thing, or a collection of indivi-
dual things.
(6) An Abstract Term represents an attribute or attri-
butes, considered apart from the individual object of
which it may be the attribute. Thus " man " is concrete,
"humanity" is abstract; "living being" is concrete,
" life " is abstract ; " generous" is concrete, " generosity "
is abstract.
3. (a) A Positive Term implies the presence of some
attribute or group of attributes.
(fc) A Negative Term implies the absence of the attri-
butes included in the corresponding positive term. Thus
metallic, compound, light, are examples of positive terms;
of which the corresponding negative terms are non-
metallic, element, darkness.
4. {a) A Connotative Term is one which represents an
individual thing, or group of individual things, together
with one or more of their attributes. " Animal " is a
connotative term, as it implies the attribute "animality";
so also is " mountain," which implies the attributes
" height," etc.
(b) A Non-Connotative Term signifies an individual
thing only, and does not imply any attribute. Thus :
Whiteness, London, are examples of non-connotative
terms.
CLASSIFICATION OF TERMS. 25
The student must carefully consider this distinction of terms.
Think, for example, why " mountain " is called a connotative term,
but Snowdon, the name of a particular mountain is not. Now the
name " Snowdon " might suggest to anyone with sufficient geo-
graphical knowledge, all the attributes implied in the term
" mountain." But a word is not connotative because it may suggest
facts or attributes which are otherwise known, but only when it
actually implies them. Many logicians have overlooked this, and
have considered proper names connotative. In answering a question
in an examination it would be wise to give your reason for consider-
ing a proper name as " non-connotative."
5. (a) An Absolute Term is a name which is complete in
itself, i.e., which in its meaning implies no reference to
anything else ; as gas, sound, tree, etc.
(6) A Relativb Term is a name which not only denotes
some object, but also implies in its signification the
existence of some other object called the correlative.
Thus when we use the term friend or father for some man,
we imply the existence of some other person or persons to
which the man stands in the relation of friendship or
fatherhood.
These definitions of the various Tcinds of terms must he
thoroughly understood, and the student must be well exercised
in the classification of terms. When the appended examples
are attempted it will be foimd a more difficult task than might
be supposed. The main difficulty will be found in deciding
whether an abstract term is general or singular. Some logicians
argue that all abstract names are singular. Thus the adjective
" red " is the name of red objects, but it implies the possession
by them of the quality " redness," and this quality has one
single meaning. It is much simpler, however, to consider
some abstracts general on the ground that they are names of
attributes of which there are various kinds or subdivisions; e.g.,
the word colour which is a name common to whiteness, red-
ness, ^etc, or the term whiteness in respect of the various
shades of whiteness to which it is applied in common. But
26 EXERCISES.
just because the point is a disputed one, you should give your
reason for classifying abstract terms as general or singular.
A further difficulty arises in dealing with terms that are
equivocal, i.e., capable of being used in several senses. Indeed,
some writers make a further classification of terms, as Univocai*
(terms which can only suggest one meaning) and Equivocal
or Ambiguous (terms which may have two or more meanings).
An equivocal term is really two or more terms with identical
spelling, and should be so treated. Thus the term "force " is
.equivocal, as it might mean an army or that which causes
motion, etc., and each meaning demands a distinct classifica-
tion of the word. It is better, therefore, to say at once if a
term is equivocal or univocal, and then proceed.
EXEKCISES ON CHAPTEE IV.
J. Discuss the grammatical parts of speech from a logical point
of view.
2. May terms he classified as categorematic and syncategore-
matic ? Give reasons for your answer,
3. Describe a ''collective term.'' Illustrate the difficulty of
distinguishing these from general or abstract tei-ms.
4. Classify the following terms : donkey, reagent, red, redness ^
London, sugar. Mikado of Japan, intensity, also, vexation, blind,
emotion, darkness, foot, Westminster Abbey, uncle.
5. Point out the ambiguity, if any, of tlie following terms : vice^
hydrogen, peer, paper, sense, minister, tea-cup, interest.
6. Distinguish between the meaning of the terms abstract a7id
concrete, and show the applicability of these terms (1) to parts of
speech, and (2) to arithmetic. Say what is the use of the distinction.
CHAPTER V.
The Denotation and Connotation
of Terms.
If the question were asked "What is an animal?" we can
imagine two forms of answer being given : (a) an exact
definition of the term; (6) an enumeration of the various
classes of animals. The first answer might be expressed
thus: — "An animal is a sentient, organised being." This
definition tells us what must be the attributes of anything
in the universe to which the name " animal " can be rightly
applied. Such a definition is said to mark the connotation of
the term. On the other hand the latter definition which
proceeds to enumerate all the different classes of animals is said
to mark the denotation of the term.
A Term, therefore, in Logic is considered to discharge a
double function: —
1. Connoting the attributes of .things.
2. Denoting individual things.
Notice that a term is a word which signifies a mental idea
or concept. But in Logic we do not speak of the connotation
or denotation of a concept. When speaking of concepts we
use the words intension and extension.*
The intension of a concept corresponds to the connota-
tion of the term signifying the concept.
The extension of a concept corresponds to the denotation
of its related term.
The connotation of a term (or the intension of its corre-
sponding concept) signifies the attributes implied in the
meaning of the term.
* Some writers, however, speak of the intension and extension of terms,
and even the denotation and connotation of concepts.
88 RELATION OP CONKOTATION AND DENOTATION.
The denotation of a term (or the extension of its corre-
sponding concept) signifies the number of individual things
to which the term is applicable in the same sense.
The student is invited to reflect upon these definitions. It will
then be seen that an important logical truth is involved. Every
common term like man, bird, etc., stands for a number of individual
things (different individual men or birds), and a quantity of
attributes (rational being, feathered biped, etc.) Thought as
expressed by "terms" is thus a kind of quantity, and all our
affirmations and assertions about terms are really a comparison of
quantities. If I say "men are animals," I mean that "men "are a
quantity of things contained in a greater quantity of things called
"animals."
Now the two particular kinds of quantity we are consider-
ing (connotation and denotation) have a mutual relation. For
a moment's reflection will show that the wider or greater the
denotation of a term becomes, the narrower or smaller must be
its connotation. Thus compare the two terms "animal" and
"man." The term animal embraces far more individual
things under it than the term man, therefore its denotation is
greater. But the term man implies a larger number of
attributes than the term animal. For everything that you can
say of animal you must say of man, but you also say of man
certain things which you cannot say of all animals. Therefore
the connotation of the term man is greater than that of the
term animal.
As a fairly correct general rule it may be said that as
the denotation of a term is increased, the connotation is
diminished, and vice versa.
In other words the greater the number of individual things
included under a common term, the fewer will be the number
of attributes which can be predicated of the whole of them.
This is expressed by the Logical rule that the connotation
and denotation of a term (or the intensive and extensive
quantity of a concept) are in inverse ratio. The greater
THE PRBDICABLES. 29
the denotation, applicability or extent of a term — the less
must be its connotation or comprehcnsiYe quantity. The
maximum of the one must in all cases be the minimum of
the other, and vice versa.*
Now observe when two common terms are so related
that the whole connotation of the one is included within the
greater connotation of the other — the term which has the
greater connotation is called the " Species," and the one which
has the smaller, or included connotation, is called the '< Genus."
Thus taking the two related terms " man " and " animal," the
term " man " implies all the attributes that the term "animal"
implies, as well as some further ones peculiar to itself.
" Man " has the larger connotation, therefore " man " is a
species of the genus " animal."
In the proposition " Man is an animal " we assert that
" man " the species is included in " animal " the genus. Every
affirmative proposition makes some such assertion respecting
the subject of the proposition. The following problem, there-
fore, arises : " Can the predicates of all propositions he
classified in relation to their subjects under certain definite
heads ? " Logic attempts this by the Doctrine of the Pre-
dicables.
The predicables, then, are a classification of all the possible
relations of the predicate to the subject of a logical pro-
position. The following is the usual form of this classification :
II. Genus
2. Species
3. Difierentia
crtjuiuames are eitner -\ 4. Proprium Y of the subject,
(property)
5. Accidens
(accident)
These five heads of the predicate require consideration.
* This doctrine of "Connotation and Denotation being in inverse ratio " is'
given in accordance with traditional logic. It is open to much criticism^
and is only a "fairly correct general rule."
30 GENUS AND SPECIES.
1. Genus is a common term, signifying a wider class which
is made up of other narrower classes, e.g., animal, triangle.
2. Species is the name given to the narrower classes,
included in a genus, e.^/., Vertebrates, Invertebrates; equilateral
triangle, etc.
Genus and species, then, are relative terms, and must be
considered together. A genus would be meaningless apart
from two or more species into which it is divided. A species
would be equally meaningless apart from the genus in which
it is contained.
The student will notice that the same term may be at the
same time a species of the next more general class, and a
genus to the less general classes included under it. Thus take
the term" triangle." Triangle is a species of the genus "recti-
linear figures," whilst at the same time it is a genus of the
different kinds of triangles : equilateral, isosceles, etc.
From this it follows that every term may be both a genus
and a species. But the technical language of Logic implies,
however, that this is not universally the case. It implies that
there is a genus which is not a species of any higher genus ;
and that there is a species which is not a genus to any lower
species. For Logic speaks of : —
1. The highest genus ;
2. Intermediate genera or species ;
3. The lowest species.
The highest or most general genus, i.e., which can have
none above it, is such a one as " Being." This is called "the
summum genus." The lowest relative species, which can have
none below it, is the name of any individual thing. This is
called " the infima species."
Any highest genus broken up into its component species,
and these component species in turn regarded as genera again
broken up into their component species, and the process
^ UNIVERSITY
OF
TREE OP PORPHYRY.
31
repeated until you cannot proceed further, {i.e., when an infima
species is reached) is called a " Predicamental Line." A process
such as this is illustrated by the ancient " Tree of Porphyry " :
Substance
(a summum genus).
Jorporeal.
1
Incorporeal.
Body.
Animate.
1
Inanimate.
Living Being.
Sensible.
1
Insensible.
Animal.
1
Rational.
1
Irrational,
Man.
Socrates.
Plato, and other individual men.
Here Substance is the Summum Genus and Man is the Infima
Species (i.e., man cannot be divided into any smaller species, but only
into individual men).
Each of the intermediate genera down the middle line (Body, Living
Being, Animal), is called a subaltern genus or species, and the nearest
genus to every term of which that term is itself a species, is called the
proximuni genus.
32 DIFFERENTIA.
3. Differentia. — It has already being seen that a species has
a larger connotation than its corresponding genus {i.e., the
species implies more attributes). Now take any term used as
a species and compare it with its next, or proximate, genus.
The excess of the connotation of the species over the conno-
tation of the genus is called the " Differentia " of the specieSr
Thus:—
Genus+Differentia= Species.
Referring to the Tree of Porphyry, "Living being" is a
species of the genus "body." " Animate " is the attribute
which forms the differentia of the species " living body,"
thus : —
Body + Animate = Living body
(genus) (differentia) (species).
4. Property (Proprium). — By property is meant any
attribute which is common to every individual in a given class ^
but which is not necessary for distinguishing that class. This
will be clear from the following illustration. Take the term
"triangle." A triangle is a figure bounded by three straight
lines. " Three-sided " is the differentia of a triangle. But
triangles have many other properties, e.g., " three-angled,"
" all their angles equal to two right angles," etc.
5. Accident (Accidens). — An accident is an attribute which
has no necessary connection with the term to which it belongs.
Thus the size of a triangle — i.e., big or little — is an accident.
Size does not at all affect what Euclid proves concerning
triangles.
Accidents are usually divided into
Separable accidents — e.g., how a man is dressed ;
Inseparable accidents — e.g., the colour of his hair.
SXEBCISES.
EXERCISES ON CHAPTER V.
1. Define differentia^ property, and inseparable accident, giving
examples. How far may these distinctions be interchanged.
2. To which of the predicables would you refer the predicates in
the following propositions, and why : —
(a) All men are animals.
(b) Mr. Gladstone was a great statesman.
(c) The three angles of a triangle are together equal to two
right angles.
(d) All ducks are web-footed.
(e) John ruled badly .
(f) Alkalies by their union with acids form salts.
3. Explain clearly the connotation and the denotation of a term.
What determines the connotation and denotation of terms ? Have all
terms a denotation and connotation ?
4. Arrange the following terms in their order of extension: —
Vertebrate, human, substance, child, organism, schoolboy.
5. Explain the terms intension and extension as applied to terms
in Logic, and distinguish gemis and species, illustrating your
explanation by the terms cart, eagle and man.
6. Distinguish between denotation and connotation, and show the
importance of the distinction in teaching.
7. Give the genus, tlie differentia, a proprium and an accident of
silver, Darwinian, square, house.
8. ^^ A generic term denotes a larger number of objects tlmn a
specific term ; but it connotes a smaller number of attributes."
Explain this statement and illustrate it by examples.
CHAPTER VI.
Definition and Division of Terms.
The definition of a term is the explicit statement of the
connotation of the term.
Since every definition of a term must take the form of a
proposition, it would be more convenient to have considered
the logical doctrine of definition when we are discussing
propositions. But it is usual to consider the subject at this
stage of our study.
In a definition that which is defined is always the subject
of a proposition. The predicate must declare with sufficient
precision what the subject means. In other words, the
predicate must show forth the attributes which separate the
subject in question from all other subjects.
All definitions are propositions, but all propositions are not de-
finitions. Only those propositions are definitions in which the
predicate so makes clear the attributes of the subject, as to separate
it from all other subjects with which it might be confounded.
The subject and predicate of a definition are, therefore,
exactly co-extensive. The difference between them is this : —
what was latent — wrapped up, as it were, in the subject — is
fully unfolded or analysed in the predicate. Logic asserts that
this result is achieved when the predicate of the defining
proposition exposes the proximate genus and the differentia of
RULES OF DEFINITION. 36
a term. For the genus implies all the attributes of the term
considered as a species of the genus; whilst the differentia
displays those attributes which distinguish the term as a
species. In Logic, then,
The definition of a term = proximate genus + differentia.
Notice that there are some terms which are incapable of logical
definition, e.g., a summum genus, all proper names, etc. The former
has no proximate genus, the latter have such a multiplicity of
attributes that we can only mention a number of them sufficient for
the practical purpose of recognition. This enumeration, however, is
" description " not definition.
The student must not confound logical definition with
"*' dictionary definition." In the latter all that is done is to
substitute one word for another, assumed to have a similar
connotation, on the ground that the new word is more familiar
or intelligible than the one for which it is substituted.
There are certain simple rules which Logic lays down to
which propositions must conform to entitle them to be regarded
as good logical definitions.
1. The definition must bring into view the essential, dis-
tinguishing attributes (differentia) of what is defined."^
2. The definition must be adequate, and applicable
exclusively to what is defined.
3. We must not define by negations.
i. The definition must be expressed in unambiguous,
intelligible language.
Defiinition is a most important subject. Avoid confusing
the definition of names with the definition of things. The
definition of a name is the settlement of what the name shall
be, by which a thing or a concept shaU be designated. Any
man is entitled to determine this as he pleases, so long as he
adheres consistently to the name he has connected with the
* Obviously, to merely name properties or accidents can never be a logical
■definition.
36 LOGICAL Division.
concept or thing. Sounds or signs on paper, are in themselves
indifferent to meaning. Each or any may be used to express
any meaning that has been agreed upon by those who use
the word. Definition of a thing, is not thus arbitrary. These
definitions depend on what is involved in the essential nature
of the thing defined. Men are apt to confound definitions oi
names with definitions of things, and to confuse both with that
full analysis of the attributes implied in our concepts which it
is the province of logical definition to bring into light. We are
frequently asked to accept definitions of names as if they were
the true definitions of things. Because we agree to employ
a certain sound to express some meaning, it does not follow
that the meaning so expressed corresponds to the essential
attributes of the things signified by the sign.
Logical Division. — Diyision is the analysis of the
denotation of a term.
It is always expressed in the form of a proposition, the
term divided being the subject, and the exposition being
the predicate.
There are other familiar kinds of division with which
logical division must not be confounded, e.g.,
{a) Partition, which is the act of dividing some physical
whole mto its constituent parts, e.g., ship=hull, mast,
sails, etc. ; man=head, trunk, limbs, etc.
(6) Distinction of ambiguous or equivocal terms, e.g.,.
Humanity =(1) human nature, or (2) the human race
collectively; Vice=(l) a moral fault, or (2) a mechanical
tool.
(c) Enumeration of individuals, e.g., naming aii ihe
books in a library.
Logical division expounds the denotation of a term not by
enumerating individuals. This would in most cases be im-
possible. No one could enumerate all the different men
RULES OF LOGICAL DIVISION. 87
included under the term " man." It proceeds by mentioning
only the smaller groups denoted by the term.
Collective and singular terms cannot be divided into smaller
groups, and, therefore, cannot be logically divided.
