Christian Culture and Philosophy Series
GENERAL EDITOR

Donald A. Gallagher, Ph.D.
Professor of Philosophy
Villanova University

CPS
The colophon for the Christian Culture and Philosophy Series
consists of the Greek letters X and P, symbolizing Christianity, em­
braced by C, a variant of S, representing the Greek word sophia.

THE SCIENTIFIC ART OF

AN INTRODUCTION TO THE PRINCIPLES
OF FORMAL AND MATERIAL LOGIC

EDWARD

D.

SIMMONS

Associate Professor of Philosophy
Marquette University

THE

BRUCE PUBLISHING

MILWAUKEE

COMPANY

NIHIL obstat:
John E.

Twomey, S.T.L., Ph.D.
Censor deputatus

imprimatur:
►p William

E. Cousins
Archbishop of Milwaukee
March 14, 1961

Library of Congress Catalog Card Number: 61-12989
© 1961 The Bruce Publishing Company
MADE IN THE UNITED STATES OF AMERICA

To my Mother and Father
Teachers Par Excellence

Preface
This text in logic is one of the first contributions to the new Christian
Culture and Philosophy Series. The book is designed generally to serve
the end of the Scries, and particularly to make available to undergraduate
students and their instructors an elementary', but scientific, presentation
of the principles of both formal and material logic. The order follows
the division of logic into the logic of the first, second, and third operations
of the intellect. In each section, significant logical relations, both formal
and material, are examined. Since any scientific inquiry requires hard
intellectual labor, it is inevitable that a scientific presentation of logic
will entail some difficulty for the student. Yet this is as it must be. Formal
logic is easier than material logic, but formal logic alone is an inadequate
instrument of the intellect for rational discourse. Since most students
take only one course in logic, it is imperative, despite the difficulties
involved, that in it they be introduced to material as well as formal logic.
Since scientific inquiry' is intellectually taxing, some might argue that
the art of logic could be acquired much more easily in an elementary
course apart from its science rather than along with it. However, the
art and science of logic are indistinguishably one. Unless the rules of
logical procedure are scientifically grounded in incontrovertible first
principles of the logical order, they cannot adequately serve as principles
either for a critique of or defense for scientific discourse. The propriety'
of a logical process could not be adequately defended by an appeal to
a rule of logic unless that rule were itself self-evident or scientifically
resolved into what is self-evident. Thus, it is an illusion to think that
the art of logic could be acquired in any adequate fashion apart from
the science of logic. Consequently, this text attempts to present scien­
tifically the basic principles of both formal and material logic. Its
proximate end is to generate in its users an intellectual habit which will
serve as an adequate instrument for rational discourse, especially in the
other sciences.
This book is not overly difficult, despite what has been said. The
point rather is that it is not easy —but neither is logic. It is rigorous —
so is logic. More to the point, it can be used successfully only by students
who are prepared to put some effort into their work — so too logic
cannot be acquired without effort. Every' attempt has been made to make
the presentation as straightforward as possible, given the intrinsically
rigorous character of the subject matter. The opening chapter is designed,
vii

viii

PREFACE

among other things, to give the student some appreciation of the nature
and divisions of logic so as to orient him for the course to follow. The
final chapter is devoted exclusively to the nature of logic. It is felt that
at least a semester’s work in logic is a prerequisite for any penetrating
analysis of the nature of logic. In both the opening and final chapters
there is a discussion of the division of logic into the logic of the three
operations of the intellect. It is according to this division of logic that
the book is divided into three parts. The opening chapter in each part
includes an investigation into the nature of the intellectual operation
from which that part gets its name. The remaining chapters in each part
take up the logical theory pertinent to the part in question. Throughout,
an effort has been made to offer sufficient examples so that the usefulness
of the logical theory under discussion can be seen in a concrete setting.
Each chapter is followed by exercises, which are designed to assist the
student to appreciate the meaning and force of the logical theory pre­
sented in that chapter. No teacher is ever fully satisfied with another
man’s exercises, and every teacher has some of his own to offer to his
students. However, the exercises suggested in this book are varied
enough, both in format and in degree of difficulty, so that every teacher
will find them to a greater or less degree of some help for his students.
It is the teacher, not the textbook, who determines the program of his
course. Any teacher may find that there arc things treated in this book
which he chooses not to include in his course. Thus, for example, a
teacher might choose to pass over the discussion of the truth-functional
proposition, or, perhaps, to omit the final chapter on the nature of logic.
Perhaps some teacher may choose to omit some of the more difficult
matter in some of the chapters. With this in mind several chapters have
been ordered so that the more difficult matter is treated separately from
the rest. Thus, in the chapter on relations between propositions, the case
of the singular proposition — which offers many difficulties — is taken up
separately. Again, since an elementary text cannot take up every question,
the author has omitted any detailed discussion of the types of analogy
and has discussed only the relations of the simply attributive categorical
proposition in the chapter on relations between propositions. z\ teacher
using this book may feel it necessary to supplement it with his own
treatment of the division of analogy into its types and/or his own
treatment of logically related modal propositions or compound proposi­
tions. Nonetheless, it has seemed to the author, after some years of
experience in the teaching of elementary logic, that the subjects treated
in this book are, for the most part, those which generally should and
can be handled in an elementary’ course covering a semester’s time.

PREFACE

¡X

Like many other textbooks, this is written within the context of the
Aristotelian tradition. Thus, it bears an understandably basic resemblance
to many other logic books. 'Hie rules of validity for the categorical syl­
logism have not changed since they were discovered by Aristotle. We are
not scandalized, then, to find them repeated faithfully from textbook
to textbook. They are not listed differently here. Yet this text does have
several distinctively different features. As we have said, it is not limited
to formal logic. Again it aims at the acquisition of the art of logic through
the science of logic and not apart from the science of logic. 'Hie second
part of the book includes a study of the hypothetical proposition.
Among the problems taken up in this chapter are the following:
Why cannot hypothetical propositions in the strict sense be truth­
functional? What is the significance of a truth-functional proposition?
How can symbols and even truthtables be profitably employed both
for hypothetical strictly taken and truth-functional propositions? In
the logic of the third operation the chapter on the demonstrative syllogism
and the consideration of self-evident propositions in the chapter on
induction represent treatments necessary for an adequate course in logic
which arc cither omitted or given scant attention in most logic textbooks.
The final chapter, on the nature of logic, is an attempt to investigate this
difficult subject matter on a level beyond that usually reached in logic
textbooks which speak of the nature of logic only at the beginning
of the book.
The book is within the Aristotelian tradition in the sense that, for
primary sources, it owes most to the Organon of Aristotle and to logicians
who have commented on the Organon and who have attempted to
develop their own logical theory from that of Aristotle. The chapters
on the categories, on the categorical proposition, on the categorical
syllogism, on the demonstrative syllogism, and on fallacious argumentation
owe most to Aristotle’s Categories, On Interpretation, Prior Analytics,
Posterior Analytics, and On Sophistical Refutations. Several chapters are
indebted to Aristotle’s Topics, especially those which treat of definition,
division, and dialectical argumentation. The chapter on the predicables
is first of all indebted to Porphyry’s Introduction to the Categories of
Aristotle. Other primary sources include the commentaries of St. Thomas
on the On Interpretation and Posterior Analytics of Aristotle, the com­
mentaries of both St. Albert and Cajetan on several of the logical works
of Aristotle, and John of St. Thomas’ Logical Art.
Tin's book could not have been written without the help of many
friends. They must share in whatever credit is forthcoming for the good
that may be in it. The author, who alone is responsible for the short­

X

PREFACE

comings, must first of all express his gratitude to Professor Yves Simon.
As a teacher beyond compare, Yves Simon has l>een the inspiration for
this book. It most certainly will not measure up to his standards, and
yet this has Ixxn the intention of the author. Many philosophy teachers,
including Professor Simon, have read preliminar}' versions of the book.
Their criticisms and suggestions have been most helpful and are most
certainly appreciated. Tins is especially true of the men in the Department
of Philosophy at Marquette University, some of whom have taught from
a preliminary version of the book. 'Hie author must also express his
gratitude to Marquette University for encouragement in this project and
for assistance in the form of reduced schedules and secretarial help, lire
author is especially indebted to Mr. William May of Tire Bruce Publishing
Company and Professor Donald Gallagher, Editor of the Christian Culture
and Philosophy Series, for their editorial assistance. A special vote of
thanks goes to Mr. James Campbell and Mr. Gerald Wahmhoff for
service above and beyond the call of duty while serving as graduate
assistants to the author during the days of composition. Finally —and
most of all — the author is grateful for the patience and active assistance
of his wife during the trying period of time in which this book was
in preparation.

Contents
Preface
chapter

I.

......

Introduction

vii

1

.....

i. The Role of Logic .....
1
The Distinction Between Sense Knowledge and
Intellectual Knowledge ....
2
in. ITic Three Operations of the Intellect
.
.
5
iv. Logic and Logical Relationships
v. The Divisions of Logic
....
9
Exercise I
.
.
.
.
.11
ii.

PART I:
chapter

II.

THE LOGIC OF THE FIRST OPERATION

Simple Apprehension: The First Operation

.

17

The Nature of Simple Apprehension
.
.
17
The Role of Abstraction in Simple Apprehension
18
in. The Nature of Signs and Their Import in Logic .
20
iv. A Comparison of the Three Operations of the In­
tellect in Terms of Some Significant Distinctions
23
Exercise II .
.
.
.
.
.24

I.

ii.

chapter

III.

Comprehension and Extension

.

.

25

I. The Comprehension of the Concept
.
.
25
ii. The Extension of the Concept
.
.
.29
in. ITie Priority of Comprehension Over Extension .
30
iv. The Inverse Ratio Between Comprehension and
Extension ......
30
v. Superiors and Inferiors
.
.
.
.31
Exercise III .
.
.
.
.
.33

IV. The Nature of the Universal .
.
.35
I. The Problem of the Universal
.
.
.35
ii. Abstraction by Way of Simple Apprehension and
Abstraction by Way of Negative Judgment
.
37
in. The Division of the Universal Into the Predicables
and the Categories .....
39
Exercise IV .
.
•
•
.
.39

chapter

xi

CONTENTS

xii
CHAPTER

I.

II.

in.
IV.

V.
VI.
VII.

VIII.

IX.

CHAPTER
I.

II.

III.
IV.
V.

VI.
VII.

CHAPTER

I.
H.
III.

IV.
V.

CHAPTER

I.

II.

in.
IV.
V.
VI.

.

.41

The General Notion of the Predicables
The Division of Universal Natures Into
Predicables
.
.
.
.
The First Predicable — Species
.
The Second Predicable — Genus .
The Third Predicable — Difference .
The Fourth Predicable — Property .
The Fifth Predicable — Accident .
The Limitations of This Division .
The Tree of Porphyry
.
.
Exercise V .
.
.
.

.
Five

.

.
.
.
.
.
.
.
.

.41
.43
.44
.46
.47
.49
.50
.51
.52

VI. The Categories

.

.54

V.

.

The Predicables

.

.

.

41

The General Notion of the Categories
.
Preliminar}' Considerations .
.
.
What Kind of Being Can Be Categorized .
The Division of Being Into Ten Categories
The Ten Categories Considered Singly
.
The Categories and the Predicables .
.
The Modes of Opposition .
.
.
Exercise VI .
.
.
.
.

.
54
.54
.
57
.
59
.
62
.65
.
66
.69

VII. Definition

.72

.

.

.

.

Definition and Division
.
.
.
.72
Nominal and Real Definitions
.
.
.73
The Four Causes
.
.
.
.
.75
Types of Real Definition ....
The Rules for Good Definition
.
.
.80
Exercise VII .
.
.
.
.
.82

VIII.

Division

.

.

.

Logical Division
.
.
.
Types of Logical Division .
.
Physical Division
.
.
.
Rules for Good Division
.
.
Codivision and Subdivision .
.
The Value of Definition and Division
Exercise VIII.
.
.
.

.

.84

.
.
.
.
.
.
.

.84
.85
.87
.88
.89
.
90
.92

xiii

CONTENTS

PART II:
chapter

IX.

THE LOGIC OF THE SECOND OPERATION

Judgment: The Second Operation

.

.

97

i. The Nature of the Second Operation of the Intellect 97
n. The Pre-eminence of the Second Operation
. 99
in. Judgment and the Motive for Assent
.
.
99
iv. Truth and Falsity .
.
.
.
.101
v. The Proposition and Its Elements .
.
.101
Exercise IX .
.
.
.
.
.104
chapter

I.
ii.
in.
iv.
v.
vi.
vn.
vm.
ix.
chapter

i.
ii.
in.
iv.
v.
vi.
chapter

i.
ii.
in.
iv.
v.
vi.
vn.

X. The Supposition of Terms
.
.
Signification and Supposition
.
.
Material and Formal Supposition .
.
The Types of Formal Supposition .
.
The Types of Real Supposition
.
.
Universal, Particular, and Singular Terms .
Collective and Divisive Terms
.
.
Possible and Actual Existence
.
.
Nonsupposing Subject-Terms
.
.
The Unity of Terms and Logical Discourse
Exercise X .
.
.
.
.

.106
.106
.107
.198
.109
.110
.112
.113
.114
.
115
.117

XI. The Categorical Proposition .
.
.120
The Difference Between the Categorical and Com­
pound Proposition
.
.
.
.120
Simply Attributive and Modal Propositions
.
121
The Division of the Proposition by Way of Quality
122
The Division of the Proposition by Way of Quantity
122
The Extension of the Predicate
.
.
.124
A, E, I, and O Propositions .
.
.
.126
Exercise XI .
.
.
.
.
.127

XII. The Compound Proposition .
The Nature of the Compound Proposition
Conjunctive and Hypothetical Propositions
'Hie Copulative Proposition .
.
Adversative and Causal Propositions
The Conditional Proposition
.
The Alternative Proposition
.
The Disjunctive Proposition
.

.
.
.
.
.
•
.
•

.129
.129
.129
.130
.132
.133
.135
.137

xiv

CONTENTS

Reduction of Alternative and Disjunctive Proposi­
tions to the Conditional .
.
.
.137
ix. Categorical Propositions Expressed as Hypothetical
139
x. The Symbolic Representation of Compound
Propositions
.
.
.
.
.139
xi. The Truth-Functional Proposition and Its Truth
Tabic....................................................................... 141
xii. A Schematic Résumé
.
.
.
.143
Exercise XII ......
144

vin.

chapter

i.
ii.
in.
iv.
v.
vi.
vu.
vin.
ix.
x.

XIII.

.

147

THE LOGIC OF THE THIRD OPERATION

XIV.

Reasoning: The Third Operation of the
Intellect
.....

i. The Nature of Reasoning .
ii. The Argument and Its Elements
in. Validity and Truth .
.
iv. The Division of Argument .
Exercise XIV
.
.

.
.
.
.
.

.
.
.
.
.

XV. The Nature of the Categorical Syllogism
I. TheDefinition of the Categorical Syllogism
ii.
The Elements of the Categorical Syllogism .
in. Direct and Indirect Conclusions
.
.
iv. The Figures of the Categorical Syllogism .
v. The Moods of the Categorical Syllogism .
vi. The Basic Principles for Categorical Syllogism

chapter

.

Propositional Relations
.
.
.
.147
The Nature and Types of Opposition
.
.148
Subalternation
.
.
.
.
.
152
The Square of Opposition .
.
.
.154
The Nature and Types of Conversion
.
.
155
Legitimate Conversions
.
.
.
.157
Obversion
.
.
.
.
.
.159
Contraposition
.
.
.
.
.161
Propositional Relations for the Singular Proposition
163
Résumé
......
166
Exercise XIII
.....
167

PART III:
chapter

Relations Between Propositions

173

.173
.174
.176
.178
.180

.