A collective term can be transformed into a common term, and
so become capable of logical division. Thus "the fourteenth
regiment" may be transformed into "soldiers of the fourteenth
regiment," and in this form may be divided into officers, privates, etc.
When we proceed to divide a term into terms expressive
of smaller groups, we seek some attribute which may be
predicated of certain members of the group, but which cannot
be predicated of the rest. This attribute is called the basis of
division {fundamentum divisionis). Of course, the same genus
may be variously divided by adopting different bases of
division. Thus in dividing the genus " triangles " we may
adopt the relative length of their sides as our basis, and so
divide triangles into equilateral, isosceles, and scalene. Or
we might adopt the size of their angles as the basis, and so
divide triangles into right-angled, acute-angled, and obtuse-
angled. But two or more bases of division must never be
confused together in the same division, or we fall into the
error called in Logic " Cross division." It would, e.g., be
cross division to divide triangles into isosceles, right-angled,
and scalene.
There are certain rules to which a logical division must
conform, viz. : —
1. Each act of division must have one and only one
basis of division, or cross division will ensue.
2. The division must be exhaustive, i.e., the dividing
members when taken together must be co-extensive with
the divided whole.
3. If the division is a continued one {i.e., embraces more
than one step), each step should, as far as possible, be a
\ CLASSIFICATION.
proximate one — in other words "proceed step by step.
e.g. :—
Figure.
Curvilinear. Eectilinear.
I
I I I
Triangle. Quadrilateral. Polygons.
I
Equilateral. Isosceles. Scalene.*
When we turn from the division of our concepts as
expressed in terms, and proceed to consider material things the
logical doctrine of division becomes a theory of logical scientific
CLASSIFICATION. The object of classification is to so arrange
the facts with which we may be dealing that we can acquire the
greatest command over them, and convey the greatest amount
of information about them in a few words.
Classification is really a branch of Inductive Logic. It is one of
the important processes subsidiary to the application of the inductive
canons. By its use we obtain a greater command over the knowledge
we possess, and are put in the right avenue for obtaining additional
information. It provides that our knowledge of things shall be so
arranged that the facts may be more easily remembered, and that we
may more readily perceive the laws by which they are governed.
♦There is a further method of division in which each step is a division into
corresponding positive and negative terms, e.g. : —
Figure.
Rectilinear. Non-rectilinear.
_J
Triangles, Non-triangles,
etc.
This is called division by Dichotomy. It is extremely cumbersome and of small
importance.
EXERCISES. 89
Logic considers all attempts at classification as either natural or
artificial. By a natural classification is meant the grouping of facta
in accordance with real natural distinctions. Thus an actual scientific
knowledge of facts is a pre-supposed requisite for a natural classifica-
tion. Different branches of science have different objects in view,
and accordingly they often adopt a special basis for classification.
The practical farmer divides plants into those which are useful, and
those which are weeds. Whilst the botanist adopts the division into
monocotyledons and dicotyledons as his basis. The student who h is
an elementary knowledge of Geology and Zoology will remember how
differently fossils are classified in the two Sciences.
An artificial classification selects some point of resemblance
amongst objects, and one which is easy to identify, and proceeds to
classify related objects upon this basis. The Linnaean system of
classification in Botany, which takes for its basis the number of
stamens and pistils in a flowering plant, is a good illustration of an
artificial system. In Zoology, where the primary basis of classification
is into vertebrates and invertebrates, we have an example of a natural
classification.
EXERCISES ON CHAPTER VI.
1. Criticise the following definitions: —
(a) Ignorance is a hlind guide.
(h) The cat is a domestic ayiimal.
( c) Enjoyment means pleasure.
(d) Tranquillity is the absence of unrest.
fe ) Alcohol is a kind of medicine.
2. Define the terms gold, coal, legal nuisance, civilization,
Cleopatra^ s Needle, bread, anger, Snowdon.
3. What do you understand by a perfect definition ; and what
jyrocesses of thought are employed in arriving at one ? Oive two or
three examples which err by being either too wide or too narrow.
4. What is the difference between (a) a description, (b) a defifMr
tion, (c) an explanation ?
5. Explain what is meant by logical division, and briefly state
its rules. Oive instances which observe, and instances which violate
the rules.
10 EXEBCIBEB.
6. Coymnent on the following as logical divisions : —
(a J Pens into quill pens and steel pens.
(b) Ireland into Ulster^ Munster, Leinster and Connaught.
(c) Animals into vertebrate and invertebrate.
(d) Colour into whiteness, blackness and blueness.
(e) Lights into artificial, blice and red lights and moonlight.
( f) Vice into an immoral act and a viechanical tool.
(g) Englishmen into rich and poor, consumptive and biliotts..
7. Show the relation between Definition, Division and Classifica-
tion.
CHAPTER VII.
Propositions.
Having completed our investigation of the logical doctrine of
"terms," we now proceed to consider the teaching of Logic
with regard to " propositions." Just as a " term " is the
outward expression for the inward (psychological) fact, which
is called a " concept," so a " proposition " is the translation
into language of the inward mental act, which is called " judg-
ment." Now, it has already been shown that a judgment is
the simplest and most elementary manifestation of a complete
thought. Every assertion or denial that we can frame in our
minds is a judgment. When this mental act is expressed in
language, we have what is called in Logic a proposition.
A proposition, therefore, may be defined as The verbal
expression of a truth or falsity, or A sentence making an
affirmation or denial.
Propositions which make simple assertions or denials,
without any condition attached, are called Categorical.
A Categorical Proposition is one which simply asserts or
denies some fact, e.g.,
All men are mortal.
No men are infallible.
Notice that in a categorical proposition we bring together
two terms, and connect them by the copula. For logical
purposes this copula is always the present tense of the verb
•" to 6e," with or without the negative particle " not.''
i9 PROPOSITIONS :
In ordinary language, of course, our categorical judgments
are expressed in various ways. But Logic considers that
3very simple assertion or denial can be expressed in one
general form, and, for logical purposes, the assertion or denial
must be reduced to this form. Hence the student must
become accustomed to expressing the ordinary forms of simple
assertions and denials in the precise form required by Logic.
There is no doubt that the logical form of an assertion will
often appear awkward and " wordy," compared with ordinary
conventional modes of expression, but the advantage gained
by the precise exposition of our assertions is of the highest
logical importance. Take as an illustration the assertion,
"John was the brother of Eichard." Li order to get the
present tense of the verb "to be " as the copula of this
sentence, it must be expressed in some such form as : — " John
Is a person who was the brother of Eichard." This trans-
formation sometimes causes a Uttle perplexity. Take, for
example, the following sentences : —
(1) The bell will toll to-morrow.
(2) None but the brave deserve the fair.
(3) It does not rain.
(4) Fire!
These ordinary conventional sentences, when transformed
into simple categorical propositions for logical purposes^
become —
Subject.
Copula
Predicate.
(1) The tolling of the bell
is
an event which will
happen to-morrow.
(2) No not-brave persons
are
deserving of the fair.
(3) Eain
is not
falling.
(4) This property
is
on fire.
CATEGORICAL AND CONDITIONAL. 48
Observe, that when a sentence is being thus transformed for
logical purposes, and divided into its logical elements (subject,
copula, predicate), if any one of the elements has been omitted in the
conventional form, it must be supplied in the precise logical form.
Thus the exclamation " Fire 1 " is sufficient, for practical purposes, to
convey definite information, but until its subject and copula have
been supplied, it is useless for logical purposes.
A categorical proposition, then, is one which makes an
unconditional assertion or denial. When the assertion is
expressed as a proposition displaying its logical elements, the
copula is in all cases the peremptory ** is " or " is not." But
many of the assertions or denials that we are making con-
stantly are of such a nature as to forbid the employment of
the unconditional " js " or "is not." To a large proportion of
our judgments some condition' or other is attached. Now,
Logic draws a sharp distinction between judgments which are
unconditional and those to which some condition is attached.
The former are categorical, the latter conditional. We shall
be chiefly concerned with categorical propositions, but it is
needful to mention the two kinds of conditional propositions
which are most common.
Conditional propositions are usually distinguished as Hypo-
thetical and Disjunctive.
1. Hypothetical ^propositions have a conjunctive condition
The following are examples : —
(a) If A is B, then also C is D.
(6) If Logic exercises the intellect, it ought to be-
studied.
(c) Where ignorance is bliss, 'tis folly to be wise.
Example (a) and similar examples, where symbols (A B, etc.) are
used, are called abstract examples ; (h) and (e) are called concrete
•zamplea.
44 PROPOSITIONS :
2. Disjunctive propositions have an alternative condi-
tion, e.g. : —
{a) A is either B or C.
(6) He is either a knave or a fool.
(c) All men are either good or bad.
Sometimes we find propositions conditioned, at once conjunctively
and disjunctively, e.g. : —
If A is B, then O is either D or E.
If a man becomes a soldier, then he must serve either at home
or abroad.
Besides this obvious division of propositions into categorical
(unconditional) and conditional, Logic further distinguishes
them by their quality and their quantity.
The quality of a proposition is determined by the copula.
The copula may be either " is " or " is not."
In the former case the proposition is affirmative, in the
latter it is negative.
A is B (affirmative) (1)
A is not B (negative) (2)
But we may also assert —
AU A is B,
or only, Some A is B.
The distinction of propositions, according as the affirmation
or denial is made of the whole or only a. part of the subject, is
what is meant by determining the quantity of a proposition.
Propositions, in which the assertion or denial is made of
the whole of the subject, are called universal propositions.
Propositions, in which only part of the subject is affected are
called PARTICULAR propositions.
Notice carefully, that in universal propositions, the subject
of the proposition is distributed^ i.e., taken in its full denota-
tion.
In particular propositions the subject of the proposition is
undistributed, i.e., the extent of its denotation is indefinite.
QUANTITY AND QUALITY. 46
Particular propositions are usually expressed in the form
Some A's are B.
Some A's are not B.
The word "some" is absolutely indefinite; it may mean
" few " or *' many," or indeed "all." In Logic it is the^
equivalent of " one at least."
The student should also carefully note that in universal
propositions the subject may be either : —
{a) An undivided, whole class, of every member of which
the predication is made, e.g,^ "Men are mortal"; i.e., All
men and every individual man ; or
(6) An indivisible individual, indicated by a proper
name ; e.g., " John is mortal."
Propositions, which have a proper name for their subject, are
sometimes called Singular Propositions. In most cases they may be
considered only a sub-class of Universals. But instances arise which
may cause perplexity. Thus : " John is sometimes eloquent," might
be considered as universal with a somewhat complex predicate.
(The student will have found, ere this, that in expressing proposi-
tions in logical form, the predicate is often very complex). The pra
position in its full logical form would be: " John is a speaker who is
sometimes eloquent." This is a true universal. On the other hand
the proposition might be rendered : " Some of John's speeches are
eloquent," in which case the subject is particular, not universal.
These various ways of dividing propositions may now be
collected, thus : —
Propositions are divided
1. On the basis of their quality into {a) affirmative^
(6) negative.
2. On the basis of their quantity into (a) universal,.
(6) particular.
The distinctions of quality and quantity are considered aS
applying only to categorical propositions. To some extent the
same distinctions can be applied to conditional propositions^
46 POUR PEOPOSITIONAL FORMS.
But to attempt- this would be quite beyond the scope of this
elementary treatise.
From this we gather that all categorical assertions or
denials may be grouped under four general forms. For, when
our assertions are expressed in logical form, we affirm that the
subject is, either
(1) In its whole logical extent, or
(2) In part of its logical extent,
contained under the logical extent of the predicate ; or, on the
other hand, the proposition excludes either
(3) The whole logical extent of its subject, or
(4) Part of the logical extent of its subject,
from the logical extent of its predicate.
This fourfold division answers to a combination of the
divisions of propositions on the two bases of quality and
quantity.
Every categorical proposition, true or false, that can be
made on any subject whatever must find its place under one
of the following heads : —
1. Universal affirmative, usually denoted by the symbol A.
2. Universal negative, ,, ,, ,, ,, E.
3. Particular affirmative, ,, ,, ,, „ I.
4. Particular negative, „ ,, ,, ,, 0.
The symbols A, E, I, O, are taken from the Latin words affirmo
and nego. A and I are the first two vowels of the former word, E and
O the vowels of the latter word.
The student should carefully consider the following simple
examples of the four forms of which, in each case, an abstract
example, a concrete example, and a diagrammatic illustration
are given. Notice the meaning of "is "in the prepositional
forms. " Is " means " is contained in " ; " is not " means " is
not contained in." " All X is Y " thus means " All X is
contained in Y."
FOUB PROPOSITIONAIi FORMS.
47
Form A. — Universal Affirmative.
All X is Y.
All gold is yellow.
Form E. — Universal Negative.
No X is Y.
No man is infallible.
Y
Form I. — Particular Affirmative.
Some X is Y.
Some men are wise.
48 four proposition al forms.
Form 0. — Particular Negative.*
Some X is not Y.
Some men are not wise.
The folloiuing ohsei'vations on this fourfold form of Pro-
positions are of the utmost importance : —
Form A. — The subject is distributed, i.e., taken in its
full extension : the predicate is not distributed. When
. we assert that " all gold is yellow," we mean that gold,
at all times and in all forms, is yellow; therefore, the
term " gold " is fully distributed. But the predicate is
not distributed. For the proposition asserts only that
amongst an indefinite number of yellow things, gold is
always one.
Form E. — Both the subject and the predicate are
distributed. When we assert that " no man is infallible,"
we mean that the two terms "man " and "infallibility ''
are mutually exclusive. The attribute of infallibility
cannot be predicated of any man in the whole universe.
*Ia the diagrammatic illustrations the shaded parts always represent the
subject of the proposition. The student must note that the proposition only
contains information about the part shaded. Thus in the diagram representing
the proposition " Some X is Y," our information is confined to the shaded part
of X entirely We could not assume therefrom that some X is not Y. The
proposition only asserts that some portion of X is included within Y. As a
matter of fact sJl might be, but the proposition does not say so. These
iiagramraatic representations of propositions are called Euler's Circles, and are
open to much criticism.
SIGNS OF QUANTITY IN LOQIO. 49
Form I. — Neither the subject nor the predicate is
distributed. When we assert that " some men are wise,"
we mean that amongst men there is an indefinite number,
forming an equally indefinite proportion of those beinga
of whom the attribute of wisdom may be predicated.
Form 0. — The predicate only is distributed. When we
assert that " some men are not wise," we mean th^ an
indefinite number of men are excluded from the whole
definite class of beings, of whom the attribute of wisdom
may be predicated.
These observations may be summarised : —
Form A distributes its subject only.
,, E distributes both its subject and its predicate.
,, I distributes neither its subject nor its predicate.
,, O distributes its predicate only.
The student will notice that "this," "each," "every,"
"all," "no," and "some" are the only signs of quantity
recognised by Logic. In ordinary speech many others are
used, but they must be reduced to one of the signs given above
before they can be considered in a logical reference.
Note particularly that expressions like "few," "many," or
such fractional terms as " three-fourths " are all considered
equivalent to " some." In short, " some " really stands for
" some at least " ; and beyond that, the word is altogether
indefinite. " Any " and similar expressions must be con-
sidered as equivalent to " every."
Cases will sometimes arise in which it is a matter of uncertainty
whether a given expression is intended to be taken as a universal or
a particular. This is especially so in current sayings and proverbs,
e.g., " Knowledge is power," " Haste makes waste." Such cases can
only be determined by a careful survey of the facts the expressions
are supposed to summarise.
60 BZEBCISE8.
EXERCISES ON CHAPTER VII.
i. Define a logical proposition ; and enumerate with examples^
the various lands of propositions.
2. What do you understand as the exact meaning of the logical
copula ?
3. What are the signs of quantity recognised by Logic ? How do
they compare with those used in grammar ?
4. Give the logical equivalent of each of the following expressions :
" All are Twt** ; " Only tliese are " ; *' All except one " ; " Scarcely
any " ; " Few are not.''
5. RediLce each of the following to strict logical form, and indicate
wJietlier the proposition is A, E, I, or O : —
(a) All birds have two wings.
(b) All his shots except two hit the mark.
( c) Tlie more tJie merrier.
(d) TJiere's not a joy tlie world can give like that it takes
away.
(e) All that glitters is not gold.
(f) He jests at scars who never felt a wound.
(g) None fail to remain poor who are both ignorant and lazy.
6. The following sentences are somewhat ambiguous. Make at
least two logical propositions of each : —
(a) All are not clever who read much.
(b) Some of the guests behaved disgracefully.
(c) All the books cost a sovereign.
7. What logical 2^roposition is iinplied in each case, wlien tlie
following are declared to be false : —
(a) Honesty is tlie best policy,
(b) All men are liars.
(c) Sonie horse dealers are honest.
8. Express in the simplest logical form you can the sense of the
following passages : —
(a) It never rains but it pours.
(b) You cannot have your cake and eat it.