182
.182
.183
.184
.185
.188
.
189

CONTENTS

XV

vu. An Objection to the Categorical Syllogism .
.191
vni. The Expository “Syllogism” ....
193
Exercise XV .
.
.
.
.
.194
chapter

I.
ii.

in.
iv.
chapter

i.
ii.

in.
iv.
v.
vi.
vu.
vin.
ix.
x.

xi.
chapter

i.
ii.

in.
iv.
v.
vi.
vu.

XVI. The Rules for the Categorical Syllogism .
196
Tire General Rules for Any Figure .
.
.
196
Tire Special Rules for Each Figure of the Categorical
Syllogism .
.
.
.
.
.205
The Valid Moods of the Categorical Syllogism . 208
The Reduction of the Syllogism to the First Figure 213
Exercise XVI
.
.
.
.
.218
XVII. The Hypothetical Syllogism
.
.
222
The Nature and Types of the Hypothetical
Syllogism ......
222
The Figures and Moods of the Hypothetical
Syllogism .
.
.
.
.
.223
The Simple Conditional Syllogism .
.
.225
The Case of the “Disguised” Categorical Major .
227
The Reciprocal Conditional Syllogism
.
.229
The Inclusive Alternative Syllogism
.
.
.231
The Exclusive Alternative Syllogism
.
.232
The Disjunctive Syllogism .
.
.
.234
The Reduction of Alternative and Disjunctive
Syllogisms to the Conditional Syllogism .
.236
Hypothetical Syllogisms With a Multimembcrcd
Major
.
.
.
.
.
.237
Major Premises With Compound Components
.
239
Exercise XVII.......................................................... 240

XVIII. Complex Syllogistic Patterns
.
.
243
Complex Syllogisms .....
243
The Abbreviated Syllogism .
.
.
.243
The Syllogism With a Justification for Its Premises 245
The Polysyllogism
..... 247
The Sorites ......
248
The Dilemma
..... 251
Combinations of Complex Syllogisms
.
. 253
Exercise XVIII
.
.
.
.
.254

XIX. Demonstrative and Dialectical Discourse
i. The Syllogism Materially Considered
.

chapter

. 258
. 258

CONTENTS

xvi

n.
ni.
IV.
V.

VI.
VII.

VIII.

CHAPTER
I.

II.

III.

IV.
V.

VI.
VII.

CHAPTER
I.

II.
III.

CHAPTER
I.
n.

III.

IV.
V.

VI.
VII.
VIII.
IX.

258
260

Science and Demonstration ....
Prescicntific Knowledge
....
The Requirements for the Premises of
Demonstration
.....
Universality, Pcrscity, and Convertibility .
Types of Demonstration ....
The Middle Term in Demonstration
Dialectical Argument
....
Exercise XIX .
.

261
263
264
270
271
276

XX.

.....

280

The Insufficiency of Deduction
The Types of Induction
....
The Self-Evident Proposition
The Types of Self-Evident Propositions
The Induction of Self-Evident Propositions .
Mediate Induction .....
The Problem of the Inductive Enumeration
Exercise XX ......

280
281
282
284
286
287
290
294

XXI. Fallacious Argumentation

297

Induction

.

The Notion of the Fallacies .
.
.
The Fallacies of Language ....
Fallacies Beyond Language ....
Exercise XXI
.....

297
298
300
306

XXII.

309

The Nature of Logic

309
310
312
312

The Purpose of This Chapter
Art and Science
.....
Logic as an Art
.....
Logic as a Science .....
The Logic of the First Operation, the Logic of the
Second Operation, and the Logic of the Third
Operation ......
Formal and Material Logic ....
Logic as a Common Mode ....
Doctrinal Logic and Logic in Use .
The Question of Symbolic Logic .
Exercise XXII
.....

316
317
319
320
321
323

Index

325

THE SCIENTIFIC ART OF LOGIC

CHAPTER

I

Introduction

I. The Role of Logic

There arc several ways in which a man can come to know something
he has not known before. Tire most obvious of these can be described as
an immediate discovery' based upon a direct sensory experience. Suppose,
for example, that a man does not know that his new neighbor has
several children. He will certainly learn this if he sees his neighbor’s
children for the first time. Another way he might learn this is to be
informed of it by another neighbor who himself has seen the children.
To come to new knowledge in this manner, that is, on the testimony of
another, is to learn by way of faith. But there is still another way he
might learn that his new neighbor has several children. If, for example,
he secs several tricycles and bicycles of varying sizes in his neighbor’s
yard, he will know that in all probability his new neighbor has several
children. To leam something in this fashion, that is, as following from
something else which is already known, is to know by way of reasoning.
All men have a certain native ability to reason. No one would find it
difficult to conclude in the situation cited above that the man’s new
neighbor has several children. This native ability is known as natural
logic, and this suffices for most people in many instances of reasoning.
Nevertheless, natural logic is more or less limited and subject to error.
For example, it would ordinarily enable any man to know that the
first of the following three reasoning processes is correct. It might or
might not help him know that both the second and the third are incorrect.
1. Every animal is subject to death.
But every man is an animal.
Therefore, every man is subject to death.
2. Every animal is subject to death.
But even' man is subject to death.
Therefore, every man is an animal.
1

2

THE SCIENTIFIC ART OF LOGIC

3. Every animal is subject to death.
But some animals are not men.
Therefore, some men are not subject to death.
Natural logic may be strengthened by the art of logic. This art perfects
man’s intellect, enabling him to proceed easily and without error in
reasoning, even if the reasoning process is complex and difficult. A man
who has the art of logic knows without hesitation that the conclusions
in the second and third examples given above simply do not follow. He
knows this certainly, and he is able to explain the precise reason why
the conclusions do not follow. On the other hand a man who has merely
the native ability to reason might erroneously think that the conclusions
follow. Even if he knows that they do not he is hard pressed to explain
precisely the defect in the reasoning processes.
St. Tliomas Aquinas has defined logic as “the art which directs the
very' act of reason, that is, the art through which a man may proceed
with order, ease, and correctness in the act of reason itself.”1 We shall
sec later on that logic is both a science and an art. It is called the rational
science because it is not only reasonable, which is true of every science,
but because it bears directly upon the act of reason itself. It is called the art
of arts because it directs the reason, which is the source of all the arts.
We shall sec that the art of logic is an instrument of the intellect in all
its rational activities and especially in the rigorous procedures of scientific
inquiry.
We have distinguished between natural logic, which is a native endow­
ment, and the art of logic, which can be acquired. From now on we shall
speak of acquired logic simply as logic, for our concern shall be with the
art of logic rather than with natural logic.
II. The Distinction Between Sense Knowledge and
Intellectual Knowledge

Logic is a perfection of the intellect, enabling us to proceed properly
in the act of reasoning. In order to understand what this means, and
especially to appreciate the limits of logic, we must distinguish between
sense knowledge and intellectual knowledge. Human knowledge is com­
plex, being both sensory and intellectual. Logic, however, is directly
concerned only with intellectual knowledge. In the discussions which
follow, it may sometimes be convenient to speak as though the intellect
or the senses know. The student should not be misled by this. It is
1 Commentary on the Posterior Analytics of Aristotle, Bk. I, les. 1.

INTRODUCTION

3

always the man who knows, either through his senses or through his
intellect.
It is not easy to explain the difference between sense knowledge and
intellectual knowledge. The reason, perhaps, is that any explanation must
be exclusively an example of intellectual rather than sense knowledge.
Yet it is relatively easy to point out this difference. What we see, hear,
smell, taste, and feel all comes under the heading of sense knowledge;
what we understand is intellectual. Through our senses we become aware
of the individually concrete, phenomenal characteristics of physical things.
The intellect, on the other hand, cuts through the kaleidoscopic pattern
of sensible characteristics to an understanding of underlying meanings.
The intellect penetrates to the natures of things, to the “whatness” or
quiddity or essence. Fido, for example, appears to the senses as something
black and white, noisy, smelly, darting from place to place. Intellectually
Fido is understood to be a dog. Fido’s color can be seen, his barking
heard, his odor smelled, and his movement seen. These concrete char­
acteristics present themselves immediately to the senses. They make
their impression upon the senses only as individual, fluctuating data in a
concrete pattern of phenomenal or surface qualities. Fido’s nature as
dog differs significantly from all his concrete characteristics which we
know through our senses. It does not present itself immediately to the
senses. In fact, it is not known immediately, and it cannot be grasped
by any sense at all. It is known only by some suprasensory power which
cuts through the data of sense knowledge to the nature underlying them.
The nature which makes a dog what it is cannot be seen, or heard, or
tasted, or smelled, or felt. Yet the intellect, a suprasensory power, can
know the nature which underlies what can lx; seen, heard, tasted,
smelled, and felt in what we choose to call "dogs.” One point should be
added lest our example prove more misleading than illuminating. Fido’s
actual color, the noises he makes, and the odors which are peculiarly
his own are known on the sense level as concrete, phenomenal data.
Yet each of the sensible modifications of Fido’s intelligible nature is a
meaningful aspect of a reality which can be understood on a suprasensible
level. Fido’s concrete coloring is grasped as a visible characteristic on
the level of sense knowledge, but whiteness and blackness arc intelligible
objects just as surely as dogness itself.
In seeing the difference between sense and intellectual knowledge it
may be helpful to distinguish between various types of objects. For
every consideration there is an object toward which the mind is directed.
No man can totally exhaust any object in one consideration. Accordingly,
we must distinguish between the thing being considered and that par­

4

THE SCIENTIFIC ART OF LOGIC

ticular aspect of it properly and precisely known in the consideration in
question. The thing in general is called the material object, whereas the
precise aspect under which it is known is spoken of as the formal object.
Take our knowledge of Fido, the dog. Here Fido is the material object
both of the act of seeing his color and of the act of understanding or
thinking his nature. Fido’s color is the formal object of the act of
seeing, and his nature or essence as dog is the formal object of the act
of understanding or thinking. The basic difference between sense knowl­
edge and intellectual knowledge is a difference in formal object. Sense
knowledge bears precisely on the phenomenal aspect of reality, and
intellectual knowledge bears precisely on the meaningful aspect of things.
One important difference between sense knowledge and intellectual
knowledge is this. 7bc object of sense knowledge is singular or par­
ticular whereas that of intellectual knowledge is, for man, characteristically
universal. An object is said to be particular in this context inasmuch as it
is incommunicably limited to one individual, and, in fact, to that
individual in a given place at a given time. What I sense in anything
belongs to that thing only in the place in which it is while being sensed
and only at that time at which it is being sensed. A snapshot representing
some person is adequately identified only in terms of a given individual
in a given place at a given time. 'Phis is why we identify snapshots by
name, date, and place. An object is said to be universal, on the other
hand, inasmuch as it is applicable to many and is free from the restric­
tions of time and place. The nature or meaning man is applicable to an
individual man in this place at this time. But it is also applicable to
other men and to this man and other men in any place at any time.
Thus what we grasp intellectually when we know a nature, essence, or
meaning differs radically from which we know in an act of sense knowl­
edge. The intelligible object is universal, communicable, and unchanging
whereas the object of the senses is singular, incommunicable, and in a
state of flux.
Although the objects of the senses and the intellect are radically different
from one another, they do not exist in separation from one another. The
sensible and the intelligible are differing aspects of the same corporeal thing.
The intelligible aspect is somehow shrouded by the sensible, and we can
get to know it only by first knowing the sensible and by intellectually
discerning it in the sensible, as it were. The intellectual act of mentally
separating the intelligible from the sensible is called abstraction. We
shall say more about abstraction in the next chapter. For now, one further
point remains to be noted. The intellect makes its object universal since
it knows its object by way of abstraction. However, the point of departure

INTRODUCTION

5

for abstraction is the singular, and the intellect is able to reflect upon its
act of abstracting and through this to the singular itself. Thus, the object
of the intellect remains directly and characteristically universal, but the
intellect can know the singular in an indirect and reflexive way.
III. The Three Operations of the Intellect

We distinguished between sense knowledge and intellectual knowledge
in the preceding section in order to clarify the notion of logic. Logic
is an art which perfects the intellect. Logic docs not perfect the senses. As
a matter of fact the senses cannot be intrinsically perfected because they
do not need to be perfected. The senses function as well as they can
apart from any acquired determination. However, the intellect is capable
of further, acquired perfection, as is shown by the fact that we can
distinguish between natural and acquired logic. We shall see in this
section that there are three different operations of the intellect. Logic
perfects the intellect in a sense for all three.
We have said that logic is a perfection of the intellect which strengthens
it for the act of reasoning. But reasoning presupposes two prior intel­
lectual operations, namely, judgment and simple apprehension. Before
a man can reason he must judge, and before he can judge he must
apprehend. The intellect must be well ordered on the levels of appre­
hension and judgment before it can be well ordered on the level of
reasoning. Hence, if logic is to perfect the intellect for the act of rea­
soning—and it does —it must also perfect the intellect for appre­
hension and for judgment to the extent to which this is possible
and to the extent to which this is necessary ultimately in the interests
of sound reasoning. Tims, in order to understand the nature of logic
and to appreciate its full extent we must consider what the three opera­
tions of the intellect arc.
In the first section of this chapter we considered a reasoning process
which is clearly correct. Let us use it here as an illustration of the three
operations of the intellect.

Every animal is subject to death.
But every man is an animal.
Therefore, every man is subject to death.

Before we can conclude, in this argument, that every' man is subject
to death, we must already have seen that every' animal is subject to death
and that every man is an animal. And before we can affirm, on the one
hand, that every animal is subject to death and, on the other, that every

6

THE SCIENTIFIC ART OF LOGIC

man is an animal, we must first know the meanings of animal, subject to
death, and man. Thus, reasoning presupposes a prior operation, called
judgment, and judgment in turn presupposes the operation called simple
apprehension.

Simple Apprehension
Simple apprehension is an intellectual act whereby we understand an
object without affirming or denying anything about it. The object of
simple apprehension can be referred to as an essence, a quiddity, a
nature, or a meaning, so long as these terms are understood to signify
any intelligible object. When I think man in an act of simple apprehen­
sion, I do no more than think this meaning. I neither assert that man is,
nor that man is not, nor that man is such and such or not such and such.
Obviously, this is a truncated type of knowing. It may be that we are
only seldom psychologically aware of such a moment in knowledge. Yet,
as we have seen, this is an operation of the intellect necessarily prior to
both judgment and reasoning.
Judgment
Judgment is the act of the intellect affirming or denying one thing of
another. It is also called the act of composition or division, because one
thing is united with another in the act of affirmation or separated from
the other in the act of denial. To think man is to apprehend, and to think
subject to death is to apprehend. But to think Man is subject to death
is to unite one object to another in an affirmative judgment. In similar
fashion, to think coal is to apprehend, and to think white is to appre­
hend; but to think Coal is not white is to separate objects in a negative
judgment. By comparison to the act of simple apprehension, judgment
is a relatively complete act of knowing. In judging, the intellect takes a
stand on the way things arc, so that a judgment is cither true or false.
In simple apprehension, the intellect simply registers an intelligible aspect
of things without making any assertions about it. Neither man nor
subject to death is true or false. However, Man is subject to death is
true, and Man is not subject to death is false.
Reasoning
Reasoning is the act of the intellect proceeding to new knowledge by
way of something already known. Some judgments are so related to one
another that together they give rise to a new judgment which is different
from cither of them. Every' animal is subject to death and Every man is
an animal taken together yield the conclusion Every' man is subject to

INTRODUCTION

7

death. The judgment which is the conclusion in this reasoning process
is different from either of the two judgments from which it follows. Yet
we cannot simultaneously understand these two judgments without under­
standing the conclusion which follows from them. Although the con­
clusion is actually different from either of them, it is potentially contained
in them. In the act of reasoning the intellect moves from previous
knowledge to new knowledge potentially contained in the old. This
movement of the mind from the known to the unknown is spoken of
as rational discourse.

IV. Logic and Logical Relationships

Logic is the art which perfects the intellect for the act of reasoning.
Reasoning is an operation in which the intellect orders the objects it
knows in such a way as to proceed in an orderly fashion from old knowl­
edge to whatever new knowledge is contained in the old. Because of the
nature of the intellect and the reality it is suited to know, there are
determinate rules which govern the act of reasoning. These rules are
neither obvious nor self-evident. They can be acquired only with diligent
study. One who knows what these rules are and why they are what they
are, and only he, can reason easily and without error no matter how
complex the reasoning called for. Only he can adequately defend himself
when he reasons, and only he can attempt a competent critique of the
reasoning processes of others. The art of logic is the habit of mind per­
fecting the intellect for the art of reasoning. Thus the art of logic is
acquired when we understand the rules which govern rational discourse,
and this, indeed, is the purpose of the course in logic.
These rules are determined by the logical relationships existing in the
mind between objects known intellectually. As a matter of fact, these
rules arc simply the canonical statement of the natures of these logical
relations. In order to understand these rules, we must study these logical
relationships. In our final chapter we shall discuss the nature of logic
more completely than is possible here. 'Hiere we shall speak of logical
relations as second intentions, and we shall discuss logic as indivisibly
the art of rational discourse and the science of second intentions. Here
it is sufficient to indicate briefly by way of example the nature of logical
relations and to stress their significance for discourse.
Let us return once again to the first two of the three reasoning
processes we have already used for illustration:

1. Every animal is subject to death.

8

THE SCIENTIFIC ART OF LOGIC

But ever}’ man is an animal.
Therefore, every man is subject to death.

2. Every animal is subject to death.
But every man is subject to death.
ITierefore, every man is an animal.
In the first example, animal is used as though it had a certain relationship
to man and subject to death. In the technical language of logic it is used
as though it were a middle term. And in the second example, subject
to death is used as though it were a middle term in reference to the
others. As a matter of fact, animal is related as a middle term to the
other objects, and this is one of the reasons why the first conclusion
follows from its premises. But subject to death is not related to the
others as a middle term. When it is used as though it were, correct
procedure is violated and thus the second conclusion is invalid. A
logician knows that the first example is a good reasoning process and
that the second is defective because he knows the nature of the rela­
tionships determining the mies which govern reasoning. Knowing what
a middle term is, he knows that animal is correctly ordered in reference to
the other objects in the first example and that subject to death is badly
ordered in reference to the other objects in the second example. He knows
that the first reasoning process is correct —and he knows why it is
correct. And he knows tliat the second is defective — and why. It is not
necessary that the student understand at this time what a middle term
is for this illustration to be effective. But it is necessary for him to note
that the property of being a middle term is a logical relationship which
can belong to one object in reference to others. If an object is actually
related to others as a middle term, it can be used validly in a reasoning
process. If an object is not so related, then its use as a middle term will
involve error. This much is all that is necessary to indicate by way of
example the significance of logical relations for rational discourse and
the place of logical relations in the logic course.
The property of being a middle term is just one of the relationships
belonging to the objects in our example of a correct reasoning process.
We could point out many more. For example, animal is not only related
to the other objects as middle term, it is related to man as predicate and
also as proximate genus and to subject to death as subject; and man is
related to animal as subject and to both of the others as minor term.
However, these logical relations are as unfamiliar to the beginning student
as is the property of being a middle term. Here it is sufficient to say that

INTRODUCTION

9

these properties, along with many others, have their part to play in
determining the status of the reasoning process in question. These are the
things we shall study in logic, namely, logical relations such as the
property of being a subject, a predicate, a proximate genus, a minor term,
and a middle term. They are not easily understood, partially because,
unlike the objects with which we are most familiar, they exist only in
the mind. Yet they must be studied, because they must be understood.
It is only by understanding the logical relationships which determine
the character of logical discourse that one can acquire the scientific art
of logic.
V. The Divisions of Logic

Our study of logical relationships is divided from one point of view
into the logic of the first operation, the logic of the second operation,
and the logic of the third operation. It is divided from another point of
view into formal logic and material logic.
We have seen that the third operation of the intellect presupposes
the second, and that the second presupposes the first. It is not surprising,
then, that there arc relationships of logical importance not only on the
level of reasoning itself but on the level of judgment and simple appre­
hension as well. Because of this, logic is conveniently divided into the
logic of the first operation, the logic of the second operation, and the
logic of the third operation of the intellect. Each division of logic, of
course, is concerned with those logical relations proper to that level.
The third operation of the mind is ordered to the second because truth
is secured in the latter. Yet logic as a whole is primarily ordered to the
third operation because it is principally in the act of reasoning that the
intellect can go wrong and is in need of perfection.
The division into formal and material logic has the following basis.
In evcryr process of reasoning there are two aspects. The first, which is
based on the form of the process, consists in the arrangement of the
argument. The second, which is based on the matter, is concerned with
the meaningful content of the argument. /\ reasoning process is said
to be valid or consistent when its conclusion follows from its premises,
that is, when the premises cannot be true without the conclusion being
true. The question of validity or consistency is one of form alone.
Regardless of the meaningful content of the reasoning process, so long
as the premises are correctly disposed in reference to one another the
conclusion follows and the reasoning is valid. So long as the premises
arc badly disposed in reference to one another the conclusion docs not

10

THE SCIENTIFIC ART OF LOGIC

follow and the reasoning is invalid. Both of the following two reasoning
processes arc valid:

1. Every intellectual being is free.
But every man is an intellectual being.
'Iliercfore, every man is free.
2. Every elephant is a horse.
But ever}’ dog is an elephant.
Therefore, ever}’ dog is a horse.

No statement in the second example is true, whereas ever}' statement in
the first is. Each is valid because each involves a legitimate formal
pattern. As a matter of fact the two arguments involve an identical
formal pattern, which might be symbolically represented as follows:
Every A is B.
But every C is A.
Therefore, every C is B.

The following reasoning process, which we have already used as an
illustration in another reference, is invalid, although each of its statements
is true:
Every animal is subject to death.
But every man is subject to death.
Therefore, every man is an animal.

The inconsistency of this logical pattern may be more clearly evident to
a beginning student in a reasoning process with the same form but with
true premises and a false conclusion, as in the following:
Every Hoosier is an American.
But ever}’ Badger is an American.
Therefore, ever}' Badger is a Hoosier.
Validity is the end or goal of formal logic, which studies those logical
relations which determine the formal status of logical discourse. A
knowledge of these relations should result in the ability to reason validly.
However, it is not sufficient simply to be able to reason validly, for a
valid process may not be true. We should be able to reason validly with
truth and, where possible, with certainty and full explanatory force. In
other words, we should be able to reason not only validly but with pro­
bative force as well.
Valid reasoning processes may differ significantly in probative force

INTRODUCTION

11

because of their meaningful content. We have already noted that validity
is indifferent to truth. Valid reasoning processes are possible although
no statement in the process is true. Even when all the statements are
true, the reasoning processes may have different probative force because
of differences in matter. Consider the following valid reasoning processes:
1. Every mother loves her child.
But Susan is a mother.
'ITicrcfore, Susan loves her child.
2. Everyone capable of speech is rational.
But every man is capable of speech.
Therefore, every man is rational.
3. Everyone who is rational is capable of speech.
But every man is rational.
'Ilicrcfore, every man is capable of speech.

The first reasoning process argues validly to a probably true conclusion
from a premise true only for the most part. The second argues validly
to a certainly true conclusion establishing the fact of rationality in man
from one of its effects, namely, the capability of speech. The third argues
validly to a certainly true conclusion from premises establishing not
only the fact of the capability of speech in man but also the explanation
of this attribute in terms of its cause, namely, rationality. The three
reasoning processes are equally valid, but they differ significantly from
one another in terms of probative force.
Truth, certainty, and explanatory force are the end of material logic,
which deals with those logical relations determining the material, or,
perhaps better, truly scientific status, of logical discourse. A knowledge
of these relations, in addition to a knowledge of those which are of formal
import, should equip a man to reason with total effectiveness.
The purpose of logic in general is totally integral reasoning. This in­
volves soundness both on the side of form and matter. Hence we shall
concern ourselves, on each succeeding level of intellectual operation, with
logical relations both in formal and material logic.
Exercise I (A)

Before beginning your logic course, try the following brief test. The logical
processes involved are relatively simple. Yet there is a good chance that natural
logic will not be sufficient for the average student to handle the test easily and
without error. If this is the case for you, clearly you can benefit by your
course in logic. Time yourself. A student of average competence should be

12

THE SCIENTIFIC ART OF LOGIC

able, after completing his course in logic, to get a perfect score in no more
than six or seven minutes. This preliminary test in no way indicates the full
scope of the course to follow, but it should nevertheless serve to alert most
students to their need for a reflexive study of logical processes.
In the following arguments pick out the conclusions which consistently
follow from their premises and those which do not follow. For the latter
give the reason why they do not follow from their premises.
1. Since Americanism is opposed to Communism and Communism is op­
posed to Fascism, it follows that Americanism is opposed to Fascism.
2. Since no triangle has five sides, neither can any square have five sides,
for no square is a triangle.
3. No canary is four-legged, because no canary is a dog and every dog is
four-legged.
"•
4. The immaterial is inconsequential; thus God is inconsequential, since
God is immaterial.
5. Since all Communists are threats to the integrity of our country, then
all juvenile delinquents are Communists, because all juvenile delinquents
are threats to the integrity of our country'.

6. Since no rectangles are three-sided, it follows that some plane figures are
not rectangles because some plane figures arc three-sided.

7. All men are intelligent beings, and all intelligent beings are possessed of
freedom of choice; hence, all beings possessed of freedom of choice arc
men.
8. One cannot bum the candle at both ends and retain one’s health. Ac­
cordingly, since John has not retained his health he must have burned the
candle at both ends.
9. Since no syllogisms arc inductive, and some syllogisms are probable argu­
ments, tiren some probable arguments are not inductive.
10. Since it is true that all Hoosiers are Americans it follows that:
10.1. It is False that all Americans arc Hoosiers.
10.2. It is true that all who arc not Americans are not Hoosiers.
10.3. It is true that some Hoosiers are Americans.
10.4. It is true that no Hoosiers are non-Americans.
10.5. It is false that some Hoosiers are not Americans.
Exercise I (B)

1. Define the following significant terms: natural logic, the art of logic,
sense knowledge, intellectual knowledge, material object, formal object,
the singular or particular (object), the universal (object), simple appre­
hension, judgment, reasoning, logical relation, formal logic, validity,
material logic, truth, certainty, probability, explanatory force.
2. If all men are endowed with natural logic, why is a course in logic
necessary?

INTRODUCTION

13

3. If logic is acquired at the end of the logic course, how can the course
escape the charge of being illogical?
4. What is the difference between a material object and a formal object?
Use this distinction to contrast sense knowledge with intellectual knowl­
edge. Suggest several other applications of the distinction between the
material object and the formal object.
5. Discuss the intimate relation between sense knowledge and intellectual
knowledge in man.
6. Explain the interrelation between the three operations of the intellect.
How do the three operations of the intellect serve as a principle of
division for logic?
7. What is the subject matter of the science of logic? What is its end?
8. Distinguish between validity and truth.
9. Explain the difference between formal logic and material logic. Inter­
relate this division of logic with the division of logic determined by the
three operations of the intellect.

PART I

The Logic of
the First Operation

CHAPTER

II

Simple Apprehension: The First Operation

I. The Nature of Simple Apprehension

Simple apprehension is that intellectual act whereby we know or
understand an object but do not assert anything of it. Through this act
we know an intelligible object which we can call a nature, an essence, a
quiddity, or even a meaning. In simple apprehension, however, we make
no assertions about this intelligible object. That is, we do not judge
whether it exists or not, we do not affirm or deny of it any determinations
whatsoever, and we do not affirm or deny it of anything. For example, we
can apprehend what man is without affirming Some man exists or Even­
man is capable of speech or John is a man. In our act of apprehending
man we think simply man without affirming or denying anything about
this object and without affirming or denying it of anything. It is difficult
to appreciate the nature of this operation, for we cannot always easily
recognize it in our own personal psychological experience. We can catch
ourselves thinking John is a man or Some men are not tall much more
readily than we can catch ourselves thinking man or tall. However, an
analysis of the operation of judging, which essentially involves either
affirmation or denial, shows that wc must admit that simple apprehension
is a distinct and prior operation of the intellect. Surely we cannot affirm
man of John or deny tail of man unless we first know- man and tall
through acts of simple apprehension.
In comparison to the object of a judgment, the object of simple
apprehension is simple or incomplex. When we make a judgment we
know one thing as affirmed or denied of another, whereas in simple
apprehension wc know- something without making any affirmations or
denials about it. Nevertheless, on its own level the object of simple
apprehension can be spoken of as simple or complex. It is simple if the
nature it represents is simply one, and it is complex if that nature is a
composite of several natures. Both man and philosopher are objects of
17

18

THE LOGIC OF THE FIRST OPERATION

simple apprehension. The same is true for both stone and pebble. How­
ever, man differs from philosopher, and stone from pebble. Man and
stone are objects whose meanings are realized in what is distinctly one
nature, whereas philosopher and pebble are objects whose meanings are
realized only in a combination of distinctly different natures. Philosopher
is equivalent in meaning to man plus the quality of philosophy. Pebble
is equivalent in meaning to stone plus smallness. A philosopher is a
philosophical man and a pebble is a small stone. Man and stone are
examples of simple or incomplex objects of simple apprehension, whereas
philosopher and pebble are examples of complex objects of simple
apprehension.
Philosophical man and philosopher are identical in meaning, just as
small stone and pebble are identical in meaning. Nevertheless they differ
in the manner in which they arc conceived. Philosophical man and small
stone are complex objects mentally expressed in a complex fashion;
philosopher and pebble arc these same complex objects expressed simply.
The same reality is signified by both philosophical man and philosopher,
and the same is true of small stone and pebble. But the mode of signifying
is different in each instance. And just as complex objects can be conceived
incomplexly, so incomplex objects can be conceived complexly. Man and
rational animal are identically the same incomplex object as far as what
is signified is concerned. However, rational animal is thought complexly
while man is conceived incomplexly.
Simple apprehension is not so simple after all. It is “simple” appre­
hension only in the sense that it fastens on a nature or essence without
affirming or denying anything whatsoever about that nature. The object
apprehended may or may not lie simple. 'Hie objects of simple apprehen­
sion can be: either (1) incomplex in themselves and incomplexly con­
ceived (e.g., man), or (2) incomplex in themselves and complexly con­
ceived (e.g., rational animal), or (3) complex in themselves and incom­
plexly conceived (e.g., pebble), or, finally (4) complex both in themselves
and as conceived (e.g., small stone). Man is clearly only an object of
simple apprehension, but so too is honest, courageous, idealistic politician
fighting for the common good against the enemies of the community
despite the apathy of the citizens.
II. The Role of Abstraction in Simple Apprehension

Physical things involve an intelligible aspect as surely as they involve
a set of sensible characteristics. As a matter of fact, everything involves
many different intelligible aspects, including its own basic nature plus

SIMPLE APPREHENSION: THE FIRST OPERATION

19

the natures of each of its many attributes. However, as noted in the
preceding chapter where we distinguished sense from intellectual knowl­
edge, the intelligible aspects of a thing are shrouded in its sensible
aspects. This is true in the sense that the intelligible aspects of anything
arc not immediately evident to the intellect, whereas its sensible qualities
are immediately evident to the senses. However, because the sensible
characteristics of anything are proportioned to its nature (e.g., snow is
regularly white), they do, in a way, manifest that intelligible nature.
Through the instrumentality of sufficient sensory experience, a man is able
to penetrate to underlying intelligible natures. This intellectual penetra­
tion of the sensible is abstraction. As physical things exist outside the
mind, they are immediately or actually sensible and only mediately or
potentially intelligible. Abstraction makes the potentially intelligible
actually intelligible so that it can be simply apprehended.
There is a double force to the abstraction which yields the object of
simple apprehension. First, intelligible objects as known are not limited
to one individual but are universally applicable to many. The reason is
that they are abstracted from the individuating characteristics which
mark off one individual from another. The object man is mentally sep­
arated from the individuating characteristics of any individual man. Thus
man is equally applicable to Man', George, and Joe. In addition, the
object of simple apprehension is also abstracted from the act of existing.
Take the object man as our example again. This object is understood
apart from the act of existing which makes it actual in any particular
man such as Mary, George, and Joe. There is thus a distinction between
what a thing is, its nature or quiddity or essence, and the act of existing
which makes it be. The first operation of the intellect bears only upon
quiddity or essence. It does not bear on the existence, without which,
of course, no nature is at all. In simple apprehension a nature is known,
not as actually existing, but only as open to existence. Although the
nature known in an act of simple apprehension has been abstracted from
actually existing things, our knowledge is not false. For we do not think
that the nature does not exist, we simply think that nature without
affirming or denying its existence, just as we think it without thinking
about the individuating notes under which it exists in reality. The nature
man is found in the real order only in individual men, each with his
own individuating characteristics. Again, this nature is found only with
the act of existing in the real order. But man is a legitimate object of
the intellect even when apprehended apart from individuating notes
and the act of existing, for neither affects in any way the character of
this nature as an object of the intellect. Because the nature known in