'e) Unless help arrives we are beaten.
(d) Many are called, but few are chosen.
9. Say whether the following is a categorical or hypothetical
proposition, and why : — Trespassers will be prosecuted.
CHAPTER VIII.
Immediate Inference.
The whole of our study thus far has been a preparation for
the investigation of inference or reasoning. Inference, in its
wider meaning, is the derivation of one proposition from one
other proposition or from two other propositions. Thosa
cases in which a conclusion is evolved from some one pro-
position, without the help of any other, are called Immediatb
Inferences. Thus, when we say " All animals are organised
beings," we are able to infer directly from this that any
particular animal is an organised being, and, again, that "no
unorganised beings are animals." Every single assertion or
denial that can be made will yield quite a number of other
^propositions, which differ from the original proposition in
logical quantity or quality, or both.
An Immediate Inference, then, is the inferential derivation
of a new proposition from some one given proposition.
The number and variety of conclusions which can be immediately
derived from any single propc«ition, will be quite surprising to one
who is not familiar with this kind of exercise. Take, for example,
the following A (universal affirmative) proposition :— •• All X is Y."
OS OPPOSITIOlf.
What Inferences can be immediately derived from this ? Proceed
thus : All X is Y ; No X is not-Y ; Some X is Y ; Some X is not
not-Y ; No not-Y is X ; All not-Y is not-X ; Some not-X is not-Y ;
Some not-X is not Y.
Thia will be clearer if a concrete example is given :— " All men are
mortal." From this we may infer : " No men are not-mortal " ;
" Some men are mortal " ; •• Some mortal beings are not not-men " ;
•' No not-mortal beings are men," etc.
Now, without considering whether the examples just given
are exhaustive, or whether all the conclusions are of practical
importance, we will proceed to discuss the more important-
forms of Immediate Inference under the following heads : —
I. Immediate Inferences of Opposition.
II. „ „ „ Conversion.
III. „ „ „ Permutation.
I. Inferences of Opposition. — Propositions are said to be
opposed to each other when they have the some subject and
predicate respectively, but differ in quantity or quaHty, or
both.
Of the several kinds of opposition, that known as Contra-
dictory Opposition is the most perfect and of the greatest
logical value. This kind of opposition is an application of the
" law of the excluded middle," viz., that, of two contradictory
propositions, one must be true and the other false. This
occurs when an A proposition is contradicted by an proposi-
tion ; or an E proposition is contradicted by an I proposition.
A.— All X is Y.
Contradictory =0. — Some X is not Y.
E.— No M is N.
Contradictory = I.— Some M is N.
OPPOSITION. 63
Taking either of these pairs of propositions, we see at once
that both cannot be true and that they cannot both be false.
Therefore, if either of the two propositions is affirmed to be
true, we immediately infer the falsity of the other.
Contrary Opposition is that which exists between an A
and an E proposition, having the same subject and predicate.
In this case, both propositions may be false, but both cannot
be true, e.g. : —
A. — All men are good.
Contrary =E. — No men are good.
This kind of opposition is of much less logical value. If we
know that one proposition is true, we may immediately infer
the falsity of the contrary. But if we know that one proposi-
tion is false, we cannot infer the truth of its contrary.
Sub-Contrary Opposition is that which exists between an
I and an proposition, which both have the same subject
and predicate : —
I. — Some men are wise.
Sub-contrary 0. — Some men are not wise.
In this case both of the propositions may be true, but both
cannot be false. If we know that one of them is false, we can
immediately infer the truth of its sub-contrary.
Subaltern Opposition is that which exists between a
universal and a particular proposition, i.e., propositions which
both have the same subject and predicate, but differ in
quantity : —
A. — All men are mortal.
Subaltern I. — Some men are mortal.
. From any universal prop^ RJtio^ "^"^ ^°'^ immo/liQfoly in^^y
t he truth of any pa/ticular proposition of the same quality
(an I from an A, or an from an E), but not Yice versa*
64
OPPOSITION,
An ancient square sets forth these various relations of
opposition thus : —
A Contrartes £
N.B. — Propositions must always have the same subject
and predicate before we can place them in opposition.
There should now be found no difficulty in determining
what inferences can be immediately drawn from the known
truth or falsity of any one of the four ordinary propositional
forms.
For convenience the student is advised to commit the following to
memory :—
Contradictories cannot both be true, nor can they both be false.
Contraries may both be false, but both cannot be true.
Sub-contraries may both be true, but cannot both be false.
Subalterns may both be true and both false. If the universal is
true so is the particular ; but the truth of the particular does not
imply the truth of the universal.
CONVERSION. 65
II. Inferences of ConveFsion. — By conversion is meant the
immediate inferring of a new proposition from a given proposi-
tion, in which the subject of the given proposition forms the
predicate of the new proposition, and the predicate the subject.
Thus from "No stones are organised beings" — is obtained by
conversion, " No organised beings are stones."
The remarks in Chapter VII. on the distribution of the
subject and predicate in the four propositional forms, are of
great consequence here. For, in converting a proposition, care
must be taken that the two terms are used in precisely the
same extent in the new (or inferred) proposition as they were
in the original proposition. Now, in converting an E or an I
proposition, no difficulty arises. "No X is Y" distributes both
its subject and its predicate. Hence, we may at once say " No
Y is X." So, also, the I proposition " Some M is N " distributes
neither its subject nor its predicate. Thus, we can immediately
say " Some N is M." But in the A proposition " All S is P,"
the subject S is distributed, but the predicate Pis undistributed.
If we converted this into " All P is S " we should distribute P
in the new proposition, whereas it was not distributed in the
original proposition. This we may not do. From " All S is
P" we can only infer " Some P is S." Hence we say that A
propositions can only be converted " by limitation " {per
accidens).
Summarizing these points we learn that : —
From an A proposition we can infer an I proposition by
" conversion by limitation."
From an E proposition we can obtain another E proposi-
tion by simple conversion.
From an I proposition we can infer another I proposition
by simple conversion.
Lastly, we have to consider the case of O (particular
negative) propositions. Can these be converted ? Take, for
instance, " Some X is not Y." Here X, the subject, is not
56 PERMUTATION.
distributed. If we convert the proposition and say " Some Y
is not X," we distribute X in the new proposition. But, in
conversion, we may never distribute a term in the new proposi-
tion, which is undistributed in the original proposition. Hence,
we conclude that O propositions cannot be converted.
Practice in drawing immediate inferences by the conversion
of given propositions is a most valuable test of the student's
progress in logical study. Both in ordinary discourse and in
examinations most ludicrous results follow from not observing
the rules of legitimate conversion. One examiner says that
when he has asked for the converse of the proposition '• None
but the brave deserve the fair," students have said with perfect
seriousness: "The fair deserve none but the brave," or " No
one ugly deserves the brave." The error in such cases arises
from the fact that the student has omitted to put the given
sentence into exact propositional form, as logic requires. If
this were done the sentence would become : — "No one who is
not-brave is deserving of the fair," and this is a simple E pro-
position, and may therefore be converted simply into " No one
deserving of the fair is not-brave," or, expressed more con-
ventionally, " No one deserving of the fair is a coward."
III. Inferences of Permutation. — Of this kind of immediate
inference there are several forms : —
(a) By Obversion. — Here we infer a new proposition, having
for its predicate the contradictory of the predicate, e.g. : —
Original proposition. — All X is Y.
Inference by Obversion. — No X is not-Y.
We may always obvert a proposition, if at the same time
we change its quality. The rule of obversion is usually given
thus : Substitute for the predicate term its contrapositive, and
change the quality of the proposition.
Contrapositive is a mediaeval word for the opposite of a term.
Thus " not-A " is the contrapositive of " A." It is convenient to use
this word so that " contradictory " may be used exclusively of
propositions.
PBBMUTATION.
67
Thus, All X is Y yields No X is not-Y.
No X is Y „ AU X is not-Y.
Some X is Y „ Some X is not not-Y.
Some X is not Y „ Some X is not-Y.
(b) By Contraposition. — In this case we infer a new pro-
position which has the contrapositive of the original predicate
<or its subject, and the original subject for its predicate, e.g. : —
Original proposition. — All X is Y.
Contrapositive. — No not-Y is X.
Immediate inference by contraposition is sometimes called
the converse by contraposition.
From A, E and of the propositional forms we may infer
A contrapositive, but not from I.
Original Proposition.
All X is Y.
No X is Y.
Some X is not Y,
Contrapositive.
No not-Y is X.
Some not-Y is X.
Some not-Y is X.
In drawing immediate inferences accuracy is all important.
The exposition in this chapter has been illustrated by symbols,
but if the principles have been duly grasped it will not be
difficult to apply them to concrete examples. In doing so the
student must always reduce the sentences given as examples
to strict logical form, if they are not already in that condition.
The great importance of this subject makes it advisable that
several worked examples should be presented for the reader's
consideration.
58 WORKED EXAMPLES.
1. '^ hat immediate inferences are derivable from, the pro-
position ^^ All really happy men are virtuous " ?
(a) The Truth of the Subaltern: "Some really happy
men are virtuous."
(6) The Falsity of the Contradictory : " Some really happy
men are not virtuous."
(c) The Falsity of the Contrary: " No really happy men
are virtuous."
{d) By Conversion : " Some virtuous men are really
happy."
(e) By Ob version : "No really happy men are not
virtuous."
(/) By Contraposition : "No not- virtuous men are
really happy."
2. Give the Converse, the Obverse and the Contrapositive
of the following propositions : — (a) The longest road comes to
cm end; (b) UnasTced advice is seldom acceptable. (Each of
these propositions must first be reduced to logical form.)
{a) This sentence=" The longest road is limited." This
is a universal affirmative.
Its Converse is : " Some (one) limited thing is the longest
road."
Its Obverse is : " The longest road is not unlimited."
Its Contrapositive is: " No unlimited thing is the longestr
road."
(6) This sentence =" Some unasked advice is unaccept-
able." This is a particular affirmative proposition.
Its Converse is: "Amongst (some) unacceptable things
is unasked advice."
Its Obverse is : "Some unasked advice is not acceptable."^
The sentence being an I proposition it has no contra^
positive.
WORKED EXAMPLES. 69
8. Convert and contraposit the proposition^ " For every
wrong there is a legal remedy.^'
The proposition reduced to logical form is : " Every
wrong is capable of a legal remedy."
Its converse is : " Some things capable of legal remedy
are wrongs."
Its contrapositive is : "Nothing incapable of legal remedy
is a wrong."
4. What 'eductions are possible from the proposition,
*' Amethysts are precious stones " ? {N.B. — " Eduction " is a
term frequently used for " Immediate inference?^) *
The given proposition is a universal affirmative, " All
amethysts are precious stones," and may be treated as the
proposition in the first-worked example.
60 EXERCISES.
EXEECISES ON CHAPTEE VIII.
1. Explain and illiLstrate by exarnples the difference between the
converse and the contradictory of a proposition ; and say when and
under what conditions the converse of a proposition is or is not
necessarily true.
2. Explain with illustrations, the difference between the contrary
and the contradictory of a proposition.
3. Explain why a universal negative proposition admits of the
conversion of its terms. ^^ All equilateral triangles are equiangular. ''^
Say whetlier the terms of this proposition are convertible. If not,
why not ?
4. Give thi converse, the contradictory and contrary of " All A is
B " ; " Some men are wise."
5. Give the contradictory and the converse of : —
(a) Two blacks don't make a white.
(b) James struck John.
(c) Three-fourths of the candidates passed.
6. Assign the logical relation between each of the following pro-
positions with the proposition " All crystals are solids " ; —
(a) Sorne crystals are solids.
(b) No crystals are not solids.
( c) Some solids are crystals.
7. What is mediate inference ? Give where possible the converse,
the obverse and the contrapositive of: —
{a) (said Hudibras) : " I smell a rat."
{Jo) Where no oxen are, the crib is clean.
( c) Only protestant princes can occupy the English throne.
8. What is opposition ? Which of the forms of (ypposition has tJie
greatest value and why ?
CHAPTEB IX.
Mediate Inference.— The Syllogism,
Immediate inference is the derivation of a new proposition
from some given proposition. However useful this exercise
may be, the new proposition is always recognised as only a
different way of expressing the original proposition. Mediate
inference professes to give a conclusion of a much more
fruitful kind. In every example of a mediate inference, two
propositions, and two only, are implied. In these two pro-
positions the conclusion to be drawn is potentially contained,
and out of these two propositions the conclusion is actually
drawn by reasoning. The two propositions given are called the
premisses of the conclusion.
It will be seen afterwards that in ordinary discourse the
two premisses and the conclusion are seldom fully expressed.
One of the premisses is generally left to be understood, but,
in spite of this, it is implied in the reasoning. When, however,
the two premisses and the derived conclusion are fully and
formally stated, the expression is called a Syllogism. Formal
Logic assumes that, in every instance in which we draw a new
and fruitful conclusion, the reasoning when fully expressed
must take the form of a Syllogism,
62 THE SYLLOGIBM.
A Syllogism, then, is a conclusion expressly evolved from
two propositions called its premisses.
Ea^ch of the premisses of a syllogism must once have been a con-
clusion from two other more remote premisses, unless one of the
premisses is the statement of a truth which is axiomatic in its
nature. All that we know, inferentially, about the universe, is known
in the form of a vast number of conclusions drawn from other
premisses. Knowledge is thus a net-work of conclusions, suspended
ultimately upon a few axiomatic or self-evident truths. All arguing
Implies that there are certain remote premisses or assumptions,
bearing logically on all questions, and about which the disputants
must be agreed.
It is worthy of observation that some persons, who are not acute
reasoners, are yet able to see truth at a glance. Others are subtle
and ready reasoners whose natural intuition (insight) is small.
Argument and insight are often found in inverse ratio. It has been
remarked that, generally, women are more strongly endowed with
insight, and men with reasoning power.
The following is a simple form of a Syllogism : —
All men may be educatedo ,^
5- (Premisses.)
Savages are men. }
Therefore, Savages may be educated. (Conclusion.)
Notice that in this example there are three propositions.
Of these the first two are the premisses, and the last the
conclusion. There are also three terms : " men," " savages "
and " educated," and the last two of these appear in the
conclusion. The term which forms the predicate of the con-
clusion ('* educated ") is called the major term, and the term
which forms the subject of the conclusion (" savages ") is
called the minor term. The term which appears in both the
premisses, but which does not appear in the conclusion
(" men ") is called the middle term. Further, the premiss
which contains the major term (*' All men may be educated ")
is called the major premiss; and the premiss which contains
the minor term ("All savages are men") is called the minor
premiss. These general definitions hold good for all kinds of
syllogisms.
BUIiSS OF THE SYIiLOaiBM. 6S
From these definitions it will be easy to see that a syllogism
is the logical comparison of the two terms which appear in the
conclusion, by means of a third, or middle, term.
Logic lays down three fundamental rules which apply to
every variety of syllogism.
1. Each syllogism must have three, and only three,
terms ; it must have three, and only three, propositions.
2. Of the three terms thus involved in every syllogism,
the middle term {i.e., the term common to both premisses)
must be taken universally {i.e., it must be distributed), at
least in one of the premisses ; and neither of the other
terms, i.e., the major or the minor, can be taken univer-
sally in the conclusion, unless it was taken universally in
the premiss in which it occurred.
3. No conclusion can legitimately be drawn if both the
premisses are negative ; or if both are particular ; and, if
one of the premisses is particular, the conclusion must be
particular; or, if one of the premisses is negative, the
conclusion must be negative.
Notes on the Canons, or Eules of the Syllogism.
Rule I. -We require to add that the terms must be used
throughout in exactly the same sense. Owing to the ambiguity
of words it sometimes happens that a syllogism will seem only
to contain three terms when in reality there are four, i.e., one
of the terms has been used in two distinct senses. In the
fallacy quoted in the early part of this work we have an
example of this :
He who is most hungry eats most.
He who eats least is most hungry,
Therefore he who eats least eats most.
64 RULKS OF THE SYLLOGISM.
In this example a little reflection will show that terms arer
not being used throughout in the same sense, and that there
are in reality more than three terms involved.
Rule 2. — The middle term must be once distributed, other-
wise it cannot be a medium for comparing the other twa
terms. It must be either wholly in, or wholly out of one of
the other terms before it can be the means of establishing
a connection between them.
If we use a diagrammatic illustration of the Syllogism, th«
necessity of the distribution of the middle term is obvious.
Thus let the Syllogism bo
All M is P.
All S is M.
.-. All S is P.
Here M is the middle term, and it is shown to be wholly in P.
If, however, M were not wholly distributed we should have
to represent it partially within P. Consequently, we should
not be able to say whether S was contained in the part of M
within P or in that part of M which is without P and of
which we are not supposed to know anything.
A syllogism with an undistributed middle term is the most
common form of erroneous reasoning.