20

THE LOGIC OF THE FIRST OPERATION

simple apprehension (e.g., man) is abstracted from the act of existing
as well as from individuating characteristics, it is applicable not only to
many actual men (e.g., Mary, George, and Joe) but to many possible men
as well (e.g., Mary’s great-great-grandchild, who does not actually exist but
would be man if he did).
III. The Nature of Signs and Their Import in Logic

In order to understand more fully the three operations of the intellect
we must say something about the nature and types of signs and explain
their significance for logic. The notion of a sign is familiar to us. It is
something which stands for, or points to, or gives knowledge of some­
thing other than itself. Signs can be divided from one point of view into
natural and conventional and from another point of view into formal
and instrumental signs. The reason, as we shall see more fully, why the
logician is interested in signs, is that one kind of sign brings about
knowledge, whereas another communicates it.
As its name indicates, a sign is natural whenever the connection be­
tween it and the thing it signifies is determined by nature, i.e., when it
is in the nature of something to point to something else. A common
example, that of smoke as the sign of fire, serves to illustrate the character
of a natural sign. A fire is not the same as the smoke that it causes, but
it is the nature of fire ordinarily to produce smoke. Hence, smoke is
a natural indication of the presence of a fire, which may or may not
be directly observed. Smoke can be understood as giving knowledge of
something other than itself by pointing to fire on the basis of the
natural relationship it has to fire. Other examples include falling
barometric pressure as a natural sign of stormy weather, spots as a natural
sign of measles and fever as a natural sign of infection, a smile as a
natural sign of pleasure, and artistic works as a natural sign of rationality.
A sign is conventional rather than natural whenever the connection
between it and what it signifies is the result of man’s arbitrary' imposition.
For example, a green light is a conventional sign to indicate legal
passage; an “A,” excellence in classroom achievement; and an upraised
right thumb on the part of an umpire, an “out” in a baseball game.
Each of these is a sign, because each gives knowledge of something other
than itself by pointing to this something else. However, none of these
is natural. The connection between them and the things they signify is
not natural. In each case the sign signifies what it does only because
it has been chosen to signify this and is understood to do so. A red light
could signify “go” and a green light could signify' “stop”; an “A” could sig­

SIMPLE APPREHENSION: THE FIRST OPERATION

21

nify unsatisfactory schoolwork and any other letter could be taken to stand
for excellence. And so on for the rest of our example. Granting the arbitrar}'
character of conventional signs, we do not want to deny that some
conventional signs are naturally suited to their role. Given the explosive­
ness of the color red it follows that a red light is better suited to signify
“stop” than a blue or yellow or green light. Since “A” is the first letter
of the alphabet (though this itself is, of course, something arbitrar}')
“A” is better suited to indicate excellence than “B,” “C,” “D,” or “F.”
Nevertheless, these remain only conventional signs. Their aptness, how­
ever, is itself a natural sign of the intelligence of the men who chose
them to be used as they arc conventionally used.
It is not as easy to explain the difference between formal and instru­
mental signs as it is to explain that between natural and conventional
signs. Examples of both natural and conventional signs arc easily under­
stood and readily available, but ordinarily students have no experience
of formal signs to fall back upon and can clearly understand only
examples of instrumental signs. Formal signs signify without first being
known themselves. Instrumental signs signify only after being known
themselves. Significantly all the examples we have so far given of signs
are instrumental. These are the signs with which we arc most familiar.
We know these best, because, by definition, these signs arc themselves
known before they give knowledge of something else. ITic red light, for
example, must be seen before a motorist knows he must stop, and many
a motorist has tried to excuse himself to a traffic officer for not stopping
when he should have by pleading that he has not seen the red light.
Any sign which gives knowledge of something other than itself
without first being known itself is a formal sign. This means that wc
come to learn of these signs only indirectly and by reflecting on our direct
experience of things. Paradoxical as it might seem, human knowledge is
impossible without formal signs. To see why this is so, consider once more
our distinction between sense and intellectual knowledge; by way of the
former wc grasp the sensible characteristics of things and by way of the
latter we grasp intelligible natures. Neither can be understood in terms of
sheer physical possession. Wc do not physically possess what we know
either sensibly or intellectually, but represent it to ourselves in either
a sensor}' or intelligible way. When I sense the phenomenal characteristics
of something, those characteristics are present to me in an image which
enables me to know them as they exist in the thing. But what I sense
is certainly not the image through which I sense, but the sensible char­
acteristics of the thing. Experience testifies to this, yet this cannot be
explained save in terms of an image which represents within me what

22

THE LOGIC OF THE FIRST OPERATION

is sensible in the thing. Since the image gives knowledge of the sensible
aspect of the thing, it can be called a sign. Since it signifies without
first being known itself, it is a formal sign. We do not say that the image
can never be known. It can be known reflexively, just as we are knowing
it at this moment. But it is not known at the moment that it functions
as a sign. It is not known at the moment it presents the sensible
object to the knower, by interiorly representing it. At that moment it
is a pure sign with nothing more to it than the object it signifies, so
that this object and nothing more is known. Just as sense knowledge
requires a formal sign, called the sensory image, so intellectual knowledge
requires a proportionately more excellent formal sign. The object of
simple apprehension is mentally expressed in a formal sign called the
mental word or formal ^concept; what is known on the level of judgment
is mentally expressed in a formal sign called the mental proposition;
and what is known on the level of reasoning is mentally expressed in a
formal sign called the mental argument. In each of these cases the formal
sign interiorly expresses the object of whatever cognitive operation is
in question: sensing, simply apprehending, judging, or reasoning. In each
case knowledge is impossible without the formal sign, but in no case is
the formal sign itself directly known. Formal signs, of course, arc natural
and never conventional since they are, as signs, the natural representations
of what they signify.
We said that signs arc of interest in logic because knowledge is brought
about and communicated through signs. We have seen how knowledge is
brought about through formal signs. But what kind of a sign is used
for the communication of knowledge? Knowledge, of course, is ordinarily
communicated by way of language, and words are the fundamental
elements of any language. A word, in brief, is a graphic or oral expres­
sion conventionally designed to signify some meaning. Words are signs
of meanings. They are instrumental signs, since they must be seen or
heard before they can function as signs. And they are conventional signs,
since the relation between them and the natures they represent is arbitran-,
the work of men. If, for example, I want to let someone know that 1
am thinking about man, I must say “man.” Because others understand
what “man” stands for, they will know what I am thinking. I might
say “1’homme” or even “homo,” and if others know French and Latin
cither of these words senes as well as “man” to signify the meaning
man. As a matter of fact we might have devised a code in which “horse”
stands for the nature man and “man” stands for the nature thunder.
and so on. Words are clear-cut examples of instrumental signs which are
conventional rather than natural. Ilius, since knowledge is communicated

SIMPLE APPREHENSION: the first operation

23

by words taken either singly or in combination, it is clear that knowledge
is communicated by instrumental and conventional signs.
Strictly speaking, the logician is only indirectly concerned with formal
signs and words. rlhe art of logic directs the ordering of the objects of
intellectual knowledge. Hence the logician is directly interested in ordering
these objects, not the formal signs through which they are known. Yet,
since these objects arc known and expressed only through formal signs,
the former cannot be ordered properly unless the latter be likewise
ordered. Nevertheless the logician is content to leave the direct investiga­
tion of formal signs to psychology or epistemology. As a matter of fact
the logician’s concern is not even directly with the objects signified
by the formal signs. The logician does not study the objects — leaving
them to the sciences concerned with real things —but the logical rela­
tions belonging to them as known. The logician is even less concerned
with the oral or written expression of intelligible objects. He is not
directly concerned with the way in which what is known is spoken or
written, but with the way in which it is ordered within the mind. He is
neither a linguist nor a grammarian. Yet he cannot escape language and
grammar. Tire linguistic and grammatical expression of his thought
reflects its interior status. He can learn from teachers only by listening
to their words, and he can teach his students only by speaking to them.
Hence he must be interested in the oral or written expression of the
objects of intellection. Yet it is clear that the ultimate verification of
his statements must be found within the realm of objects and their
interrelationships and not in the sphere of grammatical expression. On
this point it is enough to remark that words are conventional signs and
that the logician seeks to know the canons of rational discourse demanded
by the nature of the reason.
IV. A Comparison of the Three Operations of the Intellect
in Terms of Some Significant Distinctions

We have distinguished between the object of the act of knowing and its
two expressions, mental and verbal. The object of simple apprehension
is a nature or quiddity', about which nothing is either affirmed or denied.
The mental expression or formal sign of this object is called a concept,
since its formation within the mind is analogous to biological conception.
The nature or quiddity known in simple apprehension is the object
signified by this concept. Because this object is the objcct-of-thc-concept,
it has been called the objective concept. The mental expression of
the objective concept, which is the concept in the strict sense, is

24

THE LOGIC OF THE FIRST OPERATION

called the formal concept. The formal concept is sometimes referred
to as the mental word, the mental term, or the idea. The verbal
expression of what is known on the level of simple apprehension is a
word or combination of words spoken of as an oral or written term.
We have noted already that the formal sign for the second operation
is called the mental proposition, whereas that of the third operation is
called the mental argument or argumentation. The corresponding verbal
expressions on the levels of judgment and reasoning are spoken of as
oral or written propositions and argumentations.
Throughout this book we shall be concerned with both mental and
verbal terms, propositions, and arguments. For the most part our interest
will be focused upon the objects they express and the logical relations
ordering these objects one to another. Thus, when we speak about terms,
propositions, and arguments, we will be referring primarily to the objects
of the first, second, and third operations of the intellect. For the same
reason, whenever we speak of the concept without distinguishing between
formal concept and objective concept we shall be speaking primarily of
the latter, although the name “concept” belongs first of all to the former.
Exercise II

1. Define the following: simple apprehension, judgment, reasoning, abstrac­
tion, essence, existence, sign, natural sign, conventional sign, formal sign,
instrumental sign, mental term (or formal concept), objective concept,
mental proposition, mental argumentation, verbal term, verbal proposition,
verbal argumentation.
2. Compare these objects of simple apprehension as to complexity and
incomplexity, both in themselves and as conceived: giant, science, scien­
tist, pretty girl, girl, horse, colt, rational animal, four-sided polygon, tall
man with dark brown hair, and policeman.
3. Contrast simple apprehension with judgment in terms of the complexity
of their respective objects.
4. What is the twofold force of the abstraction proper to simple apprehen­
sion? What is its significance for the universality of the objective concept?
5. What is the difference between a natural sign and a conventional sign
naturally suited to be used as it is used?
6. How do formal signs differ from all other signs? What are the only
examples of formal signs?
7. Classify the following as signs: ringed moon as a sign of stormy weather,
“dog” as a word-sign of the nature dog, the idea dog in the mind, the
sensory image of a given dog, a ruddy complexion as the sign of health, a
cross as the sign of Christianity, “!” as a sign of exclamation, a camel
as the trade-mark for a certain type of cigarct, “SOS” as a sign of peril,
and a moan as a sign of pain.
8. In which of these three are we most interested in logic: the mental
term, the verbal term, or the objective concept?

CHAPTER

HI

Comprehension and Extension

I. The Comprehension of the Concept

Every objective concept has both a definite intelligible content and
the ability of being said of many subjects. The content of the concept
can be called its signification or comprehension. The sum of the subjects
of which it can be said can be called its extension. We shall see that the
extension of a concept depends upon its comprehension and that these
two are inversely proportioned to one another.
Two concepts can differ comprehensively in at least three different wap.
First of all, they can be so radically different as to involve no common
element. This is the way man differs from pretty, or running, or bigger.
Later wc shall sec that concepts of this kind are in different categories.
Second, concepts may differ in meaning while admitting a common
element. This is the way man differs from horse, or from maple tree,
or from stone. To be a man is not to be a horse, but both arc animal.
To be a man is not to be a maple tree, but both are living. And to be
a man is not to be a stone, but both are corporeal. In all of these
examples there is a common element which partially explains their
comprehension. Concepts differing from one another in this way are
sometimes said to be disparate concepts. Finally, concepts may differ
when one is an element which partially explains the comprehension of
the other. Tin’s is the way man differs from animal, or from living body,
or from body. To be a man is to be a certain kind of animal, and to be
an animal is to be a certain kind of living body, and to be a living body,
of course, is to be a certain kind of body. Thus animal is part of the
comprehension of man; living body is part of the comprehension of
animal and man; and body is part of the comprehension of living body,
animal, and man. Concepts which differ in this way can be said to be in
the same logical line. They can lx: compared as the more general (animal)
to the less general (man), or, perhaps better, as the indeterminate
(animal) to the determinate (man).
25

26

THE LOGIC OF THE FIRST OPERATION

Our consideration of the way concepts can differ helps us to see more
clearly what comprehension involves. The second and third ways, in
particular, point out that a concept is a total meaning and that its
comprehension consists in a number of intelligibly distinct notes. Thus
it follows that a concept is more or less rich in content as it involves more
or less intelligible notes. Any concept can be analyzed into a number
of meaningful notes. Taken together, these notes give it its meaning.
For example, the concept parallelogram is a total meaning which includes
several intelligibly distinct notes. A parallelogram is a four-sided, recti­
linear, plane figure with opposite sides parallel. The concept square is
also a total meaning with several intelligibly distinct notes. It is a richer
object than parallelogram, for it includes all the notes of parallelogram
and more, because it is an equilateral, rectangular parallelogram. Evidently
four-sided, rectilinear, plane figure, etc., arc intelligibly distinct notes
although they need not exist in separation from one another. Plane figure
is intelligibly distinct from rectilinear and four-sided since many plane
figures (c.g., circles) are neither rectilinear nor four-sided. And rectilinear
is distinct not only from plane figure but also from four-sided, since
some rectilinear figures (e.g., triangles) arc not four-sided. Nevertheless
these notes are not existentially distinct in any square. This is an im­
portant point. They are completely misunderstood if they arc regarded
as physically distinct aspects of reality somehow joined together in things.
As one geometrical entity, any square is wholly a parallelogram and wholly
equilateral and rectangular. It is not first a parallelogram to which rec­
tangularly and equal sides are added. However, we can think of what
makes any parallelogram a parallelogram without thinking about what
makes it a square or any other kind of parallelogram. And this is true,
although there is no parallelogram which is not a square or some
other specific type of parallelogram. Furthermore, whatever makes a
parallelogram a square is intelligibly distinct from whatever makes it
a parallelogram. Taken together, both these intelligibly distinct notes
enter into the meaning of square and are called its comprehensive
notes. The total signification or comprehension of any concept is, of
course, the sum of these notes. They arc intelligibly distinct even though
they need not be physically distinct. And the distinctions we draw between
them in the mind are based upon differences in things. The intelligible
distinction between plane figure and rectilinear holds even though every
parallelogram is both rectilinear and a plane figure. This is clear since
even' circle is a plane figure without being rectilinear.
Concepts which are complex in themselves, such as pebble, include
a number of more elementary intelligible notes. The reason is that they

COMPREHENSION AND EXTENSION

27

represent a plurality of different natures. Pebble, for example, is a total
meaning made up of the meanings stone and small. Yet even concepts
incomplex in themselves involve distinct intelligible notes. Thus man is
a total meaning in which we can discover the following comprehensive
notes: substance (that kind of a being apt to exist in itself and not in
another), material (having parts outside of parts), living (able to operate
vegetatively), sentient (able to operate on the sensory level), and rational
(able to reason). Animal is an incomplex concept in the same logical
line with man although its content is less rich than man. Tlie compre­
hensive notes of animal are substance, material, living, and sentient. Brute
is an incomplex concept sharing a common element with man. Just as
man adds rational to animal, so brute adds its own distinctive difference
to animal. Running is a concept whose comprehension is completely
different from man. They share no common element, not even the most
general, for just as man is ultimately reduced to substance, so running
is reduced to action.
It is legitimate to speak of comprehensive notes as intelligibly distinct
from one another. But we must be careful not to think that each of
these notes is on a par with one another. One is always more general
than the other. The more general is always related to the other as matter
to form, that is, as the determinable to the determining factor. The more
general is open to the more precise, and the latter can be understood
only in the light of the former. Thus, substance and material are two of
the comprehensive notes of man, of animal, and of living body. Together
substance and material constitute the concept body. However, they do not
constitute the concept body as two parts on a par with one another.
Substance is the more general note which is open to further determination
by way of material (or its opposite immaterial). Substance expresses
indeterminately what body expresses determinately, and body docs so
because it adds to substance the determining form or difference, material.
Material is less general than substance since some substances are not
material, and material presupposes substance since everything material is
fundamentally substantial.
The comprehensive notes of a concept belong to it at least implicitly
or virtually. We may or may not explicitly think about the distinct
intelligible notes which belong to a concept. Yet the concept is what it
is because of these comprehensive notes, and in apprehending it we
grasp them. If we know them explicitly, as we do when we completely
define the nature known, we are said to know them distinctly. There is
the temptation to look upon simple apprehension as an operation whereby
the intellect immediately and intuitively understands essences in the

28

THE LOGIC OF THE FIRST OPERATION

fullness of their intelligibility. This is not the case. Simple apprehension
itself must be preceded by sufficient sensory experience to warrant the
abstraction of the universal from the singular. And one act of simple
apprehension differs greatly from another. The natural movement of
the intellect is from the general to the particular and from the confused
to the distinct. The general notion of body is understood long before
animal is grasped. And animal is understood before man. Our knowledge
of man is obscure at first, that is, we grasp its comprehensive notes only
implicitly before we finally reach a complete definition. It is significant
that the first operation of the intellect is perfected in defining its object.
The act of defining is, in its most perfect instance, the act of making
explicit the comprehensive notes of the object we are defining. We
cannot perfectly define many objects because we cannot fully make
explicit the notes we grasp confusedly in the object. Later we shall say
that rational animal is a perfect definition of man. But we shall not be
able to define brute perfectly. We know that, like man, brute is animal,
and that instead of being rational, brute is something else. But what is
the difference which is added to animal to make brute? We are unable
to name it. Instead we use the negation of rational in its place, for surely,
whatever it is, it is nonrational. Nonrational animal suffices to point out
brute as distinct from man. However it falls short of being a perfect
definition of brute because it docs not set forth, in a positive way, the
comprehensive notes of brute. There is an opaqueness to the human
intellect, testified by our inability to give a perfect definition of many
objects.
In the strict sense only those notes which arc intrinsic to the nature
or essence conceived are comprehensive notes. In a less strict sense, any
attribute which flows necessarily from an essence and belongs always and
exclusively to that essence can be called a comprehensive note even
though it is not properly intrinsic to the essence. We shall speak about
this kind of attribute later as a property. Substance, material, living,
sentient, and rational arc the comprehensive notes of man in the strict
sense because they arc the meaningful elements which together neces­
sarily constitute the meaning man. However, because man is what it is,
certain properties always belong to every man and only to men. For
example, by reason of his essence or nature, every man is able to talk,
to laugh, to fashion and use tools.1 Since the only cause of these abilities*
’ Throughout this book we shall refer frequently to such abilities as the ability to
talk, the ability to laugh, and the ability to use tools. These will ordinarily be under­
stood as taken radically, with no concern for their actual exercise or even the immedi­
ate possibility of their actual exercise. Thus a dumb man will be thought to have the

COMPREHENSION AND EXTENSION

29

is human nature, only men can talk, laugh, fashion and use tools. These
abilities and other properly human characteristics are sometimes spoken
of as comprehensive notes of the concept man. However, they are compre­
hensive notes only in a secondary sense. They have an intimate and
necessary connection with the comprehensive notes of man in the strict
sense. Yet there is a basic difference between them. A man is a man
because he is rational, because he is sentient, and so on. He is a man
not because he is able to talk or laugh. On the contrary, he is able to
talk and laugh precisely because he is a man. Whenever we speak of
comprehensive notes we shall ordinarily refer to comprehensive notes
strictly taken, i.c., to those meaningful notes intrinsic to the concept
which taken together necessarily give us the total meaning of the concept.
II. The Extension of the Concept

Because a concept is abstracted from individuating characteristics and
the act of existing, it can be said of many. Logicians call this property
the extension of the concept. Extension is the sum of those subjects
,cither singular or universal) in which all the comprehensive notes of
a concept arc realized. Thus John, Peter, Paul, Mary and all other indi­
vidual men fall within the extension of man, since the comprehensive
notes of man, namely, substance, material, living, sentient, and rational
are realized in each of them.
The extension of a concept is not measured by adding up units or by
"counting heads.” There is, in fact, an infinite number of subjects within
the extension of any concept, and this because the concept is abstract.
It is, as wc have seen, abstracted both from individuating characteristics
(hence man can be said of both John and Mary) and from the act
of existing (hence man can also be said of Mary’s great-great-grandson,
who doesn’t exist but would be man if he did). There is a limit to the
number of actually existing men, past, present, and future, but no limit
to the number of logically possible men. The same is true of any other
concept, abstracting, as it must, from actual existence. Any concept in­
cludes within its extension an infinite number of subjects of which it
can be said. Nevertheless, wc can still speak of one concept having greater
or less extension than another concept as long as we consider extension
in terms of an infinite multitude and compare one infinite multitude
with another. For example, man can be said of an infinite multitude of
actual and possible men, while animal can be said of an infinite multitude
ability to talk — in the sense that his nature is such as radically to ensure this ability —
even though there arc physical impediments to the exercise of this ability.

30

THE LOGIC OF THE FIRST OPERATION

of actual and possible men and brutes. Since the subjects of which
man can be said are only some of the subjects of which animal can be
said, the extension of animal can be regarded as greater than that of man.
III. The Priority of Comprehension Over Extension

Although comprehension and extension arc both properties of the
concept, comprehension has a logical priority over extension. We can
distinguish between the actual content of the universal concept and the
universality which enables it to be said of many subjects. The concept
is first of all a meaning constituted by the sum of its comprehensive
notes. It can be said of many subjects only if its comprehensive notes are
found in them. Thus extension is defined in terms of comprehension, a
clear indication that the latter is of primary importance.
A different opinion on the universal is held by the nominalists. In
their view the universal concept is merely a collection of those subjects
which answer to the name or term in question. Nor is there any necessary
reason why one term is said of the members of any given group, for
there is no nature which is shared commonly by the members of the
group. According to the nominalists, John and Peter and Mary are men
simply because they arc called men. The truth is obviously quite opposite.
John and Peter and Mary arc called men because they are men, i.e.,
because each is substance, material, living, sentient, and rational.
IV. The Inverse Ratio Between Comprehension and Extension

Comprehension and extension are readily seen to be inversely propor­
tional. The more notes in a concept, the richer it is in meaning and the
fewer the subjects of which it can be said. Correspondingly, the fewer its
notes, the more deficient it is in meaning and the greater the number
of subjects of which it can be said. Square is said of fewer geometrical
entities than parallelogram, parallelogram of fewer than quadrilateral
figure, quadrilateral figure of fewer than rectilinear figure. Sign is said of
more things than instrumental sign, while instrumental sign is said of
more things than conventional sign, and conventional sign of more things
than term. The more determinate a concept is comprehensively the
more restricted it is extensively. The comprehensive notes can be likened
to clues in a detective story and the extensive subjects to suspects. 'Hie
more chics that are uncovered the fewer the suspects. Each succeeding
clue narrows the field of suspects accordingly. Similarly, the addition of
a comprehensive note to a concept yields a new concept more determinate

31

COMPREHENSION AND EXTENSION

and rich in content than the original and correspondingly more restricted
in extension. For example, add to the concept substance the note material,
and the concept body results. Body is said of whatever substance is said
of, with the exception of spiritual substances. Then add living to body,
and the concept living body or organism results. Organism is said of
whatever body is said of, with the exception of nonliving bodies.
The inverse ratio between comprehension and extension can be seen
graphically in the following schema which completes the illustration
begun in the preceding paragraph:
CONCEPT
Mon

COMPREHENSIVE NOTES
Substance — Material — living — Sentient — Rational

Animal

Substance — Material — living — Sentient

Organism

Substance — Material — Living

Body

Substance — Material

Substance

Substance

^Spirits

Men

Brutes — Men
Plants — Brutes — Men

Nonliving bodies — Plants — Brutes — Men

— Nonliving bodies — Plants — Brutes — Men

EXTENSIVE SUBJECTS

Naturally the inverse ratio between comprehension and extension is
significant only in reference to concepts in the same logical line.
V. Superiors and Inferiors

Those subjects, whether universal concepts or individuals, which con­
tain the comprehensive notes of a given concept and thus fall within
its extension arc called inferiors of that concept. The concept itself, in
reference to these inferiors, is called a superior. Superior concepts are also
called logical wholes or potential wholes. Inferiors arc spoken of as sub­
jective parts.
These names are given from the point of view of extension and not
comprehension. As a matter of fact, in virtue of the inverse ratio between
comprehension and extension, these inferiors are superior and these
superiors inferior from the point of view of comprehension. From the
same point of view logical or potential wholes arc really parts, and sub­
jective parts arc really wholes. But from the point of view of extension
the concept with fewer comprehensive notes is superior, whereas the
concept with more comprehensive notes is inferior. From this vantage
point the less comprehensive concept can be considered a whole contain­
ing the more comprehensive concept as a part (of its extension) and
the superior is well described as a potential whole. Take rectilinear plane

32

THE LOGIC OF THE FIRST OPERATION

figure as an example. Rectilinear plane figure is superior to triangle and
to quadrilateral figure, since its comprehensive notes are found in both.
Rectilinear plane figure is actually neither triangle nor quadrilateral figure,
but rectilinear plane figure is, as it stands, potential to both the note
three-sided and the note four-sided. Hence, though actually neither,
rectilinear plane figure is potentially both triangle and quadrilateral figure.
It is aptly described as a potential whole in reference to these. The
inferiors are also well described as subjective parts. Whenever superiors
and inferiors are universally affirmed of one another, the inferior must be
used as the subject of the proposition. I can say Every triangle is a recti­
linear plane figure, but I cannot say Ever}' rectilinear plane figure is a
triangle.
Objects which are inferior in one relationship may be superior in
another. Thus man is inferior to animal but superior to Mary, and
parallelogram is inferior to quadrilateral figure but superior to square. This
is true of all universal inferiors, for they can always be considered superior
at least in reference to the singulars within their extension. However, no
singular inferior can ever be a superior, since singulars as such have no
extension.
There is one difficulty which should be mentioned here. We have said
that the superior can be said of its inferior. Thus man can be said of
Mary. Animal, organism, body, and substance can all be said of Man­
teo. There is no difficulty yet, for man, animal, organism, body, and
substance are all in the same logical line. All these concepts arc hier­
archically arranged in so far as they are more or less determinate expres­
sions of Mary’s essence. However, able to laugh and able to talk can also
be said of Mai}', and perhaps also pretty', tired, sitting down, and tapping
her foot. These can be said of Mary because these too arc superiors in
reference to her. Mar}' can be a subjective part of pretty just as she is a
subjective part of man. But she is not an inferior of pretty in the way she
is an inferior of man. The difficulty arises because pretty, tired, able to
talk, etc., arc completely different from man, animal, organism, body, and
substance. They have no common clement with these, and must, as we
shall see, be reduced to different categories. But if there is no element
common to both these groups of concepts, how can Mar}- be their
inferior since extension depends upon comprehension? To answer this
difficulty we must distinguish between the nature basic to an individual
and the many individuating characteristics possessed by an individual
as modifications of that basic nature. Later we shall refer to this as the
difference between substance and accident. It is true to say that an
individual has a nature basic to him. TTius we say Man- is a man. It is

COMPREHENSION AND EXTENSION

33

also true to say that in an individual that basic nature is modified by
individuating characteristics. Thus we say Mary is pretty. The compre
hensive notes of man are realized in Mary as constituting her very
essence. The comprehensive notes of pretty’ are also found, but they do
not pertain to her essence. They do, however, enter into the definition
of one of her nonessential attributes. Mary is inferior to man by way
of an essential relationship; Maty is inferior to pretty by way of a
nonessential or accidental relationship. Maty is inferior to man because
she is essentially man. She is inferior to pretty l>ccausc one of her
individuating attributes includes within it the comprehensive notes of
pretty. Man is an essential superior and pretty a nonessential superior in
reference to Maty. Later we shall distinguish between essential and
nonessential predication. Essential superiors are predicated of, or said of,
their inferiors essentially, and nonessential superiors arc predicated of
their inferiors nonesscntially. Nonessential predication is sometimes spoken
of as denominative predication.
Exercise III

1. Define: comprehension, extension, inferior (or subjective part), singular
inferior, superior (or logical or potential whole).
2. What is the property of comprehension? Of extension? Which is prior to
which? Explain.
3. Discuss three ways in which concepts can differ comprehensively.
4. How are comprehensive notes “added” to one another to constitute the
comprehension of a concept?
5. What docs it mean to say that the extension of a concept involves an
infinite multitude? Why does the extension of a concept involve this
infinite multitude?
6. What is the inverse ratio between comprehension and extension? Why
does this inverse ratio apply?
7. Does the addition of a comprehensive note to a concept make that concept
more meaningful or does it make for a new concept? Explain.
8. What is the difference between knowing an object confusedly and knowing
it distinctly?
9. Explain the precise significance of the terms, “logical or potential whole”
and “subjective part.”
10. How does an essential inferior differ from a nonessential inferior?
11. Arrange the following sets of concepts according to decreasing superiority;
explain the application of the inverse ratio in each case and name the
members of the sets as superiors or inferiors of one another:
11.1. Substance, dog, organism, brute, spaniel.

34

THE LOGIC OF THE FIRST OPERATION

11.2. Athlete, pitcher, ball player, baseball player, Grover Cleveland
Alexander.
113. Hearing, ability', cognitive faculty, sense, external sense.
11.4. Moral virtue, justice, good habit, habit, perfection.
11.5. “Horse,” sign, instrumental sign, term, conventional sign.
12. Prepare sets similar to those in No. 11 explaining the same points in
reference to them as you did for those in No. 11.