BULES OF THE SYLIiOGISM. 65
A term must not be distributed in the conclusion that was
Qot distributed in the premisses. Obviously, if an assertion is
not made about the whole of a term in the premisses, we cannot
make it of the whole of the term in the conclusion without
going beyond what has been given. When this rule is broken
in the case of the major term, it is called the Illicit process of
the Major ; and in the case of the minor term, Illicit process of
the Minor. If we were to admit that a term might be taken
universally in the conclusion, which was not so taken in the
premisses, we should be admitting that the " part is greater
than the whole."
Rule 3. — Two negatives cannot yield a conclusion. For
two negative propositions are really a declaration that no
connection exists between the major and minor term and
the term by which they were to be compared — in other
words there is no middle term, and no Syllogism can be
formed with two negative premisses. That two particulars
cannot give a valid conclusion, and that the conclusion follows
the weakest premiss are corollaries from the previous rules.
The general rules of the Syllogism depend upon one great
cajaon, viz. : " Two terms that logically agree with the same
third term, must logically agree with each other ; and two
terms, one of which agrees while the other disagrees with the
same third term must logically disagree with each other."
The ultimate principle of reasoning thus defined, is expressed '
its most general forms in the "Dictum de omni et nullo " of Aristotl*
"Whatever is predicated aflBrmatively or negatively of any class,
must, on pain of involving inconsistent (contradictory) thought, be
predicated of whatever is contained under that class." Aristotle
regarded this as the axiom on which all syllogistic inference is based.
Every conclusion drawn in a syllogism, where the above
general rules have been observed, is an affirmative or negative
proposition deduced by means of a minor (or applying) premiss
66 THB STIJ^OGISTIC MOODS.
from a more general proposition that is assumed to be true,
and in which the conclusion was virtuaUy contained.
It can also be shown that there must be four, and need not
be more than four syllogistic forms. For, the general (major)
proposition, which virtually contains the conclusion must be
universal (either A or E), and the applying (minor) premiss
must bring either the logical whole, or a part only, of its subject
into comparison with the middle term. The minor premiss,
therefore, will be either A or I. The general rules of the
syllogism decide the conclusion. Hence, we may say that
every reasoning may be exhibited by one or other of the
following combinations of the four propositional forms : —
AAA. AIL EAE. E I O.
These letters, of course, tell us the quantity and quality of
the two premisses, and the conclusion of the syllogism which
each triplet forms. The arranging of the symbolic letters in
different ways is called the Mood of the sylloginm. Thus, AAA
represents a syllogistic mood in which both the premisses and
the conclusion are universal afi&rmatives. E I represents a
syllogistic mood in which the major premiss is a universal
negative, the minor premiss a particular affirmative, and the
conclusion a particular negative. The following are examples
of four forms of syllogism : —
Mood AAA.
All men may be educated.
All savages are men.
.'. All savages may be educated.
All M is P. All S is M. .-. All S is P.
[N.B. — ^This alone of all forms of syllogism gives a universal
affirmative conclusion, and is, therefore, the one most convenient
for expressing scientific reasonings with their universal afidrma
tive conclusions.]
.^^,
THE 8TLL0GIBTIC MOODS. 67
Mood AIL
All educating influences are good.
Some difficulties are educating influences,
.*. Some difficulties are good.
All M is P. Some S is M. .-. Some S is P.
Mood E A E.
No Europeans are cannibals,
All Englishmen are Europeans,
.'. No Englishmen are cannibals.
No M is P. AU S is M. .-. No S is P.
Mood E I 0.
Whatever is followed by remorse is not desirable,
Some pleasures are followed by remorse,
.'. Some pleasures are not desirable.
No M is P. Some S is M. .•. Some S is not P.
The student is advised at this point to transform some
simple arguments from the form in which they are ordinarily
used, into precise syllogistic form. Consider for example the
following : —
1. " There are no foreigners amongst the wounded^ so no
Frenchman received a wound."
Here we have given a major premiss and a conclusion. In
order to express the statement in syllogistic form we must
supply the minor premiss. The passage may then be written
as a Syllogism in E A E : —
No foreigners are wounded
(All Frenchmen are foreigners)
.'. No Frenchmen are wounded.
HB EXEBCISSS.
2. " ^0 war is long popular ; for every war increase*
taxation : a/nd the popularity of anything that touches the-
pocket is short livedo
This may be ^vritten as a Syllogism in E A E thus : —
Nothing that increases taxation is long popular.
Every war increases taxation.
.*. No war is long popular.
8. " For some wars there has been no justification ; for
they have been harmfully aggressive, and such aggression-
is without excuse.''
This may be expressed as a Syllogism in E I O, thus :—
No harmful aggression is justifiable,
Some wars are harmfully aggressive,
,•. Some wars are not justifiable.
EXEECISES ON CHAPTEB IX.
1. Wliat is understood by a proposition, a premiss, a conclusion^,
and a syllogism ? Give an example of each.
2. " From negative premisses you can infer nothing.'' Explain
a/nd illustrate this statement.
3. Show how logical form as displayed in the syllogism tends Uy
clearness of thought.
4. Give a clear explanation of the rule concerning the middle
term of a syllogism.
5. Enumerate the cases in which no valid concltision can bt
drawn from two premisses.
6. Supply a premiss that will make the folloiuing reasoning
correct : *' There is no Englishman among the wounded, so no
officer can have received a wound."
7. Put the following argument into syllogistic form : — " How can
anyone maintain that pain is always an evil, who admits that
remorse involves pain, and yet may sometimes be a real good ? "
CHAPTER X.
The Figures of the Syllogism.
In one or other of these four syllogistic forms all our reason-
ings might be expressed, just as all our judgments could be
expressed in one or other of the prepositional forms. But
Logic takes cognisance of many other syllogistic forms besides
these four. The question may suggest itself — why should we
add to these four forms, if they are sufficient for the unabridged
expression of all sorts of reasonings ? The answer is that the
addition is one of practical convenience. It will be found
that many of the concrete reasonings of ordinary life, though
capable of being expressed in one of the four syllogistic forms,
yet find a more convenient and natural expression in one or other
of the additional forms. The way in which the additional
syllogistic forms are obtained is by varying the jposition of the
terms in the prermsaes. The four syllogistic forms already con-
sidered have certain features in common. Thus, in each case the
middle term is the subject of the major premiss and the predicate
of the minor premiss. Also, the middle term is distributed in the
major premiss but not in the minor premiss. On account of
this similarity the four syllogistic forms already given are
classed together, and constitute what is known as Figubb I. oi
the syllogism.
70
FIGUBBB OF THE SYLLOQIBM.
But we can frame a series of syllogisms which violate none
of the general rules of the syllogisms, in which the relations of
Figure I. are varied. Thus we may have the middle term as
the predicate of each proposition. In these cases the middle
term will always be of greater logical extent than either of the
other two. The syllogisms which exhibit these characteristics
are classed together as Figure II.
Just as there were four valid moods under Figure I., so
there are four valid moods under Figure II. The student should
construct concrete illustrations by reference to the following
abstract examples of each of the moods of Figure II. : —
AEE.
All P is M.
No S is M.
No S is P.
A 0.
All P is M.
Some S is not M.
Some S is not P.
E AE.
No P is M.
I All S is M.
! No S is P.
EIO.
I No P is M.
I Some S is M.
' Some S is not P.
Notice that in Figure II. the conclusion in each mood is a
negative one. Hence, this figure is the most convenient for
expressing argumentative objections and refutations.
When the middle term is made the subject of each premiss,
and is, therefore, of less logical extent than the other two
terms, we get a series of syllogisms which are grouped together
as forming Figure III. But for reasons that will afterwards
appear, we can form six valid moods of this figure. Thus : —
A A I.
All M is P.
AU M is S.
Some S is P.
AIL
All M is P.
Some M is S.
Some S is P.
E AO
No M is P.
AU M is S.
Some S is not P.
EIO.
No M is P.
Some M is S.
Some S is not P.
• I A I.
Some M is P.
All M is S.
Some S is P.
AO.
Some M is not P.
All M is S.
Some S is not P.
FIGURES OF THE SYIiLOGISM.
71
Notice that a particular conclusion only is obtained in each
mood of Figure III. Hence this mood is well fitted for pro-
pounding examples argumentatively, or for establishing some
particular or indefinite conclusion.
There is yet a further group of syllogisms, known as
Figure IV, in which the middle term is the predicate of the
major premiss, and the subject of the minor. This figure has
five moods, viz : —
A A I.
All P is M.
All M is S.
Some S is P.
AEE.
All P is M.
No M is S.
No S is P.
E AO.
No P is 11.
AllMisS.
Some S is not P.
I A I.
Some P is M.
All M is S.
Some S is P.
EIO.
No P is M.
Some M is S.
Some S is not P.
The fourth figure is clumsy and unnatural, and is omitted
altogether hy many logicians. It is worth while to notice the
following results, obtained from a comparison of the conclusions
in the various moods of the four figures : —
A (universal affirmative) conclusions can only be
Obtained in one figure and in one mood of that figure.
E (universal negative) conclusions can be obtained in
three figures or four moods.
I (particular affirmative) conclusions can be obtained in
three figures or in six moods.
(particular negative) conclusions can be obtained in
each of the four figures or in eight moods.
From this it follows that A conclusions are the most difficult
to establish, and the easiest to overthrow. O conclusions, on
the other hand, are the easiest to argue for, but the hardest to
72 PIGUEES OP THE SYLLOGISM.
disprove. Or, more generally, universal and definite conclusions
are most easily overthrown, and particular and inaelmite con-
clusions are most easily maintained.
Special Rules of the Figures of the Syllogisms.
In addition to the general rules to which all syllogisms must
conform, logicians have deduced certain simple rules applicable
to the different Figures.
In the First Figure, —
{a) The major premiss must be universal.
(6) The minoi: premiss must be affirmative.
In the Second Figure, —
(a) The major premiss must be universal.
(6) One premiss and the conclusion must be negative.
In the Third Figure, —
{a) The minor premiss must be affirmative.
(6) The conclusion must be particular.
In the Fourth Figure, —
(a) When the major premiss is affirmative, the minor
premiss must be universal.
(6) When the minor premiss is affirmative, the con-
clusion must be particular.
(c) In negative moods, the major premiss must be
universal.
These special rules of the figures do not introduce new
material, they are only a concise statement deduced from
results previously obtained.
EXBBCISBB. 78
EXERCISES ON CHAPTER X.
1. What are the figures of the Syllogism ? Examine whether
J A I, E 1 are valid or invalid in each of the figures.
2. Which figure is most convenient (1) for overthrowing an
udversary^s conclusion ; (2) for establishing a negative conclusion;
(3) for proving a universal truth.
3. Gfive the special rules of the Figures.
4. Express the following argument by a Syllogism of tlie third
figure :—Some things which have a practical worth are also of
theoretical value : for every science has a theoretical as well as a
practical value.
6. What moods are good in the first- figure and faulty in the
second, and vice versa ? Why are they excluded in one figure and
not in the other ?
6. From which syllogisms can you infer universal, particular^
negative inferences, or none at all ?
7. Enumerate briefly the conditions of a valid deduction.
8. Construct a Syllogism in I A I to prove tiiat some taxation is
necessary.
CHAPTEB XI.
The Reduction of Syllogisms.
It has already been observed that all reasonings may find
their expression in one of the four moods of Figure I. The
other figures are often convenient for special purposes, but
inasmuch as the first figure is considered the most direct and
perfect mode of expressing our reasoning, Logic shows how
any syllogism of Figures II., III., and IV. (called the indirect
figures) may be transformed into one of the moods of the first
figure. This process is called the Beduction of Syllogisms.
There are fifteen moods altogether in the three indirect figures,
and thirteen of them may be reduced : (1) by the conversion of
one or more of the three propositions in the syllogism to be
reduced, (2) by the transposition of the premisses, or (3) by
both of these processes.
[A 0, Figure II. and GAG, Figure III. are exceptions^
and will be considered separately.]
The reduction of the syllogisms of the indirect figures into
direct or first figure syllogisms is one of the most profitable
exercises in formal Logic. The process is not nearly as easy
as might appear. To ensure accuracy and rapidity in the
process an ingenious mnemonic has been used by logicians for
more than 500 years. This mnemonic has been called *' the
magic verse of Logic," and certainly the words of which it is-
REDUCTION OF SYLLOGISMS. 75
composed are more full of meaning than any similar com-
bination ever made. The usual form of the mnemonic, which
must be learnt by heart, is as follows : —
Barbara, Celarent, Dari% i^erioque, prions,
Cesare, Camestres, Festino, Barolco, secundae,
Tertia, Darapti, Disamis, Datisi, Felapton,
BoJcardo, Ferison, habet, Quarta insuper addit,
Bramamtip, Camenes Dimaris, Fesapo, Fresison.
[The words in italics are the significant words, the others
being only connectives.]
The following is the key to this famous mnemonic. Every
mood in each of the four figures is represented by a different
word. In the case of the indirect figures (II., III. and IV.),
the mnemonic tells us to what mood of the first figure the
various moods of these indirect figures are to be reduced. It
gives us, also, full information as to how the reduction is to be
performed.
1. The vowels in each word give the quantity and
quality of the syllogism which the word represents.
Thus Barba.ra=a syllogism of Figure I., mood AAA.
Cesa.re=a syllogism of Figure II., mood E AE.
2. The initial letters of the words in Figures II., IIL
and IV. tell us that a syllogism, represented by a word
with that initial letter, may be reduced to the syllogism
of the first figure, which is represented by a word having
the same initial letter. Thus, the syllogism of the fourth
figure, represented by the word Camenes, may be reduced
to the syllogism of the first figure, represented by the
word Celarent.
3. The letter " s " occurring in a word performs a double
function. If it occurs in the middle of a word as in Cesare
it means that, in the process of reduction, the proposition
represented by the previous vowel is to be simply con-
\ REDUCTION OF SYLLOGISMS.
verfced. Thus, in reducing Cesare (Figure II.) to Celarent
(Figure I.) the major premiss must be simply converted : —
TNoPisMl fNoMisP.
Cesare •] AU S is M I = Celarent -! AH S is M.
[No S is Pj [NoSisP.
When "s" occurs at the end of a word, it tells that
the conclusion of the new syllogism requires to be con-
verted in order to get the conclusion in the form given in
the original syllogism.
4. When the letter " p " occurs in the middle of a word
it tells us that in the process of reduction the preceding
proposition is to be converted per accidens (limitation).
Thus, in reducing Darapti of Figure III. to Darii, Figure I.,
the minor premiss must be converted " per limitation."
fAUMisP ] TAUMisP.
Darapti-! All M is S i-=Darii-<{ Some S is M.
[ Some S is P J 1 Some S is P.
When "p" occurs at the end of a word it signifies that
the conclusion of the new syllogism must be converted
per limitation in order that the new conclusion miay
a-ppear in the same form as the conclusion in the original
syllogism.
5. When the letter " m " occurs in a word it tells us
that the premisses will require transposition in the process
of reduction, i.e., the minor will become the major. Thus,
bs reducing Cawiestres (Figure II.) to Celarent (Figure I.)
the major and minor premisses exchange places : —
C AllPisM.*) ftNoMisS.
Camestres -j No S is M.f \ = Celarent -j ^All P is M.
I No S is P. J I No P is S.
This reduction also illustrates the use of the final "s"
in Camestres. For in Celarent we have " No P is S " as
the conclusion, and by applying the meaning of the final
^*s," we convert the conclusion and so obtain "No S is
BEDUCTIO AD ABSUBDUM. 77
P," which is the form of the conclusion in the original
Camestres proposition.
6. There is still the significant letter "k" to be con-
sidered. It occurs in Baroko and Bokardo. It will be
remembered that the moods A (Figure II.) and A Q
(Figure III.) were reserved for exceptional treatment.
At the time when these mnemonic lines were constructed,
contrapositives were not recognised. In consequence of
this, a somewhat roundabout method had to be employed
in reducing syllogisms in Baroko and Bokardo. The
process is known as Beductio ad Ahsurdum — a process
quite familiar to students of Euclid. Suppose an argument
in Baroko is proceeding. The two disputants agree about-
the premisses A and in Figure II., i.e, : —
All P is M,
Some S is not M,
but one of the disputants will not accept the conclusion.
Some S is not P. How, then, shall we show that the
conclusion is the only valid one ? We may say that if
the conclusion, "Some S is not P," is incorrect, then its
contradictory assertion must be correct, viz., All S is P.
We will assume, for the sake of argument, that All S is P.
We had previously agreed that All P is M. Combining
therefore, these two,
All P is M,
All S is P,
we draw the conclusion that All S is M.
But we agreed in our original premisses that Some S
is not M. Therefore, the conclusion reached is absurd
and impossible. When we convince an opponent in this
way, by showing he cannot admit the premisses and deny
the conclusion without contradicting himself, we are said
to use the Beductio ad Ahsurdum.
The letter "k," therefore, tells us when this method is
to be used. The position of the letter indicates that in
78 EXBBCISEB.
this process of Beductio ad Ahsurdum, the first step is to
omit the premiss preceding it, and substitute in its place
the contradictory of the conclusion. We then obtain two
premisses in the corresponding mood of Figure I., which
yield a conclusion contradicting the premiss omitted.
But, since two contradictories cannot both be true, and
since the truth of the original minor was granted, we reject
' the new conclusion and infer the truth of the original
conclusion.
It may seem that this method of Reduction has no connection
with those methods which converted syllogisms of the indirect
figures to corresponding syllogisms in the first figure. The aim is,
however, the same. The reason for reducing syllogisms to the first
figure is, that the reasoning may be the more clearly seen, and that
the conclusion may be vindicated. In the Reductio ad Absurdum
the aim is also to vindicate the conclusion, but in a different manner.
Although, in the mnemonic lines, this indirect method of vindication
is contemplated only in the case of A O O and O A propositions, it
can be used in others if desired.
EXEECISES ON CHAPTER XI.
1. What is reduction ? Say briefly what purj^ose the ^process is
supposed to serve.
2. Construct an argument in Fresison and reduce it to Figure I.
3. In what moods and figures are the following syllogisms I
Reduce them.
(a) The nervous fluid will not travel along a tied nerve ;
Electricity will travel along a tied nerve ;
Tlierefore electricity is not the nervous fluid.
(6) No men are birds ;
All birds are animals ;
Therefore some animals are not men.
4. Vindicate the truth of the following argument in A O by
Beductio ad Absurdum. " Some successful persons are not industri-
ous thinkers ; for every industrious thinker is educated, but some
successful persons are not educated.
CHAPTEB XII.
Irregular and Compound Syllogisms.
In ordinary arguments it is seldom necessary, for practical
purposes, that both the premisses should be expressly stated.
One or other of the premisses is frequently suppressed. The
following, for example, is a method of argument often used : —
" That man is contemptible, for he is a coward."
In this case the major premiss, "AU cowards are con-
temptible," has been taken for granted.
An Enthymene is a syllogism imcompletely stated, i.e., one
of the three propositions forming the syllogism is taken for
granted, but not expressed. If the major premiss is omitted,
the enthymene is said to be of the first order ; if the minor
premiss is omitted, it is said to be of the second order ; and if
the conclusion is omitted, it is said to be of the tJiird order.
" That man is unhappy, for he is a miser." {First order.)
" All misers are imhappy, therefore that man is unhappy.'
{Second order.)
" All misers are unhappy, and that man is a miser."
{Third order.)
The above are examples of enthymenes of the three orders,
the full syllogism being
All misers are unhappy,
That man is a miser,
.'. That man is unhappy.
An Ej^THYJiENE frequently occurs in a very terse form. Thus,
" He must be mad to attempt that," is an enthymene. The premisses
are " All who attempt that are mad ; He is one who attempts that ;
Therefore, he is mad."
80 THB BOBITBS.
Ifc has been remarked that all our thought consists of a
chain or net work of premisses and conclusions, each premiss
being really a conclusion drawn from previous premisses.
Logic provides a nomenclature for the chains of reasoning of
which our thought consists. Thus, a Prosyllogism is a syllogism
the conclusion of which is used as a premiss in a succeeding
syllogism. An Episyllogism is a syllogism of which one or both
of the premisses are conclusions from preceding syllogisms.
The union of a prosyllogism with an episyllogism is called
a Polysyllo^ism. Thus : —
j' All M is P.
Prosyllogism. I All S is M.
(.-.AH Sis P. \
But All X is S. [Episyllogism.*
.-. AUXisPj
An Epicheirema is a polysyllogism in which the prosyllogism
is only briefly stated, after the manner of an enthymene.
Thus :—
AU S is P, because it is M,
AU X is S.
.•. AU X is P.
A Sorites is a series of propositions, inferentially connected,
in which the predicate of each is the subject of the next, and
so on indefinitely. The conclusion is formed of the first
subject and the last predicate. Thus : — All A is B, All B is 0,
All C is D, All D is E ; therefore. All A is E.
In a Sorites there are really as many syllogisms as there
are intermediate propositions between the first premiss and
the conclusion. The example just given may be exhibited as
the combination of three simple syllogisms. Thus : —
All A is B. All A is C. All A is D.
AU B is C. AU C is D. AU D is E.
.-.AUAisC. /.AUAisD. .-.AUAisE.
• Of course, the chain may be continued indefinitely
EXEBCISE8. 81
In political speeches the Sorites is a frequent mode of argument.
Take the following extract : "Free-trade is a great boon to the
working man, for it increases trade and thus cheapens articles of
ordinary consumption; this gives a greater purchasing power to
money, which is equivalent to a rise in real wages ; and any rise in
real wages is a boon to the working man." This can be exhibited as
a Sorites, but, of course, each of its general propositions must first be
expressed in precise logical form. The passage will then be found to
consist of the following propositions, joined together after the manner
of a Sorites :— Free-trade is trade-increasing ; every increase of trade
is price-lowering ; every fall in prices is money-value-raising ; every
rise of money-value is real-wage-raising; every rise in real wages is
advantageous to working men ; therefore, free-trade is advantageoiu
to working-men.
There are two special rules of the valid Sorites : —
1. Only one premiss may be negative ; and, if one
premiss is negative, it must be the last one.
2. Only one premiss may be particular; and, if one
premiss is particular, it must be the first one.
In the following argument in the form of a Sorites, there is
a breach of the second rule, and consequently an invalid
inference. " All thieves are dishonest ; all dishonest persons
are immoral ; and some immoral persons go unpunished ;
therefore, some thieves go unpunished."
EXEKCISES ON CHAPTER XH.
/. What kind of argument is the following : —
" Those who have shall not receive ; those who do not receiv§ de
not want." Is the argument valid ?
9. Define Prosyllogism and Episyllogism ; and say of what gema
of reasoning they are species. *
3. Construct a valid Sorites argument with a negative premitM.
CHAPTEE xrn.
Conditional Syllogisms.
In all the various kinds of argTiment that have so far been
considered, the propositions employed have been categorical or
unconditional. But in many of the concrete reasonings of our
ordinary intellectual life, we are obliged to use general state-
ments, to which some condition is attached. In some connec-
tion or other we are constantly using conjunctions, such as
"if," "either," "whenever," etc., and this frequent use
testifies to the number of conditional propositions and argu-
ments in daily use. When a statement to which a condition
is attached enters into an argument, the reasoning seems to
turn on the condition. Hence, an amount of complexity is
introduced into the argument. Logic recognises the fact that
conditional arguments must have a place in our inferential
thought, and exhibits the inner relation of such by the forms
of conditional syllogism.
A Conditional Syllogism is one in which the major premiss,
and that only, is a conditional proposition, and in which,
accordingly, the reasoning seems to turn on the condition. We
shall notice three forms of the conditional syllogism, viz., (I.)
the Conjunctive (or Hypothetical), (II.) the Disjunctive
(or Alternative), and (III.) the Dilemma.
CONJUNCTIVE SYLLOGISMS. 88
I. The Conjanctive or Hypothetical Syllogism. — ^
If A is B, C is D.
AisB.
.-. C is D.
If A is B, C is D.
C is not D.
••• A is not B. -
Modus jponena.
ModAia tollens.
The first proposition in these examples is a complex
proposition formed of two propositions, related in such a way
that the truth of the one follows necessarily from the truth of
the other. When two propositions are related in this manner,
they are technically known as *' the Antecedent " and " the
Consequent."
When two propositions are related as antecedent and
consequent, the truth of the consequent foUows from the truth
of the antecedent ; whilst the denial of the consequent is
virtually the denial of the antecedent. This is known as the
Law of Antecedent and Consequent.
From the statement of this law we may deduce two most
IMPORTANT COROLLARIES, viz. : —
1. The affirmation of the consequent does not justify
the affirmation of the antecedent. Granted that If A is
B, C is D, we may not argue that because C is D, therefore
AisB.
2. The denial of the antecedent does not justify the
denial of the consequent. Granted that If A is B, C is D,
we may not argue that because A is not B, therefore
C is not D.
Applying these observations to the hypothetical proposi-
tion, "If rain has fallen, the grass is wet," consider what
inference could be drawn (a) from the affirmation of the
ajitecedent, (b) from the affirmation of the consequent, (c) from
84 DISJUNCTIVE SYEiLOGISMS.
the denial of the antecedent, (d) from the denial of the
consequent.
(a) Affirmation of the antecedent, " Kain has fallen,"
yields a valid conclusion, " The grass is wet."
(6) Affirmation of the consequent, "The grass is wet,"
yields no conclusion.
Corollary (1) forbids us to conclude, *' Kain has fallen.'*
(c) Denial of antecedent, " Kain has not fallen," yields
no conclusion.
Corollary (2) forbids us to conclude, "The grass is not
wet."
(<Z) Denial of the consequent, " The grass is not wet,'*
yields a valid conclusion, " Kain has not fallen."
II. The DisjunctiYe Syllogism is one in which the major
premiss, and that only, is a disjunctive proposition.
A is either B or 0.
A is not B.
.-. A is C.
The principle which governs the reasoning here is that of
the excluded middle. If we can assume that the disjunctive
in the reasoning is exhaustive, i.e., A is either B or C and
cannot be anything else, then we may vary the general form
of the disjunctive syllogism, thus : —
A is either B or 0.
AisB.
.'. A is not C.
Bat great care must be taken in concrete reasoning that the assnmp^
Hon here involved is warranted. Consider the following example : —
Either the witness tells a lie or the prisoner is guilty.
The witness tells a lie.
.". The prisoner is not guilty.
This is not a valid argument, for the disjunction in the major premiss
Is not exhaustive. For there are other alternatives— the witness may
tell a lie, and the prisoner be guilty all the same.
THE DILEMMA. 85
III. The Dilemma. — The dilemma is a mode of reasoning
designed to show the absurdity of the logical position of an
opponent. It is a syllogism which has for its major premiss
a hypothetical conjxmctive proposition having more than one
antecedent. For its minor premiss it has a disjunctive pro-
position. It thus offers an opponent a choice of alternatives,
and the choice of either alternative leads to a conclusion which
the opponent does not like. The dilemma is expressed in
three principal forms : (1) Simple constructive, (2) Complex
CONSTRUCTIVB, (3) Dbstuctivb ; of which forms the following
are examples : —
1, Simple Constructive : —
If A is B, or if E is F, then C is D.
But either A is B or E is F :
Therefore C is D.
As a concrete example we may imagine the inhabitants
of a town, against which a hostil e army is approaching,
arguing as follows : — " If we are bombarded we shall
suffer loss, and if we surrender we shall suffer loss : but
we must either surrender or be bombarded ; therefore, in
any case, we shall suffer loss."
2. Complex Constructive. —
If A is B, C is D ; and, if E is F, G is H.
But either A is B or E is F :
Therefore, C is D or G is H.
A man in an upper room of a burning house, when
the staircase has been destroyed, might use this form of
argument in reasoning: " If I jump through the window
I shall break my neck ; if I remain here I shall be burnt
to death ; but I must do one or the other ; therefore, in
«ither case I must die.
OF THE ^'^\
UNIVERSITY /
OF /^,
86 THE DILEMMA.
8. Destructive Dilemma. — This dilemma in it&
commoner forms proceeds upon the denial of the conse-
quent, as involving the denial of the antecedent. In the
major proposition we obtain the admission that, if a
certain thing holds good, it must be followed by one or
other consequence. In the minor proposition we show
that neither of these consequences follows, and so conclude
that the antecedent is false, e.g. : —
If A is B, either C is D or E is F.
But neither C is D nor E is F :
Therefore, A is not B.
The dilemma has been known from time immemorial as
the "homed syllogism," because in the major proposition
the alternatives assn/med * to be exhaustive are opposed like
'•horns" to the opponent's position. The opponent's assertion
is, in the minor proposition, thrown off each "horn," and
finally rejected in the conclusion.
• The student should notice the expression " assumed to be exhaustive."
Dilemmatic arguments are often fallacious because all possible alterna-
tives have not been exhausted. The fallacy in this is exposed by the
construction of another dilemma equally to the point, but which gives
an opposite conclusion. This method is called " rebutting a dilemma."
Thus the complex constructive dilemma as given above might be
rebutted thus :—
If A is B then C is not D or if E is P then G is not H.
But either A is B or E is P.
Therefore either C is not D or G is not H.
Compare this with the following dilemma and its corresponding
rebutting dilemma. A mother is advising her son not to enter public
life and argues thus:— "If you act justly men will hate you, if you act
unjustly the Gods will hate you; but you must either act justly or
unjustly ; therefore public life must lead to your being hated." The sod
replies by a rebutting dilemma :— " If I act justly the Gods will love me.
and if I act unjustly men will love me ; therefore in either case I shall b»
beloved."
EXBBCI8EB.
EXERCISES ON CHAPTER XIII.
1. Assuming the truth of the statement tlmt, (a) if Ais B, C is D,
say what inference, if any, can he drawn from each of the following
further statements : (6) hut A is B ; (c) hut A is not B ; {d) hut C is
D ; {e) hut C is not D. If, in any of tlie cases, no inference can he
drawn, give the reasons. Illustrate your answer hy examples.
2. What is meant hy a disjunctive syllogism, and what conclusion
does such a syllogism yield ?
3. Express the following in the form of a dilemma : — Examhia-
tions are either needless or useless : for, if students are industrious,
they are needless ; and, if students are idle, they are useless.
4. Show that denying the antecedent or granting the consequent of
a condition involves logical fault, if tlie argument he expressed in
syllogistic form.
5. Examine the following dictum of the Caliph Omar, addressed
to the custodians of the Alexandrian Library in 640 A.D. : —
^* If your hooks are in conformity with the Koran, they are
superfluous ; if they are at variance with it they are pernicious.
But, they must either be in conformity with the Koran, or at
variance with it. Therefore, they are either superfluous or
pernicious."
CHAPTER XIV.
Fallacies of Deduction.
Thb study of logical forms, besides being a useful mental
discipline, supplies a ready test for the detection of fallacies.
Indeed, formal logic may be said to exist as a practical study
for this purpose. The syllogistic forms may be regarded
as a framework in which all our concrete reasonings may be
unfolded or displayed, and one by which their weak points
may be more readily discovered.
A Fallacy is a pcasoning apparently correct, which, neYer-
theless, InYolYes inconsistency in inferential thought. The
conclusion appears to follow from the premisses, but in reality
it does not. We do not class palpable, downright blunders as
fallacies. A fallacy is an error so wrapped up in words that
the mistake is not at once perceived, and thus tends to produce
conviction. Hence, the work of defining and exempHfying the
different kinds of fallacies is in one respect the chief end of
the science. But fallacious reasoning is so diverse that it is
impossible to exemplify every variety of it. Nor is it possible,
sometimes, to decide to what class a given fallacy ought to
be referred. For fallacies, like consistent reasonings, are mostly
expressed elliptically, and it is not always clear what the
xmabridged reasoning is supposed to be. Thus, when a person
argues ** that a country is ill-governed, because misery prevails
there," the unabridged syllogism may take two different forms
nTTBHNAL FALLACIES. 89
' neither of which is correct. (1) It may have for its omitted
premiss, " All miserable comitries are ill-governed," which no
reasonable opponent would admit ; or, (2) the omitted premiss
maybe " Every ill-governed comitry is miserable," in which
case the conclusion is invalid, for the middle term has not been
distributed in either premiss. Again we do not consider wilful
attempts to deceive as fallacies. To such attempts we apply
a stronger term. When people who know the truth but
suppress it by suggesting a wrong explanation {suppressio veri
et suggestio falsi) ^ this is moral not logical error. In short,
a dishonest intention will evade all rules of Logic.
Ordinary common sense is competent to expose most
fallacious reasonings by its own sagacity. But it not infre-
quently happens that common sense is aware of a fallacy in
the course of argumentation, without being able to say exactly
what is wrong. Arguments are felt to be wrong, but those
unskilled in logical science are puzzled how to demonstrate the
error or how to refute the fallacy. Logic suppHes the needful
help to enable students to localise and expose the error. It
makes the student familiar with the common form of unsound
inference. It keeps the attention fixed on the essential steps
of all valid reasoning. It accustoms the student to mark
accurately the exact meaning of terms used, and the relation
of these terms to one another. And it shows the necessity of
defining with precision the question in dispute. After a course
of discipline like this, the mind forms a spontaneous habit of
accurate judgment and self- consistent thought and reasoning.
Fallacies are usually divided into two classes : —
1. Internal Fallacies, where the unsound element appears
in the mode of expression. These are called fallacies ''in
dictione."
All internal fallacies may be detected even by those who are ignorant
of the matter to which the reasoning relates.
90 INTERNAL FALLACIES.
Internal fallacies are subdivided into
(i.) Purely formal fallacies, which are a breach of one or other
of the rules of Logic.