CHAPTER

IV

The Nature of the Universal

I. The Problem of the Universal

In distinguishing between sense knowledge and intellectual knowledge
we have noted that the former is of the singular whereas the latter is
characteristically of the universal. We have discussed the object of simple
apprehension both in terms of its intelligible content and of its universal
applicability to many. The universal character of our intellectual knowl­
edge is an incontrovertible fact of introspective experience. This fact,
however, has occasioned a philosophical controversy which dates back
almost to the beginning of philosophy itself. In this chapter we shall
look more closely at the universal, considering both the philosophical
problem which the fact of the universal sets up and the proper under­
standing of the universal which solves this problem. The solution to the
problem of the universal requires a distinction which should help con­
siderably to introduce the chapters which follow, on the predicables and
the categories respectively.
The object of simple apprehension is characteristically universal and
thus can be said of many.1 Man, for instance, can be said of both Mary
1 The abstraction which yields a nature which can be said of concrete subjects
mentally separates that nature from its individuating characteristics without, however,
positively cutting it off from its concrete subjects. The reason for this is that the
nature is conceived of as a whole which includes, in an indeterminate manner, the
concrete subjects in which it is realized and which are its subjective parts. However, the
same nature can also be conceived as a part which positively excludes all that goes
with it in any concrete subject. When it is conceived of in this second way the
nature cannot be said of any concrete subject. Man, white, and honest are products
of the first type of abstraction, whereas humanity, whiteness, and honesty are products
of the second. We can say Peter is honest. We cannot say Peter is honesty. However,
we can say Peter has honesty, for having honesty is equivalent to being honest. Man,
white, and honest arc examples of concrete concepts. Humanity, whiteness, and honesty
are examples of abstract concepts. To avoid confusion we must be careful to note that
both arc products of an abstraction, but that concrete concepts arc, in a sense, less
abstract and abstract concepts more abstract. Although an abstract concept cannot be
said of a concrete subject, it can be said .of an abstract subject. Peter is not honesty,
but one of Peter’s virtues is honesty. . /

36

THE LOGIC OF THE FIRST OPERATION

and Peter. Man, existing in the intellect as known in its first operation,
is truly communicable, that is, it can be shared by many. It is a universal
object with a determinate meaning in no sense limited to any individual.
But what of the real subjects of which man can be said, e.g., Man’ and
Peter? Each is a concrete individual. Each is a man only because each
possesses his or her own human nature. Mary’s human nature is not
Peter’s. Mary and Peter, as they exist independently of knowledge, are,
even as men, individuated, singular, incommunicable. How then can wc
legitimately say man of Mary, or of Peter, or of any individual man?
When I say Mary is a man, am I not identifying something universal and
communicable with what is singular and incommunicable? /\m I not
saying that Mary, unquestionably an individual man, having her own
unique human nature, is the universal man or is universally man? This
seems the case, since the nature (man) which I know is universal, and it
is this nature which I predicate of Mary. I seem to be saying that that
which is singular and incommunicable is something which is universal
and communicable.
The solution to this problem requires several distinctions. Considered
absolutely, that is, in itself, a nature is neither singular nor universal, but
is open to cither. In itself the nature man does not include universality.
If it did, it could never be realized in the individual, as an incom­
municable perfection. But we know that man is realized in this way in
Mary and in Peter. Nor does man exclude this relation. If so, it could
never be universal, as it is in the mind as an object of simple appre­
hension. Considered absolutely, that is, in itself, any nature is open to
the state of singularity which it enjoys outside the intellect and to the
state of universality which it enjoys as an object of the intellect, but in
itself it includes neither of these. Neither universality nor singularity is
a comprehensive note of any concept.
Wc can look at the universal object from two points of view, first,
as a determinate meaning with a definite intelligible content or compre­
hension; second, as related to the possible inferiors of which it can be
said. In other words, we must carefully distinguish the nature known
from the relationship of universality which relates it to many. 'The
nature and its universality are not indistinguishably one. Universality’ is
a special form which belongs to the nature precisely as known. It results
from the abstract condition of the nature as an object of simple appre­
hension. Tire nature is spoken of as universal only because it acquires
universality, which, as such, is distinct from it. To employ a distinction
introduced at the end of the last chapter we might say that the object of

THE NATURE OF THE UNIVERSAL

37

simple apprehension is universal denominatively rather than essentially.
To object to the proposition Mary is a man on the grounds that this
is to identify the universal with the singular is to ignore the distinction
between the nature which is universal and the universality which makes
it universal. It is true that Mary is a singular, and man a universal.
However, man is universal only because the mind confers universality on
it, and it is only the nature and not its universality which is said of Mary
or of any other individual man. In the proposition Mary is a man all that
is affirmed is that the nature man is really found in Mary. This nature is
considered as a meaning explained in terms of the comprehensive notes
substance, material, living, sentient, and rational. And these are real, even
though universality is not.

II. Abstraction by Way of Simple Apprehension and
Abstraction by Way of Negative Judgment

An object becomes universal to the extent that it is abstracted from
individuating characteristics. It is universal only in the mind; it is singular
in the real. Since truth is a correspondence between the mind and the
real some philosophers have objected that abstraction falsifies the object.
After all, man exists in the mind without any individuating characteristics,
but each existing man is concretely individuated. rlb solve the problem of
the universal presented in this way, wc must introduce a distinction
between two modes of abstraction. These are abstraction by way of simple
apprehension and abstraction by way of negative judgment.
In general abstraction ran be described as a mental separation. This,
however, can be done in two different ways. First of all, wc can think
one thing without thinking something else, even if both arc aspects of
the same real thing. For example, we can think man without thinking
white or six-feet tall. This is to abstract by way of simple apprehension.
Second, wecan think that one thing exists without another. This is quite
a different mode of abstraction. It is abstraction by way of negative
judgment. To abstract in the first way is simply to conceive of a nature
without simultaneously being intellectually aware of anything else. It is
to conceive a nature without being aware of those individuating and
existential characteristics which arc necessarily associated with that
nature in real things, This is to think an object separately, but —and
this is most important — it is not to think of it as though it were separate.
As long as an object can be understood apart from something else it

38

THE LOGIC OF THE FIRST OPERATION

can be abstracted from it in this way. Tins is true even though the nature
we know cannot exist apart from that from which it is abstracted. Not
everything can be abstracted from everything else in this fashion. Some
things are unintelligible apart from others and cannot even be understood
apart from them, let alone exist apart from them. These, of course, cannot
abstracted from the others, not even in an abstraction by way of simple
apprehension. Thus animal can be abstracted from man and brute because
it presupposes neither although it is open to both. But man cannot be
abstracted from animal, for man presupposes animal and cannot be
understood apart from it.
z\n abstraction by way of negative judgment is a commitment to the
existential separation of one thing from another. Tin’s is legitimate only
when that which is thought to exist apart from something else docs so
exist. It is legitimate to abstract man from stone in this way, for the
nature of stone and the nature of man never exist together in the same
thing. However, it is illegitimate to think that something exists apart from
something else when it does not, even though it can be understood
apart from the other. Thus, even though man can be understood apart
from tall, I cannot judge This man is not tall so long as this man actually
happens to be tall. To recapitulate: one thing can be abstracted from
another by way of simple apprehension as long as it is objectively (i.e.,
intelligibly) independent of the other, regardless of the existential rela­
tionship between the two; and one thing can be abstracted from another
by way of negative judgment only as long as it is both objectively and
existentially independent of the other.
The solution to our problem is clear. The object of simple apprehension
is abstracted from individuating characteristics, and it exists only with
individuating characteristics. However, it can be understood apart from
them even though it cannot exist apart from them. The abstraction
proper to the universal object is the first kind of abstraction, that is,
abstraction by way of simple apprehension. However, abstraction by way
of negative judgment would falsify the situation. The nature can be
known apart from its individuating characteristics since it docs not
depend upon them for its definition. Yet it cannot be thought to be
apart from them, for it docs depend upon them for its existence.
Ilie object of simple apprehension is never determinately the whole of
any real thing. Yet a partial view of reality is not necessarily false. The
sense of sight is not betrayed because it knows only the white of a white
thing which is also sweet and moist. Neither is the intellect betrayed
when it knows only an intelligible aspect of a thing which involves many
additional characteristics.

THE NATURE OF THE UNIVERSAL

39

III. The Division of the Universal Into the Predicables
and the Categories

A concept can be classified from the point of view of its extension
and from the point of view of its comprehension. From the point of
view of its extension or universality the universal is divided into five
types, because there are five different ways in which it can be said of
its possible inferiors. In other words, there are five different ways in
which a superior can be related to its inferiors. This docs not mean that
one concept can be related to the same inferior in five different ways.
A given concept is related to an inferior in only one way, but another
concept may be related to this inferior or to some other inferior in a
second way, and so on until we have discovered five different ways in
which a superior can be related to its inferior. Since these are the different
ways in which a universal can be said of (i.e., is predicable of) many,
these arc called the five predicables. From the point of view of its compre­
hension the universal is divided into ten types. Obviously there are more
than ten different concepts from the point of view of intelligible content.
Yet if we consider the most general classification of things, that is, the
division of being itself into its most universal types, we find that there
arc ultimately ten kinds of being. Every nature is basically one or other
of these types. They arc spoken of as the ten categories or the ten
predicaments. The ten categories divide a concept directly or essentially,
because they divide it according to differences in its essential content.
The five predicables divide a concept indirectly or denominatively, be­
cause they divide it according to the different relationships which it
takes on in the mind and which it bears in reference to its inferiors.
To illustrate, let us anticipate for a moment. We shall see that man is
ultimately that kind of nature which is substance, and that man bears the
relationship of species to its inferior Peter. As to category man is a sub­
stance; and as predicable, it is a species (in reference to Peter). Man is
essentially substance, since this is the ultimate classification of its actual
intelligible content. Man is denominatively species (in reference to
Peter), because the relationship it has (in reference to Peter) is precisely
that of species.
We shall return to the division of the universal into the five predicables
and the ten categories in the following two chapters.
Exercise IV

1. Define: universal, individual, universality, abstraction, abstraction by way
of simple apprehension, abstraction by way of negative judgment, objec-

40

2.

3.
4.

5.
6.

7.
8.

9.

10.
11.

THE LOGIC OF THE FIRST OPERATION

tivc independence, existential independence, abstract concept, concrete
concept, category, predicable.
What is the problem of the universal as it is manifested in the affirmative
proposition with a singular subject?
Distinguish between the nature taken absolutely and the nature in its
existential status.
Distinguish between the universal nature and the universality of the
universal nature.
Solve the problem of the universal as presented in answer to No. 2 through
the use of the distinctions given in the answers to No. 3 and No. 4.
Distinguish between abstraction by way of simple apprehension and
abstraction by way of negative judgment. Under what circumstances is
each abstraction legitimate? Under what circumstances would each falsify
things?
What precisely is the difference between thinking something (to be)
separate and thinking it separately? Which of these ways of thinking
applies to the object of simple apprehension?
What is the difference between objective independence and existential
independence? Which one implies the other?
What is the difference between a concrete concept and an abstract
concept? Which of the following arc concrete and which are abstract?
9.8. Honest
9.15. Growing
9.1. Clever
“
9.16. Universal
9.9. Human nature
9.2. Dexterity'
9.10. Courage
9.17. Relation
9.3. Speed
9.18. Maple Tree
9.11. Rational
9.4. Measles
9.19. Equal
9.12. Intellect
9.5. Logic
9.20. Action
9.13. Horrified
9.6. Logician
9.14. Growth
9.7. logical
For each of these supply the correlative abstract or concrete expression.
For example, happiness is abstract, and happy is its concrete correlative;
qualified is concrete, and qualification is its abstract correlative.
From what point of view is the universal divided into the ten categories?
From what point of view is it divided into the five predicables?
What is the difference between the division of the universal essentially
taken and the division of the universal denominativcly taken?

CHAPTER

V

The Predicables

I. The General Notion of the Predicables

As noted previously, there is a difference between the nature or
essence known in an act of simple apprehension and the relation of
universality given to this nature by the mind. Furthermore, we have
seen that it is in virtue of this universality that a nature can be said
of its inferiors. An analysis of the content of the natures wc know, that
is, of what those natures are in themselves, yields the ten categories, i.c.,
the ultimate types of universal natures from the point of view of their
essential content. An analysis of the universality of those natures, on the
other hand, yields the five predicables, i.c., the five different ways in
which a universal may be said of an inferior. For example, we shall
classify man in John is a man categorically as substance and predicably as
species. Man is in the category of substance because it is ultimately an
instance of substantial being in terms of its meaningful content, even
apart from its logical relation to John. Man — in this proposition — is
related to John by way of the predicable species. Man is classified in itself
or essentially as substance; it is classified in its reference to John or
denominatively as species.
II. The Division of Universal Natures Into Five Predicables

By studying some examples we can see that some universals can be
said of inferiors differently than others. Animal, able to talk, my brother,
and running can all be said, i.e., are predicable, of John, but no two of
these in quite the same way. John is essentially a man. Out of these four
possible predicates only animal is within the comprehension of the
concept man. The others arc extrinsic to John’s essence, and even they
differ one from the other. Able to talk is predicable of John as something
which belongs necessarily to all men, even though it is not within the
41

42

THE LOGIC OF THE FIRST OPERATION

comprehension of man; neither my brother nor running necessarily belongs
to all men. Still my brother is not related to John in the same way as
running is. John may be running now and not running later on, but
once he is my brother he cannot cease to be my brother. These examples
show clearly that universals differ considerably one from another as far
as their relation to inferiors is concerned.
What then are these differences? What are the different predicables?
First of all, a universal may be related to its inferior as something
comprehensively intrinsic to its essence or as something extrinsic to its
essence. If it is said essentially of its inferior it may express that essence
in whole or in part. In the former case, we have the predicable known
as species. This is the way man is said of John, for anything added to
human nature in John is nonessential. If the latter is the case, the
universal nature may be either the common and determinable or the
proper and determining part of the essence. The universal which is
predicable of its inferior as the common and determinable part of its
essence is called its genus. This is the way animal is said of John, for
animal is common to men and brutes and in itself can lx? more precisely
determined as rational or uonrational. The universal which is predicable
of its inferior as the proper and determining part of its essence is called
its difference. This is the way rational is said of John, for rational is
proper to (i.e., limited to) men and is that which determines animal
in men by contracting the genus to the species.
If a universal is essentially related to its inferior it is, of course, neces­
sarily related to it, for no subject can lack that which belongs intrinsically
to its essence. Thus species, genus, and difference are all essential and
necessary predicates. Whenever a universal is extrinsically related to a
subject, it is a nonessential predicate. However, this is not to say that
it is not necessary' to that subject. A universal which docs not belong to
the essence of its inferior may be necessary to that inferior because it is
an infallible consequent of the essence of that inferior. This type of
universal is called a property'. This is the way able to talk is said of John;
although able to talk is not one of the comprehensive notes of man.
it is an attribute necessarily present in any man. Of course not every
nonessential predicate is necessary. Any universal which is predicated of
a subject neither as one of its essential notes nor as a property necessarily
connected with that essence is neither essential nor necessary'. This type
of universal is called an accident. Running is predicable of John as an
accident, for it is neither intrinsic to his essence nor necessarily consequent
upon it. John may be running, but not all men are running, and John
need not be running and may, in fact, in a moment not be running.