(ii.) Yerbal fallacies, in which the error lies in some ambiguity
in the words used.
(i.) Purely formal fallacies are breaches of one or other of the
rules governing mediate and immediate inference. All
that is needed here is to remind the student of the
most obvious pitfalls, viz. : —
(a) Confusion of contradictory with contrary opposition of
propositions.
{b) Simple conversion of A propositions.
(c) Syllogisms with an undistributed middle.
(d) Illicit process of the major or minor (see page 65).
(e) Arguing from two negative or two particular premisses.
(/) Neglect of the rules governing conditional syllogisms.
(ii.) Yerbal fallacies. These are often mere quibbles. The
following are the chief varieties of verbal fallacies : —
(a) Ambiguity of a word {equivocation), — A word is
sometimes used in a different sense in the two proposi-
tions of a syllogism in which it occurs, e.g., " Light is
always cheering ; some afflictions are light ; therefore some
afflictions are cheering." Obviously, the middle term
" light " is used in a double sense, and there are four terms
used instead of three.
(6) Ambiguity in the grammatical structure of a
sentence ( amphibology) , e.g., ** Twice two and three." This
is ambiguous, for the answer may be either seven or ten.
" What he was beaten with was what I saw him beaten
with. I saw him beaten with my eye. Therefore he
was beaten with my eye."
(c) Composition. This is the confusion of a universaJ
with a collective term. When we assert something of
EXTERNAL FALLACIES. 91
each and every member of a class, we may infer the same
of the whole class. When we say that all the angles of a
triangle are less than two right-angles, we use the word
"all" distributively ; but, when "all" is used collectively
the sentence is incorrect. We could not say that " all the
angles of a triangle taken together are less than two right-
angles.
{d) Division. This fallacy is the converse of the fallacy
of composition. What is said collectively may not be said
of the various individuals included in the collective term.
All the angles of a triangle taken together, are equal to two
right-angles, but no individual angle of a triangle is equal
to two right-angles.
(e) Fallacy of accent. This arises from the accent or
emphasis being thrown on the wrong word in a sentence,
e.g., "And he said ' saddle me the ass ' ; and they saddled
7iiw."
2. External Fallacies. — The error here can only be
recognised by those who are conversant with the matter about
which the statement is made. These are said to be fallacies
" extra dictionem.'* It is not easy to give simple examples of
them. When the wrongful argument is stated in simple
language, the error is easily seen. But, when the error is
diluted over a speech of an hour's length, it is more difficult to
detect it. The following are the chief varieties of external
fallacies : —
i. Many Questions (plurium interrogationum). This fallacy
is committed when several questions are so combined
into one, that, if you answer "yes" or " no," you are
committed to something more than your real meaning. A
man asks : " Have you ceased ill using your mother ? "
You would not care to answer " yes " or " no." Some,
times in a court of law, questions of this kind are asked,
and a plain answer "yes" or "no" demanded. Such
92 EXTBBNAL FALLACIES.
questions should be at once broken up into their several
parts and each part answered singly.
ii. Fallacy of the Consequent, better known by the familiar
phrase "now sequitur." This is the general name
given to loose and pretended arguments, where there is no
connection between the premisses and the conclusion.
iii. The Fallacy of Accident {A dicto simpUciter ad dictmn
secundum quid) is committed when we argue from a
particular case. Thus, " To take interest upon a loan is
just, therefore I do right to exact it from my own father
in distress." The answer obviously is, " Circumstances
alter cases." The converse of this fallacy is called " A
dicto secundum quid ad dictum simpliciter.'^ In this
case a statement is made in a certain sense, and then used
in quite another. Thus, " I eat to-day what I bought
yesterday. I bought raw meat yesterday, therefore raw
meat is eaten to-day." Here the accidental quaHfication
of "rawness" is added, whereas in the original premise the
assertion is made without regard to any such accidental
qualification.
iv. The Fallacy of False Cause {j^ost Jioc, propter hoc), where
it is assumed that because one event follows another,
the former event is the cause of the later. This is a purely
inductive fallacy, and will be considered later.
V. IrreleYant Conclusion {ignoratio elenchi). This is a
most important type of deductive fallacy. The name
covers all those cases in which a conclusion is proved, which
is really not the point in dispute, but which sufficiently
resembles what was required to be proved, to be often
mistaken for it. Scarcely any fallacy is so common or so
dangerous as this. Arguing beside the point, distracting
attention by irrelevant considerations, is as frequent as it
is misleading. The incoherence of the ignoratio elenchi
lies between the conclusion offered and the proper answer
EXTBBNAL FALLACIES. 38
to the question, but involves no breach of the rules of the
syllogism. There are four varieties of this fallacy which
should be noticed.
(a) The argumentum ad hominem. This is confusion a»
to what the point at issue really is. Thus, if a new law is
proposed, it is no proper argument to urge that the pro-
poser is not the right person to bring the question forward.
When we have advice given to us, it is not logic to retort
that the preacher should practise what he preaches. If a
man is accused of a crime, it is not relevant to assert that
the accuser is just as bad. In all such cases, the argument
proceeds not upon the merits of the case, but upon the
character of the persons engaged in it.
(6) Fallacies of objections. — We commit this fallacy
when we argue that a proposal should be rejected because
it is open to objections. Such an argument is always a
fallacy, if the alternative can be shown to be open to
greater objections or difficulties.
(c) Argumentum, ad verecvm^diam. — This is an appeal to
our respect for ancient or established authority. The
fallacy lies in the as'sumption that whatever is old or well-
established must ipso facto be good.
{d) The argwnent in support of a change is the opposite
of (c). The fallacy here is the implication that all change
is progress, whereas the contemplated change may
occasion more or greater evils than would follow if no
change were made.
The Surreptitious Assumption {PetiUo principii). — ^Every
example of deductive reasoning starts from some general
principle (major premiss) about which the disputants are
assumed to be agreed. If one disputant adopts as hig
premiss a statement which the other disputant does not
accept, the question at issue remains unsettled and no
conclusion can be drawn between them. " Begging the
94 THE DETECTION OF PAIiLACIBS.
question" and " arguing in a circle " are familiar forms of
the petitio princvpii. He who argues in a circle assumes
the truth of his major premiss and by means of it reaches
a conclusion, which he afterwards uses to establish the
major premiss with which he started. Thus, an illogical
divine might argue : " We know that there is a God,
because the Bible tells us so ; and we know that the Bible
is true, because it is the Word of God." People are
especially liable to fall into this fallacy when they use a
mixture of EngUsh and classical words in the same
reasoning. For they often seem to be proving one question
by another which is identical with it, only expressed in
words derived from another language ; e.g. : —
" Consciousness is the immediate knowledge of an object; for I
cannot be said to know a thing unless my mind has been affected by
the thing itself."
The detection of fallacies is such an important branch
of logical study that a few typical fallacies are appended,
with hints as to their solution.
1. Examine the folloiving : — " Every bird comes from cm
egg ; every egg comes from a bird ; therefore, every egg comes
from an egg.''
The premisses written in logical form are : " Every bird
is an egg-product; every egg is a bird-product," i.e., there
are four terms, whereas a correct syllogism can have
only three.
For exercise, test the following in the manner above
indicated: "Knowledge is power; consequently, since
power is desirable, knowledge is desirable."
2. Is the following a valid argument ? — " To assault another
is wrong ; consequently , a soldier who assaults o/nother does
wrong.''
THE DETECTION OP FALLACIES. 96
This is the fallacy of accidents. A soldier is a man
with an accidental qualification, and we cannot argue from
a general to a special in such a case.
Examine in the same way: " Intoxicants act as a poison
to a drunkard, and everyone should avoid poison."
3. ExcMmne the following : — " He who is most hv/ngry eats
■most ; he who eats least is most hungry ; therefore, he who
eats least eats most.''
This is the fallacy of accidents ; " eats most " is taken
generally in the conclusion, but specifically in the premiss.
4. Examine: — '■'■If Jack is a good boy he will do as he is
told; he is a good hoy {for, if he will do as he is told, he is a
good boy) ; therefore, he will do as he is told.''
A petitio principii — arguing in a circle.
5. Examine : — " The sea was the place where the incidents
of my tale happened; there is the sea; therefore, my story
is true.''
An ignoratio elenchi.
6. Examine: — "J[ dog chases a tortoise : the tortoise has a
hundred yards start, but the dog runs ten yards to every one
run by the tortoise. When the dog has run a hundred yards
the tortoise will he ten yards ahead; when the dog has
covered these ten yards, the tortoise will be one yard ahead;
when the dog has covered this one yard, the tortoise will be
^th of a yard ahead, and so on. The tortoise will be
always ahead and the dog will never overtake it."
This is an ancient specimen of an ignoratio elenchi.
The argument pretends to prove that the dog will never
overtake the tortoise ; it really proves that the dog passes
the tortoise between the 111th and 112th yards.
7. Examine : — " If I am to pass this examination I shall
pass it, whether I amswer correctly or not ; if I am not to pass
it, I shall fail whether I a/nswer correctly or not ; therefore, it
does not matter how I answer the questions."
96 EZEBCISISS.
Here it is tacitly assumed that "whether I answer
correctly or not " is not a link in the fated chain of events.
It is assumed that fate does not work through correct
answering of questions, and the conclusion is merely a
repetition of this assumption. It is the venera,tion of
"Fate" that draws away our attention from the error of
this delightful petitio principii.
EXEKCISES ON CHAPTER XIV.
1. Describe any three fallacies in Logic, giving an example in
each case.
2. Point out some of the ordinary forms of fallacy employed to
mislead in argument or in oratory, and illustrate the forms named.
3. Ea^lain and illustrate the terms " redv/itio ad absurdum" and
" begging the question."
4. Examine the following :
(a) You are not what I am; I am a man : therefore you are
not a man.
(b) A fish is a cold-blooded animal and breathes by gills ;
neither of these things is true of a whale ; therefore, it is
not a fish.
5. Explain exactly the nature of the fallacies called ''accident,*'
•• argumentum ad hominem,*' and " argum^ntum ad verecundiam.**
6. Show liow a logical training enables a student to detect
fallacies.
7. Explain and illustrate by examples the following terms : —
Ambiguity ; fallacy ; premiss ; suppressio veri et surgestio falsi.
CHAPTER XV.
Inductive Logic.
The fundamental lesson of Deductive Logic has been that no
conclusion may ever contain more than was contained in the
premisses from which it was drawn. Particular premisses, wl
saw, could not yield a general conclusion. The definition of
Inductive Logic, therefore, will seem at the outset a paradox.
For Inductive Logic may be defined as the Inference from
particulars to the general, or from the known to the unknown.
It is the establishing of general laws or principles from
observed particular facts or instances. John, Thomas, etc.,
individual men, are mortal, from which the general inference
is drawn that " All men are mortal." Here we have a
general conclusion about all men, derived from an indefinite
number of particular instances. At first it seems as if
our conclusion was overdrawn. The conclusion contains
more than is given in the premisses; it seems like a
leap in the dark. Modern science consists throughout of such
general conclusions, based on particular facts. Because certain
things resemble each other in certain observed ways, we assume
that they will resemble each other in certain previously un-
observed ways. But is this general assumption warranted ? Do
any number of observed facts warrant a general universal
conclusion. If the observed facts be two, or two hundred
particular cases, are we warranted in making any assertion
beyond the number observed ? What right have we to add to
98 INDUCTION.
what we actually observe, as we certainly do, whenever we
conclude, from seeing a number of particular events occur,
that they will always occur ? We say *' all animals die," but
we have not seen all animals die. Similarly, *' all bodies
gravitate," but our experience does not extend beyond
particular instances of gravitation. Yet we are certain that
these inferences are legitimate. What, then, is the ground of
this certainty? Why are we able to conclude that '* All must
be so-and-so," because we have observed that " Some are so-
and-so *• ; that " all " bodies gravitate because " some " have been
observed to gravitate ? This is the problem of induction. For,
we must observe that this process of induction is attended with
some perplexity. In some cases one single observation is
enough to warrant a general conclusion, whilst in other cases
we hesitate to draw a general conclusion from hundreds of
observed instances. Euclid takes a single triangle and shows
that its three angles are together equal to two right-angles.
We therefore accept this as a general truth applying to
triangles of every kind and everywhere. One single instance is
sufficient to establish a general rule. On the other hand,
though every crow I have seen is a black one, I should have no
hesitation in believing someone who told me that he had seen
a grey one. Whence come the certainty in the one case and
the uncertainty in the other.
Induction is based upon one great axiom, viz. : " the
COURSE OF NATURE IS UNIFORM." In othcr words nature is not
a chaos, it is an orderly system. Any event does not follow
any other event in a haphazard way. The relation of things to
each other is governed by what we call " law." This truth is
axiomatic in its nature. It is the assumption of all Induction
and of all science. It is not a truth which we can prove, nor
does it need proof. If any one cares to deny it, we can offer no
demonstration of it, beyond showing the denier that he himsell
acts upon the assumption every hour of the day.
CAUSE AND EFFBCT. 99
This axiom is practically the assertion that things are related
to each other by law, and the one general law, everywhere
observable, is that of cause and effect. The truth, that every
fact which has a beginning has its cause, is a truth coextensive
with human experience. It is needful, then, to have a definite
notion of what is meant by cause. "We may define it as
follows : —
A cause is that which immediately precedes any ohanob,
AND WHICH, existing AT ANY TIME AND IN SIMILAR CIRCUMSTANCES
HAS BEEN ALWAYS AND WILL BE ALWAYS FOLLOWED BY A SIMILAR
CHANGE.
In this sense of the word cause is synonymous with power,
property, or quaHty. Thus : " Water has the power, property,
or quality of melting salt " is the equivalent of " Water is the
cause of the melting of salt." Each statement means that
when water is poured upon salt, the solid is transformed into
liquid. Two parts of a sequence are thus before our minds (a)
the addition of water to salt, (6) the transformation of a
crystalline solid into a liquid. These are respectively cause
and efifect. The powers, properties, qualities, or causes of
things are not to be regarded as anything superadded to the
thing. These are not the things plus their powers, but things
alone. Things are the invariable antecedents of changes in
similar circumstances. The changes occur in an order or with
a uniformity, which we believe to be regular. It is this general
fact which enables us to reason about nature and to draw
general inferences. If the changes which we see continually
happening were chaotic, without uniformity, there could be no
reasoning about them, either inductive or deductive. Now, in
our actual experience, causal connections are mixed up with
casual or merely accidental coincidences. Even the un-
scientific man remarks that some sequences repeat themselves,
whilst others do not. He watches the sequences and the
coincidences happening within his range of observation. He
100 CAUSE AND EFFECT.
makes experiments. In this way he comes, ere long, to
separate causal or constant sequences from those which are
only casual and occasional. There are many practical
difficulties to be overcome, arising from the fact that
the same effect may be produced from several causes,
and that effects are often produced partly from one cause
ajid partly from another. Gradually, he learns the efficacy
of experiment for helping him over these difficulties.
Conjectures and hypotheses suggest further experiments.
"Vary the circumstances," is a rule which comniends
itself more and more as he proceeds. Thus he puts aside
immaterial circumstances which he finds to be casually and not
causally connected with the phenomena he is investigating.
Gradually, these casual antecedents and consequents drop off,
and the true causal antecedents and consequents reveal them-
selves. Then, at last, the enquirer has discovered truth. He
is no longer an enquirer or conjecturer, but he may claim to
have established a general law or a scientific induction. The
various methods which this sketch has suggested and which
are exemplified in the various sciences, are known as the
Canons or Eules of Valid Induction. Before these are examined
in detail, it is needful to consider in detail some of those pro-
cesses which are preliminary to induction, such as observation,
experiment and conjecture.
{For general exercises on Fart II see Page 111.)
CHAPTER XVI.
Ihe Preliminaries to Induction.
OBSERVATION, EXPERIMENT, CONJECTURE.
(Hypothesis.)
In the closing sentences of the preceding chapter we have
assumed a distinction between observation and experiment.
Observations in the wide sense of the term are either simple or
artificial, i.e.j they are conducted either with or without
interference on the part of the observer. In simple observa-
tion the facts are taken just as they offer themselves. In
artificial observation or experiment the spontaneous state or
occurrence of things is modified by the observer's will. The
phenomena to be observed, the effects and causes which are
to be investigated, are put in such new circumstances as
are most suitable for the detection of their causes and effects.
In OBSERVATION PROPER, wc watcTi nature's experiments;
in EXPERIMENTAL OBSERVATION wc interfere with nature's
experiments, in order to make others of our own.
Experimental observation is obviously a powerful auxiliary
in the search for causes and effects in nature. It is simply the
outcome of the old rule, " vary the circumstances," i.e., vary
Bhe circumstances which surround the object whose causes or
effects you wish to ascertain. Each fresh experimental
102 HYPOTHESIS.
variation is a new opportunity for getting rid of the casual
or companion circumstances, and for recognising the really
constant or causal ones. Contrasting observation and experi-
ment, the latter seems by far the more potent instrument.