43

THE PREDICABLES

TTiesc, then, arc the five predicables, the five ways in which a universal
can be said of its inferiors. The division of universal natures into five
predicables can be schematically presented in the following way:
A universal may be said of its inferior either —
♦he whole of its essence---------------------------- .specie*
essentially — °s\
N.

or

the common and
\
determinable part------------- .genus
part of its
/
essence — as<
or

the proper and
determining part---------------- difference
or
necessary in so far as properly
consequent upon its specific essence-------- property

nonessentially — asç

or

not necessary, i.e., not properly
consequent upon its specific essence--------- accident

III. The First Predicable — Species

Species is defined as a universal nature predicable of many things which
differ only nonessentially and numerically in answer to the question,
"What is its essence?” Since the species of a subject is the whole of the
essence of that subject, anything belonging to that subject over and above
its species belongs to it in the manner of a nonessential or individuating
characteristic. Hence, the only differences between the subjects of a
given species will be nonessential. These subjects can differ only in so
far as one is not the same subject as the other, although both are iden­
tically the same kind of subject. The species man can be said of Mary,
Peter, and John, and of any other man. Essentially these subjects arc all
men, and they differ only in so far as one is not the same man the
other is. In other words, as far as essence is concerned they differ only
numerically.
Predicables are defined in terms of their immediate inferiors. The
immediate inferior of the species is, of course, the individual. Tire only
subjects of which any species is predicable arc distinguishable only as
different individuals within the same species. An apparent contradiction
of this point arises in the case of those subjects of a species which
represent subclasses under the species and hence can themselves be
said of many. For example, man is predicable of Eskimo as its species,

44

TIIF. LOGIC OF THE FIRST OPERATION

and Eskimo is in turn predicable of many singulars. Nevertheless, the
proper subject of the species is the individual. Whatever is added to man
to make Eskimo is a nonessential characteristic or an individuating note.
Man is predicable of Eskimo and of any other subclass of men, just as it
is of Man; Peter, and John, as the whole of its specific essence. This is
simply to say that what makes an Eskimo an Eskimo is extrinsic to the
human nature which places him within the species man. However the
concept Eskimo, precisely as a complex concept, includes within itself
more than the intelligible content of man. What makes an Eskimo an
Eskimo is accidental in so far as any Eskimo is specifically a man. But
what makes an Eskimo an Eskimo is by no means accidental to an
Eskimo precisely as Eskimo.
Since the species is the whole of the essence of its inferior, there can
be only one species for any given inferior. It may be a difficult task to
determine this species for a given inferior. Our experience of ourselves
is firsthand and sufficient to indicate that our species is man. We know
surely that the whole of our essence is explained in human nature and
that the differences between Eskimos, for example, and other men are
nonessential. But what of Fido? Is he specifically brute, or dog. or spaniel,
or cocker spaniel, or what? What about a given individual bird? Is its
species brute, or bird, or sparrow, or English spanow, or what? Here our
experience is not firsthand, nor perhaps is it sufficient for us to determine
the species accurately in each case. As a matter of fact, it is not the
business of the logician to determine the species of Fido or of some
bird or even of himself. Ilie logician discovers the general rule for
determining the species of any given inferior. The logician points out that
a predicate is the species of its subject only if it expresses the whole
essence of that subject. But the task of determining the species in any
given case is not the logician’s; his task is properly that of determining
what it is to be a species. This, he says, is to be that kind of universal
nature which can be said of many things which differ only nonessentially
and numerically. The point made here concerning the determination of
species in a given case is true in reference to each of the predicables. Tire
logician, as logician, does not classify concepts in different fields, but
he does ascertain the instrument of classification by investigating the
very nature of the predicables themselves.
IV. The Second Predicable — Genus

Genus is defined as that universal predicable of many things which
differ in species, and, like species, in answer to the question, “What is

THE PREDICABLES

45

its essence?” Unlike species, however, the genus does not include the
whole essence but refers instead to the common and determinable part
of the essence of its subject. Consequently, whatever is immediately
added to any subject over and above its genus is an essential difference
contracting the genus to a determinate species. Thus the immediate
inferiors of the genus are species, e.g., triangle is a genus immediately
predicable of isosceles triangle, equilateral triangle, and scalene triangle.
Through these species, which themselves are immediately predicable of
individuals, e.g., this or that triangle, the genus too is predicable of
individuals. Hence, though it is defined in terms of inferiors differing
in species, for these are its immediate inferiors, the genus is also predi­
cable of subjects differing only in number. To give another example: the
genus animal is immediately predicable of man and, through man, of
Mary and Peter.
Unlike species, where there is only one to an inferior, there may, in a
sense, be several genera to an inferior. The genus, as we have seen, is the
common and determinable part of the essence of its subject. This common
part may itself be wholly or partially expressed. The whole of the commoi. part of the essence is called the proximate genus, because it is the
genus closest to the species. The proximate genus lacks only that final
essential difference which is necessary to determine the species fully. Thus
animal is the proximate genus of man (and through man of Mary, John,
and Peter). All that is lacking in the concept animal to make it the concept
man is the final comprehensive note rational, which is what differentiates
men essentially from all other animals. The concept animal includes the
whole of the common part of the concept man, and thus animal is the
genus closest to the species. But what of organism, body, and substance in
reference to man or to Man' or Peter? Each of these is a genus, for each
expresses a common and determinable portion of the essence of man.
Yet they differ from animal in that in turn each expresses a more and
more common and determinable part of the essence. Animal is common
to men and brutes and must be determined by rational to become man.
Organism is common to men, brutes, and plants and must be determined
by sentient and then rational to become man. Body is still more com­
mon and further removed from man. Just as animal is called the proximate
genus of man or Mary, so organism and body are called remote genera.
Substance is the most remote or ultimate genus.
In the strict sense the species expresses the entire essence of its
subject and is predicable properly only of individuals. In a less strict
Sense, we can say that species is formed whenever a genus is contracted
by a difference. In this sense a species would be anything of which a

46

THE LOGIC OF THE FIRST OPERATION

genus is immediately predicable. This is to consider species from the
point of view of its subjcctibility rather than its predicability. Looking
at species in this way, wc can say that animal, in relation to the genus
organism, is a species. If we do this, it is clear that what is a genus in
one reference is a species in another reference and vice versa. Thus body
is a species of substance and the genus of organism; organism is species of
body and genus of animal; and animal is species of organism and genus
of man. However, substance is only a genus and man only a species. Just
as substance is called an ultimate or highest genus so man is, from this
point of view, referred to as a lowest species. From this point of view
it is also possible that what is a remote genus in one reference may be
a proximate genus in another. Thus organism is remote genus of man
or Mary, but proximate genus of animal. Substance, of course, is simul­
taneously proximate and ultimate genus of body.
V. The Third Predicable — Difference

Difference is defined as that universal which is said of many things
differing either in number or in species in answer to the question, “What
is the quality of its essence?” The difference is the proper and determining
part of the essence, which, added to the genus, contracts the genus to a
given species. When the difference is the final perfection of the essence,
i.e., when it is the essential note added to the proximate genus to yield
the species of its subject, we call it the specific difference. Rational is
related to Mary as specific difference contracting the proximate genus
animal to the species man. Sentient, which contracts organism to the
proximate genus animal, is related to Mary as a generic difference. The
same is true of living and material since they are differences respectively
of organism and body.
The specific difference, like the species, is most properly predicable
of the individual. However, it can also be said of the species by way of
the individual. Thus rational is the specific difference of Mary and also
of man. In the strict sense, of the four differences mentioned above,
namely, rational, sentient, living, and material, only rational is a specific
difference. However, just as some genera can in a loose sense be con­
sidered species (c.g., animal as species of organism), so the three generic
differences of Mary and man can be considered specific differences,
respectively of animal, organism, and body. 'Ilius sentient is a generic
difference for Mary and man but a specific difference for animal. Living
is a generic difference for Man,', man, and animal but a specific difference

the predicabi.es

47

for organism. Material is a generic difference for Mary, man, animal, and
organism, but it is specific difference for body.
Thus far we have discussed the three essential predicables, namely,
species, genus, and difference. We have seen that for any singular
inferior there is but one species, one proximate genus, several remote
genera, one ultimate genus, one specific difference, and several generic
differences. Man’s species is man; proximate genus, animal; remote
genera, organism and body; ultimate genus, substance; specific difference,
rational; generic differences, sentient, living, and material. This is a com­
plete list of the essential predicates for Mary or for any individual man.
A similar list is difficult to determine for other subjects, with the excep­
tion of some mathematical entities, because our knowledge of their
essences is less perfect. However, ultimate and remote genera and generic
differences arc fairly easily determined in all cases. Most of the serious
difficulties arise in determining proximate genus, species, and specific
difference. There is no doubt that for Fido substance is the ultimate
genus, with body, organism, and animal as remote genera and material,
living, and sentient as generic differences. It may be that brute is the
proximate genus and that dog is the species. The specific difference
probably cannot determinately be named.
VI. The Fourth Predicable — Property

Species, genus, and difference are essential predicates because they
belong to the essence of their subject. But these do not exhaust the list
of possible predicates. As a matter of fact, more often than not what we
know about a given subject can be said of that subject only as something
extrinsic to its essence. As we saw, there arc nine predicates essentially
related to Mary, each predicable either as species, genus, or difference.
But there are probably hundreds of other predicates we could apply to
Mary. We may know that she is studious, pious, pretty, John’s cousin,
able to talk, a mathematician, able to laugh, tired, hungry, nearsighted,
artistic, biped, twenty-two years old, in Boston, etc. Each of these is a
universal which can be said of Mary’. But none can be said essentially,
since none of them is within the comprehension of the concept man.
Since each expresses something outside Mary’s essence, each is predicable
of Mary only nonessentially.
None of these attributes is intrinsic to Mary’s essence, but some of
:hcm are so intimately tied up with her essence that they follow neces­
sarily from it. These are attributes which belong to all men exclusively

48

THE LOGIC OF THE FIRST OPERATION

and always. Such nonessential but strictly necessary attributes are called
properties of their subjects. Property can be defined, then, as that
universal which is said of its inferiors nonesscntially but with strict and
exclusive necessity.
Neither Mary’s ability to talk nor her ability to laugh is intrinsic to
her essence. Neither makes her to be a man. But both follow infallibly
from her essence, since they are necessarily caused by the principles
proper to her essence, ultimately by her specific difference. She is a man
not because she can talk and laugh, but she can talk and laugh precisely
because she is a man. Every' man can talk and laugh, men always can
talk and laugh, and only men can talk and laugh. Able to talk and able
to laugh, then, are universals related to Mary’ as properties. 'Ilie intimate
connection between the specific difference and the property should be
carefully noted, along with the differences between them. Very frequently,
as we have pointed out, it is difficult to determine the specific difference.
In such cases, especially when we are attempting to define a concept,
it is helpful to determine the properties of the subject. As long as we
realize that they are extrinsic to the essence we can profitably use them
to point to the essence, since they arc connected necessarily with it. Both
properties and specific differences belong always to all the members of a
species and only to the members of the species. They' differ in that the
specific difference is the final note constitutive of the essence, while the
property follows necessarily from the essence without in any way con­
stituting it. A subject is a man because it is rational; it is able to laugh
because it is a man.
Properties, strictly taken, like specific differences, are predicated of
individuals, and through them of species. But, just as there are specif:
and generic differences, so too there are specific and generic properties
And just as there are differences generic in one reference and specn|
in another, so there arc properties generic in one reference and specif:
jin another. Just as rational is the specific difference of Mary’ because
it differentiates her species from all other species, so able to laugh B
a specific property of Mary because, although nonessential, it belong’
always and only to every’ member of her species. And just as niatenx
is a generic difference for Mary because it differentiates the genus boà
from spiritual substance, so corruptible is a generic property for Man
because it belongs nonesscntially always and only to every' member of tlx
genus body'. Furthermore, just as material is a generic difference t':
Mary but the specific difference for body, so corruptible is a generic pro?
erty for Mary but a specific property for body. In similar fashion, liavi#
three angles equal to two right angles is a generic property for isosed*

THE PREDICABLES

49

triangle, since it is said nonesscntially always and only of all the sub­
jective parts of triangle, which is the proximate genus of isosceles triangle.
We might profitably suggest several other examples of property. Able
to fashion and use tools, able to take on the virtue of art, and possessed
of natural logic would each be a specific property for man. So similarly
having a boiling point of 100 degrees centigrade at sea level is a specific
property for water; having three equal angles, a specific property for
equilateral triangle; and interfertile only with members of that species,
a specific property for any given biological species. Able to know sensible
phenomena would be a generic property for man; capable of filling a
place, a generic property' for water; divisible into parts with a common
boundary, a generic property' for equilateral triangle; and able to support
noncssential modifications, a generic property for any biological species.
There is one word of caution in reference to the predicable property.
To constitute a property it is not sufficient that an attribute belong
as a matter of fact always and only to every instance of a given species.
An attribute is a property only if it is so connected with the essence of
a species that it is impossible for it not to be necessarily and exclusively
an attribute of that species. A property' is so related to its species that it
would involve a contradiction to think of that species existing without
the property. An attribute might, as a matter of fact, happen to belong
always and only to every' existing (past and present) member of a given
species without being a property' of that species. If the species could be
thought of —without contradiction — possibly as existing without this
attribute, it is something less than a property.
VII. The Fifth Predicable — Accident

Those universals which must be said of inferiors neither essentially
nor with the necessity of properties are accidents. Accordingly, accident
is defined as a universal said of many things nonesscntially and con­
tingently. T hus, sitting, John’s cousin, and biped are all said of Mary' as
accidents, for none of these is intrinsic to her essence nor is any of these
a necessary and exclusive property cither of her species or genus.
In a sense accidents are the easiest of the predicables to determine.
Yet even here we must distinguish the several kinds of accidents. Mary
is sitting, but in a moment she may not be sitting. Mary is John’s cousin
and always will be, but John is not his own cousin. Mary' is biped and
so is every other man, and they always will be. Thus, though none of
these has the necessity of a property, John’s cousin is more necessary'
than sitting, and biped is more necessary than John's cousin. Since any

50

THE LOCIC OF THE FIRST OPERATION

man may at any moment either be sitting or not sitting, sitting may be
classified as a wholly separable accident. John’s cousin, on the other
hand, is partly separable and partly inseparable. Inasmuch as some mem­
bers of the species are John’s cousins and some not, the attribute of
being John’s cousin is separable from the species; but since those who
arc John’s cousins cannot cease to be so, this attribute is inseparable
from the individual. Johns cousin, then, may be classified as an accident
separable from the species but not from individuals within the species.
Biped is wholly inseparable, since no man can be without being biped.
Biped falls short of being cither a specific or generic property inasmuch
as it is proper neither to the species (for birds arc also biped) nor to
the proximate genus (since dogs are not biped) nor to any remote genus
(since it is not true even of all the members of the proximate genus). This
subdivision of the predicable accident into wholly separable, partially
separable, and wholly inseparable accidents is helpful in that it manifests
the nature of accident in the variety of its instances. However, these
differences are of little logical significance. The important point here is
that all accidents, whatever their differences, arc alike in that each is
related to its inferior nonessentially and with something less than strict
and exclusive necessity. This characteristic, common to all accidents,
renders any accident of relatively little logical significance for purposes
of definition or proof.
Examples of accidents, especially wholly separable accidents, are easily
determined. When wc say that Mai}- is pretty, or studious, or a mathema­
tician, or in Boston, each time we predicate a separable accident of her.
When wc say she is feminine wc predicate of her an accident inseparable
from the individual but separable from the species. And when we say
that she can breathe we predicate of her a wholly inseparable accident.
VIII. The Limitations of This Division

We have said that the five predicables arc the five ways in which
universals can be related to inferiors. Strictly speaking, the universals we
refer to must be understood with certain restrictions. For a universal to
be technically one of the predicables it must be a superior, i.c., an object
which can be said of an infinite number of actual or possible inferiors
in such a way that it has identically the same meaning for each. Not even
predicate is a universal in this sense of the term. Some predicates are not
related to their subjects as superior to inferior: for example, a predicate
might be singular or it might be the definition of its subject. Othct
predicates are said of their subjects with something less than identical

51

THE PREDICABLES

the same meaning in each case: we shall refer to these later as analogous
predicates. Thus it is clear that the division of universal natures into
five predicables is not so broad a division as to encompass all possible
predicates. It applies in the strict sense only to a limited number of
predicates. However, it is highly important division. First of all, the
kind of predicate it docs divide is of major concern to us. Second, those
predicates which arc, strictly speaking, outside its scope can be themselves
classified by way of comparison with the predicables (as in the case
of objects described as quasi-genera or quasi-properties, etc.).

IX. The Tree of Porphyry

'Hie explicit listing of the predicables as species, genus, difference,
property, and accident comes to us originally from the writings of
Porphyry, an early commentator on the works of Aristotle. A schema
representing the essential interrelations between concepts from the
ultimate genus to the lowest species and even to the singular inferiors
of thi species has been reproduced through the centuries bearing the
name of Porphyry. Because of its appearance it is called the Tree of
Porphyry. It is helpful for students to use schemata of this type in
order to sec graphically the meaning and force of the predicables. 'Hie
Porphyrian Tree, you should note, has to do only with the three essential
predicables: species, genus, and difference. Use the schema in conjunction
with the explanation of these three predicables as given above in sections
III, IV, and V of this chapter.