But certain things must be borne in mind. When we are
endeavouring to ascertain what the effects of a given cause
are, we may use experiment as freely as we choose. But, in
tl^e reverse process, i.e., in ascertaining what is the cause of
a given effect, experiment is not always a safe guide. We
may take any given cause and see what effect it will produce,
but we cannot always take an effect and try experimentally
what will produce that effect. If we do so, we can only
conclude that what we discover is one way of producing the
effect out of many possible ways. So, too, in some cases we
are shut out from experimental methods. In many applications
of the science of medicine we have to rely upon observation
entirely.
Conjecture or Hypothesis. — This is an important auxiliary
in the search for the causes and effects of things. An hypo-
thesis is a provisional supposition about the true relation of
things. But we may not suggest hypotheses at random.
There are certain reasonable conditions to which hypotheses
must conform in order to be entitled to rank even as con-
ditional explanations. An hypothesis is in itself a provisional
conjecture, insufficiently supported by evidence. It is legiti-
mately made in order that we may compare with the actual
facts of the case, what would be the facts if it were well-
foxmded; and, in proportion as it yields a reasonable result
or the contrary, we may accept or reject it.
A dogmatic hypothesis is a conjecture not sufficiently
supported by evidence, which we are asked to receive as an
estabhshed truth. A suggested hypothesis is a provisional
conjecture not sufficiently supported by evidence, which we
are asked to try or test by means of the evidence.
HYPOTHESIS. 103
The conditions of a legitimate hypothesis are : —
1. It must not be abready known to be, or even strongly
suspected of being, untrue.
2. It must be of a nature to admit of proof or disproof,
for verification.
3. It must be adequate to explain aU the phenomena of
which it is offered as the explanation.
Hypotheses cannot be regarded as established general
truths about nature, until they have conformed to tae require-
ments of one or other of the Inductive Canong.
CHAPTEE XVII.
The Inductive Canons.
Thb general rules or conditions of successful search for causes
and effects are the essence of Inductive Logic. They are to
this branch of the subject what the syllogism is to Deductive
Logic. They were first laid down in Bacon's Novum Orga/num.
For the purposes of this elementary treatise it will be sufficient
if we discuss the two fundamental canons known as (1) the
Method of Agreement and (2) the Method of Difference.
1. The Canon op Method of Agreement. — "When all
the antecedents of an effect except one can be absent
without the disappearance of the effect, that one is causally
connected with the effect, due precautions having been
taken that no other circumstances have been present
besides those taken account of."
The principle involved here is obvious. Whatever can be
excluded from a sequence without affecting the phenomenon
whose causes or effects we wish to ascertain, cannot be causally
connected with it. Let a, b, c, d, e and / be circumstances
observed on some particular occasion to attend some event
which we will call x. We wish to find the cause of x. And,
to do this, we must ascertain whether any of the given
circumstances, a, b, c, etc., aje causally connected with x. To
settle this point we watch the occurrence of the phenomenon x
again and again in a well selected variety of circumstances.
In the first case we noticed that a, 6, c, d, e and / were all
CANON OF AGREEMENT. 106
present. In the next case, perhaps, a was absent but all the
rest were present. In another case h was absent but all
the rest were present, and so on. But after many variations
of the circumstances in all of which x occiirs, we may find
that one, say /, is a never-failing antecedent. All or any of
the others may be absent, but whenever x is observed we find
/, and / seems the only material circumstance. Hence we
conclude that / is causally, and not merely casually, connected
with X. whose cause we were seeking. Let us illustrate this
by a concrete example : There has been an outburst of
typhoid fever, and we wish to ascertain its cause. We take
as many cases of the occurrence of the fever as possible. We
notice the following points : —
{a) The cases occurred in different streets of the town.
(6) The ages of the victims differed.
(c) Their occupations differed, and so. on.
But amongst all the circumstances there was one which was
common to every case, viz., all the patients had drunk milk
coming from one farm. Comparison of the different circum-
stances enables us to say that (a) the locality, (6) the age of
the sufferers, (c) their occupation, are not causally connected
with the outbreak ; but that the milk, being the only circum-
stance common to all, most probably is the cause.
Observe that we say " most prohably." To be absolutely
certain we should have to know that the given circumstance
was the only material one in which all the cases
agree. Although in many cases we may be certain enough for
all practical purposes, we may not be logically certain. The
Canon of Agreement, therefore, only enables us to clear away
an indefinite number of casual, immaterial companion circum-
stances. But, it only more or less probably assures us that the
residue is alone causal. We have to allow for the plurality of
causes, i.e., several causes producing or combining to produce
the same effect.
106 CANON OF DIFFBRKNCE!.
The Method of Agreement is mainly, though not exclusively^
one of observation rather than of experiment. It is applied
more frequently and successfully to enquire into the causes of
given efifects, than into the effects of given causes. As we
have before remarked, to find the effect of a given cause,
experiment is better than observation ; but to find the causes
of a given effect, observation and its Method of Agreement are
usually safer guides than experiment.
2. The Canon or Method of Difference. — " When the
addition of an agent is followed by the appearance of a
certain effect, or when the subtraction of an agent is
followed by the disappearance of a certain effect, no other
material circumstance having been added or subtracted at
the same time or in the meanwhile, and no change having
occurred among the original circumstances, that agent is a
cause of the effect."
We have here two sets of circumstances, and we know that
they differ from one another in one, and only one essential
particular. What this essential particular is, we also know.
Now, whatever happens in the set of circumstances, in which
the given particular occurs, and which does not happen in the
set of circumstances where the given particular is absent, must
be due to the given particular. In other words it is the cause
of t\e effect observed. To take an illustration with which the
studtent will probably be familiar : a feather and a coin are
suspended in the receiver of an air-pump, from which all air
has been exhausted. They fall to the bottom at the same
moment. Air is introduced into the receiver : the feather
flutters to the bottom at some mterval after the coin. Here
the phenomenon under consideration is the retardation of the
feather. This is an observed effect of which we desire to find
the cause. The exhausted receiver is one set of circumstances ;
the receiver with air introduced is the other. The presence or
THE JOINT METHOD. 107
absence of air in the receiver is the only particular in which
the circumstances differ. The phenomenon of retardation,
oceuring in one case but not in the other, is at once described
as the effect of which " air " is in some way the cause.
It is quite possible to express the reasoning of the two
canons in the form of a syllogism. The illustration given
might be expressed in general terms as follows : —
^^ All cases of observed or experimentally produced
sequence which fulfil the conditions of the canon of
difference must he cases of constant or causal sequence
cmd not mere coincidences (Major Premiss).
These cases are cases in which the conditions required
hy the canon of difference are realised (Minor Premiss).
Therefore, these cases are cases of causal connection
(Conclusion).
In these days every teacher is obliged to pursue some
amount of experimental work, and so becomes familiar with
the logical principles involved in reasonings which obey the
conditions of the canons of induction. In the constant search
for the causes of effects, and the effects of causes, cases arise
to which the two great canons, which we have already
considered, are not directly applicable. Hence other canons
have been formulated which must now be discussed.
3. The Canon of the Joint Method of Agreement and
Difference : or, as it is more accurately called.
The Canon of the Joint Method of Agreement in Presence
and in Absence. — " If two or more instances in which the
phenomenon occurs, have only one circumstance in
common, while two or more instances in which it does not
occur have nothing in common save the absence of that
circumstance ; the circumstance in which alone the two
sets of instances differ, is the effect or the cause or an
indispensable part of the cause of the phenomenon."
108 THE JOINT METHOD.
This is the form in which the canon is usually given, but
it will become somewhat more intelligible if stated a little
more fully.
If two or more instances in which a phenomenon occurs
have only one other circumstance (either antecedent or
consequent) in common :
whilst two or more instances in which it does not occur
(though in some important points they resemble the
former set of instances) have nothing in common save
the absence of that circumstance :
then the circumstance in which alone the two sets of
instances differ throughout {i.e., being present in the first
set and absent in the second set) is the effect or the
cause, or an indispensable part of the cause of the
phenomenon.
Although this canon reads somewhat complicated and
difficult, it is as a matter of fact, only the precise statement of
a form of reasoning which is in constant use. This will appear
from a simple illustration. A man observes that whenever
he eats cucumber he suffers from indigestion afterwards.
Now by the method of agreement he might infer that the
cucumber was the cause of his discomfort. But perhaps he is
specially fond of cucumber, in which case he may endeavour
to lay the fault of his sickness upon the salmon or the cheese,
or something else that he had eaten along with the cucumber.
But if he is a wise man he makes for himself a fresh set of
instances where he has eaten cheese, salmon, etc., but no
cucumber, and if he finds that on these occasions he has not
suffered from indigestion, then he is bound to conclude that it
w^as the cucumber that alone was responsible for the indigestion.
If the canon of the Joint Method is now' read along with
this simple illustration, the student will have no difficulty
in grasping the steps in the reasoning.
THE METHOD OF CONCOMITAITr VAEIATIONS. 100
It should be noticed that the Joint Method like the Method
of Agreement rests mainly on observation, and a high degree
of probability is the utmost that can be generally inferred
by its use. It has, however, this special advantage over the
Method of Agreement, that if the second set of instances,
in which the phenomenon and its supposed antecedent
(the indigestion and the cucumber in our illustration) are
both absent, can be made exhaustive, then any hypothesis
of a plurality of causes is precluded.
The principle of the Joint Method may be summed up
in two propositions worth remembering : —
(a) That which is not followed by a given event is not
the cause ;
(6) That which cannot be left out without impairing a
phenomenon is a condition of it.
4. The Canon or Method of Concomitant Yapiations. —
" Whatever phenomenon varies in any manner whenever
another phenomenon varies in some particular manner,
is either a cause or an effect of that phenomenon, or is
connected with it through some fact of causation."
This method is in reality a special case of the previous
canons for use in cases to which they cannot be applied
in their entirety. There are, for example, certain forces
which can never be entirely eliminated, and consequently
it is impossible to obtain negative instances. Thus we cannot
entirely deprive a body of the whole of its heat. We can
therefore only reason .about the effects of heat by making
changes in the amount of heat in a given body. If when the
quantity of heat is varied we ascertain that there are con-
comitant changes in the accompanying circumstances, then we
are able to establish the relation of cause and effect between
the varying amount of heat and the attendant circumstances.
110 THE METHOD OF CONCOMITANT VARIATIONS.
Thus, if on the occurrence of friction we lind the temperature
of a body increased, say ten degrees, and ascertain that
besides the friction no other circumstance affecting the body
has changed, then we are justified in concluding that the
friction has been the cause of the rise in temperature. The
law of the expansion of bodies of heat was ascertained by this
method of reasoning.
The method of concomitant variations may now be
illustrated quite generally : —
Let A and a be the two phenomena under consideration,
and let A', a\ and A", a!' represent corresponding alterations
(whether of increase or decrease) of the phenomena. Now
suppose we have three sets of circumstances, ABC, A' D E,
A" F G, with corresponding phenomena ah c, a' d e, a!' /, g.
Now the one thing only in which the two sets of circumstances
agree throughout, is that any alteration in A {i.e., A' or A") is
followed or accompanied by a corresponding alteration in a
{a' or a!'). From this we infer that most probably A is the
cause of a. We say " most probably " for in the second case
D E might be the cause of the alteration of a to a', and in the
third case F G might be the cause of the alteration of a to a".
But if after many trials it is found that A and a always vary
together, then the probability of their causal connection
becomes more and more a certainty. Yet just because it is
nearly always possible that some unobserved cause is the real
determinant of both A, a, and also their concomitant variations,
absolute certainty can never, theoretically, be attained by this
method.
These four methods are the reasonings which are employed
when in the course of investigations we attempt to eliminate
mere casual connections present with those which are related
as cause and effect. After definite progress has been made by
the use of these methods of elimination, further problems are
greatly simplified by subtracting from any complex sequence
THE METHOD OP BBS1DUK8. 113
what has already been found to be the influence of ascertained
causes. This process of subtraction or simpHfication is known
as the method of residues.
5. The Canon or Method of Residues. — *' Subtract from
any phenomenon such part as previous induction has
shown to be the effect of certain antecedents, and the
residue of the phenomenon is the effect of the remaining
antecedents."
The reasoning here is quite simple. Suppose that the
antecedents A, B, C, D, are followed by the consequents
a, 6, c, d, and that by previous inductions it has been
ascertained that B is the cause of b, C of c, D of d. Then by
subtraction we infer that A is the cause of a.
It has been by this method that many of the elements of
Chemistry have been discovered. Quite recently it was
observed that nitrogen obtained from the atmosphere was
slightly heavier than that obtained by ordinary chemical
manipulation. This excess in weight was an instance of a
residual phenomenon for which the cause must be sought
in some peculiarity of the atmosphere. Lord Eayleigh
investigated this and so discovered argon, a new element
in the air, which had been present along with the nitrogen
obtained from the atmosphere, and which accounted for
the difference in weight.
Such, then, are the five Canons of Induction, or the logical
conditions which regulate inferences about the laws of nature,
and in particular the law of causation. They are calculated
contrivances for finding out when causal connection really
exists, and all scientific work exemplifies these rules. The
exact words of each canon should be committed to memory,
and the student should be prepared to give one or more
concrete illustrations of each method.
112 THE PLURALITY OF CAUSES.
It remains now to consider briefly the two conditions
which, separately or together, tend to frustrate the methods
in their practical application to the phenomena of nature.
These are : —
1. The fact that only comparatively few ej^ects in-
variably follow one set of antecedents alone ; and
2. It is only in comparatively few instances that a
single effect can he Tcept apart and distinguishable.
In short, there is a Plurality of Causes on the one hand, and
an Intermixture of Effects possible on the other.
1. The Plurality of Causes.— This we have already seen is
the special weakness of the Method of Agreement. Thus, for
example, if the phenomenon of heat in a given body were
mider observation we could not with certainty infer that the
particular amount of heat under observation was due to one
cause (say friction) because, by the Method of Agreement, it
had been established that friction was always accompanied by
an increase of heat. For in this particular case the definite
quantity of heat might have been the result of combustion,
the solar ray, electricity, etc., or it might have been the result
of several of these combined. The remedy for this inherent
weakness of the Method of Agreement is to multiply instances
as much as possible, and if practicable to apply the Joint
Method. The multiplication of instances enables us to
ascertain, possibly, all the causes which produce the effect, and
then it is easier to say which of these could have been present
in any special case and which of those actually present were
free to operate to produce the effect. If, then, we could
further apply the Joint Method and discover cases in absence^
the conclusion would be decisive.
2, The Intermixture of Effects. — In nature the effects of
various causes seldom remain separate and distinguishable.
INTBRMIXTUEE OF EFFECTS. 113
More frequently is it that the effects of various causes unite in
a single homogeneous total. For example, a good crop is a
single effect, but the causes which have united to produce it
are very numerous. Each cause has had its own effect, but
the separate effects are united to form one single result. In
such cases the Method of Concomitant Variations has a
peculiar advantage. For when, amid a variety of causes, one
cause happens to vary alone, we know that its effect will vary
alone also. When this has been sufficiently observed, then
cause and effect may frequently be singled out under circum-
stances ot great complication.
CHAPTER XVm.
Arguments Similar to Induction.
I. Analogy. — In ordinary life we are often as much
obliged to act upon what is probably true as upon what we
know is certainly true. Analogy is a form of reasoning which
aims only at giving more or less probable certainty. If we
find two things closely resembling each other in certain
observed ways, we argue that they will probably resemble each
other in ways which we have not observed. This is the
formula of analogy. Induction argues : " These sequences
have been found in some instances, therefore they will be
found in all instances." Analogy argues : " These two things
resemble each other in certain quaUties, therefore they probably
resemble each other in other quaUties. Some of the planets
are known to resemble the earth in certain respects, therefore
they probably resemble the earth in being inhabited." Butler's
great work on the " Analogy of Religion " argues that, because
nature and revealed rehgion have many resemblances, there-
fore it is probable that they have a common Author.
Logic lays down the following rules for good analogical
reasoning : —
1. The ratio or proportion in nimiber of resemblances
must be contrasted with the number of known differ-
ences. If the former are many and the latter few the
IKDUCTIVE FAIiliACIEB. 115
analogical conclusion is increased in probability, and vice
versa. N.B. — If one of the things about which we are
arguing is only little known, the unknown points must be
added to the points of difference in contrast with the
resemblances. Thus, the argument about the planets
being inhabited is weakened by the fact that we know
very little about them.
2. The kind of resembling and differentiating circum-
stances must be carefully considered, and the general
result compared with what we know of the laws of the
universe. Thus, in the case of the planets, we know that
life as it exists on the earth can only exist within certain
definite limits of temperature and in connection with
atmospheric air. Mercury is too hot, Saturn is too cold,
whilst the moon has no atmosphere. All the resemblances,
therefore, count for nothing when we consider the kind of
differences that exist.
II. Inductio per enumerationem simplicem. — This argument
is a kind of inductive fallacy. It argues that, because a case
happens to be true in every instance in our experience, there-
fore it is a general law or truth. Before we can assume that
a thing is universally true, because we have never known an
instance to the contrary, we must have reason to suppose
that, if there had been instances to the contrary, we should
have heard of them.
III. Post hoc, ergo propter hoc— This is the fallacy of
Induction. It is the confusion of casual with causal connec-
tion, against which the Inductive Canons sure designed to guard
us. Thus, we have a National Debt and we have national
prosperity. We are arguing post hoc ergo propter hoc, if we
ascribe the prosperity to the Debt. " After, therefore because
of " is the generic name for imperfect proof of causation from
observed facts of succession.
116 DKDUCTION AND INDUCTION.
IV. Perfect Induction is the name given to the conclusion,
when all possible cases have been duly examined, and we have
summarised the result in a general proposition. It, however.
Induction is defined as an inference from the known to the
unknown, it is obvious that "perfect induction" is really no
induction at all.
Bblation of Deduction to Induction.
Having thus briefly considered the aim and scope of
Deductive and Inductive Logic, it only remains to get
Into clear perspective the relation between the two. Some
modern logicians, seeing the vast practical importance
of Inductive Logic as the logic of the physical sciences,
have been led to doubt the value of Deductive Logic
altogether. They argue that the syllogism is only a
petitio principU. When we argue that, because all men
are mortal, Socrates, being a man, is mortal, the conclusion
was "begged" in the general proposition. But the number
and variety of fallacies of deduction which abound in ordinary
life, are a sufficient warrant to ensure the study of Deductive
Logic a permanent and important place in a liberal education.
The simplest way of expressing the relation between the two
branches is to consider Inductive Logic as the orderly state-
ment of those laws by which we arrive at general conclusions.
The general conclusions have then validity, based securely on
the principle of the uniformity of nature and the aU-pervading
law of universal causation. The general conclusions become
a sort of memoranda in which our conclusions are expressed.
But these memoranda require to be correctly interpreted and
reasonably appUed to particular cases. This is the proper
work of Deductive Logic. In short, the one is the coimter^.
purt oj the other.
BXEBCIBE8. 117
EXEECISES ON INDUCTIVE LOGIC.
o
2. Explain and illustrate the difference between the Inductive
and Deductive methods of arriving at truth. What are the chief
dangers in reasoning from analogy? Explain and illustrate the
terms " redu^tio ad ahsurdum" and " begging the question."
2. Illustrate the statement that in all discoveries of natural
science, the processes of induction and deduction follow each other
before a complete verification of a law can be obtained.
3. What is the exact difference between inductive and deductive
reasoning ? Give a simple example of each process in connection
with some subject of instruction in an elementary school course.
4. Give some familiar examples of false induction, and say what
school exercises are best calculated to encourage a habit of making a
true use of the inductive process.
5. Distinguish between analogy and induction, hypothesis and
theory. What is needed besides induction for ascertaining scientific
truth t
6. By what processes of reasoning would you prove that the earth
is round, or that the room in which you are is not empty, but filled
with something ; or, by examining a bird that it was an animal
made to live in the air ? What name would you give to the process
in the last case ?
7. Distinguish between generalisation and reasoning from
analogy, and give an instance of each.
8. Distinguish between observation and experiment, and show
how we may learn by experiment what we could not learn merely
from observation.
9. "Induction is really the inverse process of Deduction,*
Explain thin.
118 BXBRCISKS.
10. What is meant by Inditctio per enumeration szmplicem ?
11. Why is so called "perfect induction " not considered a really
inductive process ?
12. State exactly what you understand by the terms *' Gav^e and
Effect,'' ani " the Plurality of Causes."
13. What is the meaning and significance of the pri/nciple known
as " the Uniformity of Nature " ?
14. Explain the principle of the Method of Agreement and the
Method of Difference respectively, and say to what uses the two
methods are appropriate.
Appendix,
MORE RECENT DEVELOPI^IENTS OF THE
SYLLOGISM.
Since the time of Bacon and Locke, it has been largely
the fashion to consider the syllogism as a worthless instrument
for the discovery of new truths. Observation and experiment,
conducted in accordance with the Canons of Induction, has
alone been considered by some as worthy of serious attention.
But in recent years the syllogism has again attracted to itself
many sympathetic students who have exercised their
ingenuity in extending and reconstructing its traditional
forms. These newer developments are not noticed in the
body of the present work, but some brief account of the
three most important of them is now added here.
1. The Intensive or Comprehensive Interpretation of
the Syllogism. — If the student will refer to page 46 of the
present work, it will be seen that every proposition was
regarded as an assertion respecting the logical extent of the
subject and predicate of the proposition. But the terms
which form the subject and predicate have connotation
(intension) as well as denotation (extension). Consequently
a proposition may be regarded as an assertion respecting
the logical intension or comprehension of the subject and
predicate. When extension alone is in question, the assertion
is made on the relation of classes to classes. When intension
is the point of view adopted, the assertion is grounded on the
relation of attributes to attributes. An attribute contained in the
predicate of an affirmative proposition must also be contained
in the subject. From one point of view every gtnus is
120 THE NEW ANALYTIC.
seen to contain its species ; at the other point of view every
species is seen to contain its genus. The former is the point
of view of the ordinary syllogism, i.e., the syllogism in
extension ; the latter is the point of view of the syllogism in
comprehension. Any syllogism may be interpreted in either
way. In order to bring a syllogism out of extension into
comprehension, the rule is :
Beverse the premises, and then read or interpret each
proposition intensively.
Thus the following syllogism in extension —
" All M is imder P
All S is under ]\I
.-. All S is under P "
becomes, by the apphcation of the rule, a syllogism m
intension —
"All S contains M
All M contains P
.'. All S contains P."
It is not at all likely that a student of elementary logic will
be questioned on this matter, but it is quite worth while to
remember that every reasoning in the syllogistic form may
be read either way, and that the student should understand
how to translate a syllogism in extension to one in compre-
hension.
2. The New Analytic. — A much more important event in
the history of the syllogism was the revolution suggested by
Sir W. Hamilton, and known as " the new analytic of logical
forms." When we use, e.g., an A proposition, such as "All S
is P," we mean that, amongst an indefinite number of things
represented by P, all things represented by S are included.
Hamilton considered that, in the precise language of logic,
every proposition ought to say clearly all that it is meant to
express. If this were done, then the ordinary A proposition
THE NEW ANALYTIC. 121
would be written " All S is some P." Thus the foundation of
Hamilton's endeavour was the express and independent
recognition of extensive quantity in the predicates as well as
the subjects of propositions. Hence the system was described
briefly as " the quantification of the predicate." Now, if this
is carried out in each of the recognised prepositional forms
A, E, I, O, we shall obtain four entirely new ones, as shown
by the following table : —
NEW CORKESPONDINQ FORMS.
U. All S is all P
1] No S is some P
Y. Some S is all P
w. Some S is not some P
USUAL FORMS WITH PREDICATE
QUANTIFIED,
A. All S is some P
E. No S is any P
I. Some S is some P
O. Some S is not any P
The four new forms have the symbols U i| Y w, corre-
sponding to the traditional A E I 0.
Considering briefly the four new forms, it may be remarked
that two of them, U and Y, are in frequent use in ordinary
language. Thus every definition is practically a U proposition ;
e.g., *' Europe, Asia, Africa, Australia, and America are all the
continents " ; "Common salt is the same as sodium chloride."
Again, any exclusive assertion is an example of a Y pro-
position; e.g., "Graduates only are eligible for the appoint-
ment," or " Some passengers are the only survivors." Since,
therefore, U and Y propositions are in ordinary use, it would
seem a valid contention that they should receive recognition
in Logic. Of course, it is quite possible to express a U
proposition in the older forms. Thus the proposition " All S
is all P " may be resolved into two A propositions, " All S is
P" and "All P is S," which, taken together, are equivalent
to it.
Whilst there seem practical reasons for the recognition of
U and Y propositions, it must be said that the same reason
does not hold good for the new propositions q and w. They
122 THE NEW ANALYTIC.
are theoretically possible, but they are seldom if ever found
in ordinary use.
If the principle of the quantification of the predicate is
adopted, many remarkable results follow. Every logical pro-
position becomes an equation between the two quantified
terms which it contains. The relation between the terms of
a proposition thus becomes one of co-extension, and this implies
that each of the eight kinds of propositions may be converted
simply. Again, when propositions with a quantified predicate
are combined to form syllogisms, it is possible to express the
whole syllogism equationally and without figure. Thus the
ordinary syllogism —
•' All patriots are brave.
Some persecuted persons are patriots,
,'. Some persecuted persons are brave,"
may be written as an equated syllogism as follows : —
" All patriots = some brave men.
Some persecuted persons = some patriots,
.'. Some persecuted persons = some brave men."
This equational theory of reasoning can also be developed in
moods and figures as well as in unfigured syllogism. It ia
quite beyond the scope of an Elementary Logic to work
out such development. But it may be interesting to note that
in this way we obtain 3 valid figures and 108 valid moods,
instead of the 4 figures and 19 moods recognised by traditional
logic. The advantages claimed for the new analytic are : —
1. The special rules of each figure are abrogated, and
their violation ceases to be illogical.
2. Eeduction of syllogisms, like the conversion of pro-
positions, ceases to be necessary.
3. Each figure is alike capable of expressing the relation
of the terms in the reasoning, whilst each figure discharges
a function specially its own.
THE NUMERICAL SYLLOGISM. 123
Granting all these advantages, it still remains to be con-
sidered whether the forms of the new analytic are more
convenient and useful as a framework for the unabridged
expression of our assertions and reasonings, and also as an
aid in the detection of fallacies. And further, are the advan-
tages — scientific and practical — great enough to counterbalance
the inconvenience of substituting it for that analysis that has
been generally received since Aristotle ? The general answer
to these questions given by the great authorities on Logic is
in the negative.
3. The Numerically Definite Syllogism. — Some logicians
contend that, besides the definite " All " and the indefinite
" Some," Logic ought to recognise definite arithmetical
quantity. They would consider the following examples quite
legitimate logical reasoning : —
" Two-thirds M is P
Two-thirds M is S
.-. Some Sis P."
*' Seventy per cent, of M are P
Sixty per cent, of M are S
.'.At least thirty per cent, are both S and P."
In neither of these cases is the middle term distributed in
the premises, but the conclusion is correctly drawn according
to the rules of arithmetic. De Morgan contends that in
such cases it is permissible to mingle the relations of self-
consistency with the principles of arithmetic, and no doubt,
if this is admitted, the variety and complication of the
syllogistic forms will be immensely increased. But it is
properly urged, on the other hand, that where numerical
evidence of the kind contained in the propositions which form
the two illustrations above is obtainable, then the compara-
tively indefinite arguments of logic are needless and out of
place.
INDEX.
J»<c
A Propositions
Absolute terms
Abstract terms
Accent, fallacy of
Accident, fallacy of
„ separable & inseparable
Accidents
A dido secundum quid
Affirmative propositions ...
Agreement, method of ...
„ and difference, joint
method of...
Ambiguity, fallacies of ...
Ambiguity of " all," etc. ...
,, „ terms
Amphibology, fallacy of ...
Analogy
Analytic, the new
Antecedent and consequent
Appendix
Argumentum ad liominem ...
Arpumentum ad terecunduvi
Aristotle's dictum
Art and Science
Begging the question
Canons of syllogisms
Canons, inductive
Categorematic words
Categorical propositions ...
Cause and effect
Causes, plurality of
Circle, arguing in a
Classification
Collective terms
Common terms
Composition, fallacy of ...
Comprehensive interpretation of
the syllogism
Conditional syllogisms ...
Concept
Conclusion of syllogism ...
Concomitant variations,method of
Concrete terms
Conditional propositions ...
,, syllogisms ...
Conjecture
Conjunctive syllogism
Connotation
Connotative terms
Consequent
,, fallacy of the...
Contraposition
Contrary opposition . .
Contradictory opposition...
Conversion, inferences of
PAGE.
PAGE*
. 46
Copula
.. 44
. 23
Cross division
.. 37
. 24
Deduction
15, 116
. 91
Definite syllogism, the numeri-
. 92
cally definite
.. 123
i 32
Definition
.. 34
. 82
„ of Logic
.. 12
. 92
Denotation
.. 28
. 45
Description
.. 35
. 104
Detection of fallacies
.. 94
Developments of the syllogism
.. 119
. 107
Dichotomy
.. 38
90
Dictum, de omni et nullo ...
.. 65
. 49
Difference, method of
.. 106
26,90
Differentia
.. 32
. 90
Diienima ...
.. 86
. 114
Disjunctive propositions...
.. 43
, 120
syllogisms ...
.. 84
. 83
Distributed, meaning of ...
.. 44
. 119
Distribution
.. 48
. 93
Division
.. 36
. 93
„ fallacy of ... ...
.. 91
. 65
. 14
Dogmatic hypothesis
.. 102
Effects, intermixture of ...
.. 112
. 98
E Propositions
.. 46
7? Propositions
.. 121
63
Enthymene
.. 79
. 104
Epicheirema
.. 80
23
Episyllogism
.. 80
41
Equivocal terms
.. 26
99
Equivocation
.. 90
112
Euler's diagrams 47
, 48, 64
94
Evidence
17
38
Excluded middle, law of ...
20
24
Exercises 11, 18, 21, 26, 33, 39,
40,50.
23
60, 68, 78, 78, 81, 87, 96, 117, 118
91
Experiment
.. 101
Extension and intension ...
.. 27
119
83
External fallacies
.. 91
15
Fallacies, detection of
.. 94
62
Fallacy
.. 88
f 109
False cause, fallacy of ...
.. 92
24
Figures of the syllogism ...
.. 69
43
Formal Fallacies
90
82
102
Fundamentum divisionis ...
.. 37
83
Genus
.. 29
27
Grammar
.. 17
24
83
Hypothesis
.. 102
92
Hypothetical propositions
.. 43
57
53
„ syllogism ...
.. 88
52
I Propositions
.. 46
55
Identity, law of
.. 20
ii.
INDEX.
PAGE.
Ignoratio elenchi 92
Illicit process 65
Immediate inference 52
Induction 16,98,116
Inductio per enumerationem sim-
plicem 115
Inductive Logic, exercises ... 117
Inferences 13
Infima species 30
Inseparable accident 32
Intension 27
Intense interpretation of the
syllogism 119
Intermixture of effects 112
Internal fallacies R9
Irrelevant conclusion
Joint method of agreement and
difference
Judgments
Law
Laws of thought
Legitimate hypothesis, conditions
of
Logic, derivation of the word ...
Major premiss
„ term
Many questions, fallacy of
Mediate inference
Metaphysics
Method of agreement
„ ,, agreement and differ-
ence, joint
„ „ difference
„ „ concomitant variations
„ „ residues
Middle term
Minor premiss
,, terms
Mnemonic lines
Modus ponens
,, tollens
Moods of the syllogism
Negative and positive terms
Negative propositions
New analytic, the
Non-contradiction, law of
Nonsequitur
Numerically definite syllogism ...
O Propositions
w Propositions
Observation and experiment ...
Obversion
Opposition, inferences of
Particular propositions
Per accidens
Perfect figure
„ induction
92
107
13
108
14
91
61
89
104
107
106
109
111
62
62
62
75
45
120
20
92
123
121
101
56
52
45
55
74
116
PAGE.
.. 56
.. 93
7
.. 112
.. 80
.. 31
Parmutation, inferences of
Petitio principii
Philosophy and its branches
Plurality of causes
Polysyllogism
Porphyry, tree of
Positive terms 24
Post hoc, vropter hoc 115
Predicables 29
Predicate, quantification of ... 121
Premiss 61
Proper terms 23
Property 32
Proposition 13, 42
Prosyllogism 80
Proximate genus 31
Psychology 9
Quality of propositions 44
Quantification of the predicate... 121
Quantity of propositions 44
Beductio ad absurdum 77
Reduction 74
Relation of deduction and induc-
tion 116
Relative terms 25
Residues, method of Ill
Rules of the figures of the syllo-
gism 72
Rules of the syllogism 63
Science 12
Simple apprehension 15
Singular terms 23
Sorites 80
Species 29
Subaltern opposition 53
Sub-contrary opposition 53
Sufficient reason, law of 21
Summum genus 30
Syllogism 63
„ intensive interpretation
of 119
,, more recent develop-
ment of 119
„ the numerically definite 123
Syncategorematic words 23
Terms 17,22
Thought 12
U Propositions 121
Undistributed, meaning of ... 44
Uniformity of nature 98
Universal propositions 45
Univocal terms 26
Variations, concomitant, method
of
Verbal fallacies
Y Propositions
109
90
121
Printed by Henry Palmer & Co., London.
flu ElBmeiitaig & Intermediate fllgeHia.
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