THE TREE OF PORPHYRY

Substance

Material

»■ Body.

Sentient

^Animal

Nonmaterial------- Spiritual substance, etc.

Nonsentient ———Plant, etc.
Rational ^^-^^Man

Nonrational
Mary, John, Peter, etc.

Brute, etc.

52

THE LOGIC OF THE FIRST OPERATION

Exercise V

1. Define: species, genus, proximate genus, remote genus, ultimate genus,
difference, specific difference, generic difference, property, specific property,
generic property', accident, wholly separable accident, wholly inseparable
accident, partly inseparable accident.
2. The predicables directly or essentially divide universality; indirectly or
denominatively, the universal nature. Explain.
3. Explain why the predicables are divided into five.
4. What docs it mean to say that species is said of many which differ only
in number?
5. What part precisely docs logic play in determining whether a predicate
is species (or genus, or difference, etc.) of its subject?
6. Genus is defined as that which is said of many differing in species. Yet
animal is genus for both Mary and Peter, and they differ only in num­
ber. Explain.
7. How do the several types of genera differ from one another although
they remain alike in so far as each is a genus?
8. Why is difference defined as that which is said of many differing in
number or species?
9. Both property and difference are said always of all of the members of a
given species or genus and only of these. I low do they differ?
10. How does accident in general differ from property, and how do accidents
differ among themselves?
11. What is wrong with speaking of the predicables as the five ways in which
a predicate can be said of its subject?
12. Practice reproducing the Tree of Porphyry, naming each entry in its
reference to the entries above, below, and alongside of it.
13. Name the predicable relationship referring the objects on the right to
those on the left in each case:
13.1. Logic........................................................ art
13.2. Mary........................................................ tall
13.3. Body........................................................ material
13.4. Genus...................................................... predicable
13.5. "Horse” .................................................. sign
13.6. Fido ........................................................ boxer
13.7. Fido ........................................................ four-legged
13.8. Fido ........................................................ sentient
13.9. Fido ........................................................ corruptible
13.10. Fido ........................................................ hungry
13.11. Fido ........................................................ male
13.12. Fido ........................................................ son of Spotty
13.13. Fido ........................................................ substance

the predicables

53

13.14. Triangle .................................................. three-sided
13.15. Triangle .................................................. possessed of three angles
equal to two right angles
13.16. Mary........................................................ artistic
13.17. Mary........................................................ rational
13.18. Maple tree.............................................. plant
13.19. Intellect .................................................. faculty
13.20. Piety........................................................ virtue
13.21. Triangle .................................................. plane figure
. Candy...................................................... sweet
13.23. Coal ........................................................ black
13.24. Organism ................................................ living
13.25. This rock................................................ large
13.26. Tin's rock................................................ body
13.27. This rock ................................................ extended in space
13.28. Tin's maple tree...................................... belongs to John
13.29. This maple tree...................................... can function vegetatively
13.30. Species .................................................... predicable
14. Suggest as many predicates as you can think of for any given subject, and
then classify' each as to predicable relation.

CHAPTER

VI

The Categories

I. The General Notion of the Categories

There is a distinction between the nature known in simple apprehension
and its relation of universality to many inferiors. Our analysis of the ways
in which universal natures can be related to their possible inferiors has
revealed the five predicables. Now we shall consider the different ways in
which universal natures can be classified from the point of view of their
comprehensive content. This analysis will reveal the ten categories. The
ten categories are the ultimate types of natures. TTiis is to say that they
are the most general classifications of being. They divide possible pred­
icates on the basis of fundamental differences in intelligible content.
They are also spoken of as the ten “predicaments,” a name taken from
the Latin word for predicate, just as the word “category” is taken from
the Greek word for predicate. The difference between the predicables and
the categories can be seen in terms of two different questions (and their
answers) which can be asked about any universal concept. The answer
to the question, “What kind of being is known in this concept?” will
be given as one of the ten categories. 'Hie answer to the question, “How
is this concept related to a given inferior?” will be given as one of the
five predicables. To repeat an example given Ixffore in the same context:
man considered in its intelligible content is classified in the category
substance, because ultimately it is that kind of being which is apt to
exist in itself and not in another; man considered in its logical relation
to Mary or any other individual man is classified prcdicably as species,
for it expresses the whole essence of Mary and of all individual men.
II. Preliminary Considerations

Before discussing the division of being into its ten categories, there
are several preliminary points to consider. The division is not nearly so

THE CATEGORIES

55

simple as it might seem at first glance, and these points must be clear
before the division can be understood. Aristotle himself prefaces his work
on the categories with several distinctions, which have come to be known
as the antcprcdicaments, that is, “before the predicaments or categories.”
First of all we must distinguish carefully between univocal, equivocal,1
and analogous names or terms. Terms arc signs of concepts and as such
signify meanings. A term is said to be used univocally if it is used several
times to refer to identically the same meaning. To say “man” of Mary
and John, to say “animal” of Mary and Fido, and to say “triangle” of
an isosceles triangle and an equilateral triangle is each time to use a term
univocally. Mary- and John arc equally man. Mary and Fido are equally
animal. And the isosceles triangle and the equilateral triangle are equally
triangle. However, the same term may be used several times to refer to
utterly unrelated meanings. When this happens the term is said to be
used equivocally. Thus I speak equivocally when I use the same term
“sound” to refer both to the object of the sense of hearing and to a long
passage of water joining two large bodies of water. It is purely by chance
that we use the same name in these two instances. This is also the case
when I call a certain instrument for writing and an enclosure for young
children a “pen” or the external covering of a tree and a small sailing
vessel a “bark.” Sometimes one term may stand for different but related
meanings. Such a term is said to be used analogously. An analogous term
is always said in a primary sense of one of its subjects and in a secondary
sense of the others. To understand what the analogous term means in
reference to the subjects of which it is said secondarily we must under­
stand what this term means when predicated of its primary- subject. Thus
the term “proud” can be said primarily of certain men and secondarily
of peacocks. It does not mean the same thing in each case, but there is
a relationship between the two cases so that we understand the second
use by referring to the first use. Likewise the term “healthy” is primarily
said of physical organisms and secondarily of the weather, “talking” is
primarily said of men and secondarily of parrots, “good” is primarily said
of God and secondarily of creatures. As a matter of fact there arc signifi­
cant differences in analogous terms. However, we can leave the division
of analogy into its types for consideration elsewhere.
Our second preliminary consideration is the distinction between com­
plex and incomplex objects of thought. Here it is sufficient simply to
review what we have already seen. In one sense the object of simple
apprehension differs from the object of judgment as the incomplex from
the complex. In simple apprehension we merely grasp a nature or quiddity.
In judgment we know not only quiddity but the act of existing, for

■i

56

THE LOGIC OF THE FIRST OPERATION

wc make an affirmation or denial about the way a nature is. In another
sense one object of simple apprehension may differ from another as the
incomplex from the complex. From the point of view of what is known,
man and stone are both incomplcx objects of simple apprehension, while
healthy man and pebble are both complex. Man involves only human
nature and stone only stoneness. But healthy man involves both human
nature and health, while pebble involves both stonencss and smallness.
Our third preliminary consideration is twofold. It has to do, first,
with the distinction between the universal and the singular. The universal
(e.g., man or dog) is an object which can be said of many. The singular
(e.g., Mary or Fido) is incommunicable and can in no way be said of
many. Second, we must distinguish between that kind of a nature which
is such that it can exist in itself and not in another and that kind of
nature which is such that it can exist only in another as in a subject. This
is the distinction between substance (e.g., man or dog) and accident (e.g.,
tall, white, hungry). Height is always the height of something, and color
is always the color of something. Height and color exist only in so far
as they inhere in a subject. They arc accidents. One accident may inhere
in another. But there can be no accident whatever unless something
exists in itself and does not inhere in a subject. An infinite scries of
accidents is an absurdity. Human nature is the kind of being which
exists in itself and is ultimately the subject of accidents. Human nature is
a substance. Combining these distinctions we can differentiate between
singular substances, universal substances, singular accidents, and uni­
versal accidents. Singular substances (called priman' substances) neither
exist in a subject nor arc said of a subject, e.g., Man,’, Fido, this maple
tree. Universal substances (called secondary' substances) do not exist
in subjects but are said of subjects, e.g., man, dog, maple tree. Singular
accidents exist in a subject but are not said of a subject, e.g., Man’s
kindness, Fido’s hunger, this maple tree’s greenness. Universal accidents,
finally, both exist in subjects and are said of subjects, e.g., kind, hungry,
green.
A final preliminary distinction is that between real being and being of
the reason. Real being, in contrast to a being of the reason, either docs
(actual real being) or does not but can (possible real being) exist outside
the mind. On the contrary', a being of the reason is a being which can
exist only as known. It is contradictory to think of a being of the reason
existing anywhere except in the intellect as an object known. Men, dogs,
and maple trees are examples of actual real being. A better world and a
thirty-five-year-old President are examples of possible real being. All
logical relations such as genus or species, fictitious beings like two-headed

THE CATEGORIES

57

dragons, negations and privations such as blindness or deafness, and
some mathematical entities such as the square root of a minus number
arc examples of beings of the reason. Men, dogs, and maple trees may
exist in the mind as known, but they also exist outside the intellect
quite independently of our act of knowing them. A better world and
a thirty-five-ycar-old President actually exist only in the mind as known,
but there is nothing in their natures as known which would make
existence for them outside the intellect a logical impossibility. However,
genus, blindness, and the square root of minus three exist only in the
intellect as known and arc such that it wnuld be contradictor}' for them
to exist in any other fashion. Only the latter are beings of the reason;
the others are real beings, actual or possible.
III. What Kind of Being Can Be Categorized

When we speak of being as divisible into its ten ultimate types, which
arc the categories, wc must be careful to note that only certain kinds of
being can be categorized. Not all being fits into one or other of the
categories. Before investigating the division of being into the categories,
then, wc should consider these restrictions, for no division makes sense
unless we know precisely what it is w-e arc dividing. These restrictions can
be listed as six in number:
1. Only real being is categorized. Only real being, that is, being w hich
does or can exist outside the intellect can be categorized. Thus beings
of the reason such as logical relations (e.g., genus), fictional beings (e.g.,
chimera), certain mathematical entities (e.g., square root of minus
three), privations (e.g., blindness) and all other beings which exist only
as known are not categorized.
2. Only incomplex natures are categorized. This means, first of all.
that only objects of simple apprehension can be categorized. The objects
of judgment, since they are complex, cannot. Second, it means that only
the incomplex object of simple apprehension can be categorized. Objects
of simple apprehension which are complex in themselves (e.g., healthy
man or pebble) arc composites of two or more natures; each of these
in its own right fits into a given category, but the complex object formed
by their union does not. To anticipate for a moment, man is categorized
as substance and health as quality. Healthy man, therefore, cannot itself
be classified in any one category. The same is true of any artificial nature
ie.g., table or pencil), for all artificial natures are complex, since they are
composed of a material principle (one nature) plus an added artificial
form (a second nature).

58

THE LOGIC OF THE FIRST OPERATION

3. Only finite being is categorized. The categories are particular modes
of being, each representing a determinate limitation or contraction of
being. Infinite being defies limitation and cannot be categorized.
4. Only univocal terms are used in classifying natures according to
category. All equivocal terms and some analogous terms stand for a
plurality of entirely different natures. These natures may fit into a variety
of categories. This is the case when “bark” is used equivocally in relation
to trees and dogs. 'Ilie bark of a tree is categorically substance, whereas
the bark of a dog is categorically action. This is also the case when
“foot” is used analogously in reference to a part of the human body and
to a rank in a class. The foot of a body is categorically substance, whereas
the foot of a class is categorically relation. Some analogous terms do not
stand for a sheer plurality of entirely different natures. For example,
“good” can be said of a man and also of his complexion. In each case
“good” stands for something different, but for things which arc never­
theless proportionally alike. It is not the same thing for a man to be
good and for his complexion to be good. But each is good precisely
in so far as each is what it should be. Analogous terms such as “good”
stand for a perfection which is called transcendental because it “spills
over” from one category to another. Since this is the case it cannot
be said to belong to any one category.
5. Only complete natures arc categorized. Those things which are
called beings only in so far as they arc parts of a whole which is properly
the being in question are not beings; hence they arc not categorized.
In so far as they arc parts of a whole, however, they can be reduced to
the category of the whole. Because man is in the category' of substance
wc can classify his parts as substance. Thus his hair, his teeth, his blood,
etc., arc classed as substance. This is also true of his body and soul con­
sidered as parts of his nature. 'Hie comprehensive notes of any concept,
sometimes called metaphysical parts, arc categorized like that concept.
When expressed as genera they belong directly to the category', and when
expressed as differences they belong indirectly to the category. Thus,
animal and organism, like man, arc directly in the category substance;
while rational, sentient, and living are indirectly categorized as substance.
6. Only universal natures arc categorized. In the strict sense the
categories arc classifications of possible predicates. As such they represent
in the strict sense a division only of the universal nature. Since the con­
tent of a universal nature is actually found in individuals, the categories
can themselves be said even of singulars. Thus, though John is not said
to be categorized as substance, John is nevertheless a substance, lire same

THE CATEGORIES

59

cannot be said in reference to the predicables. Though man is species, it
is not true that John is a species.
IV. The Division of Being Into Ten Categories

l*he categories are the ultimate kinds of being. They represent the
basic types of being which we meet in experience. As such, they refer
to the different kinds of intelligible content which is offered to our minds
bv the natures we know in simple apprehension. At the same time, the
categories are the ultimate genera of being, that is, the broadest possible
univocal predicates which can be said of subjects. The property of being
a predicate is a logical property which finds its basis in the way its subject
exists in the real order. This means that the modes of predication follow
upon the modes of being and can be taken as reflections of them. 'Ibus
we can discover what the different categories are by considering the
differences we find in predication. Let us see what an analysis of this
kind shows. If the predicate expresses what the subject is, it is (1)
substance; if it expresses a modification of the subject, it is an accident.
If it is an accident, it will be taken either from something intrinsic to
the subject or from something extrinsic to the subject. If it is taken from
something intrinsic to the subject, it can be taken cither from that which
is absolutely intrinsic or from that which is only relatively intrinsic. An
accident taken from something absolutely intrinsic will be taken either
from the matter or from the form of the subject. That accident taken
from the matter is (2) quantity; that taken from the form is (3) quality.
The accident taken from what is relatively intrinsic to the subject is
(4) relation. An accident taken from something extrinsic to the subject
can be taken from something considered either under the aspect of a
cause or under the aspect of a measure or under the aspect of neither.
That accident belonging to a subject as cause of an effect in something
extrinsic is (5) action; while the accident belonging to a subject as
suffering an effect brought about by something extrinsic is (6) passion.
Subjects are cxtrinsically measured by time or place; thus, the accident
belonging to a subject because of an extrinsic measure will be taken
either from time or from place. The accident belonging to a subject
because of the time in which it is situated is (7) when. The accident
arising in a subject simply because of the place in which it is found is (8)
where, and because of the precise order or disposition of its parts in this
place, (9) position. Finally, the accident belonging to a subject because
of something wholly extrinsic to it which is neither a cause nor a measure,
c.g., its clothing, is (10) having possession.
Illis division into ten categories can be seen in the following schema:

Predicate signifying
what the subject is

Predicate lignifying something belonging to the
subject as a modification thereof (accident)

THE CATEGORIES

61

Since a predicate says something about its subject, we can think of
the categories as answers to ten ultimately different questions that can
be asked about a subject. These questions can be formulated in this
fashion:
1. What is the subject? (Substance)
2. How much (or how many) of this subject is there? (Quantity)
3. What kind of a subject is this? (Quality)
4. How is this subject referred to other things? (Relation)
5. What is this subject doing to something else? (Action)
6. What is being done to this subject by something else? (Passion)
7. When is this subject doing something or being acted upon? (When)
8. Where is this subject? (Where)
9. What is the posture of this subject? (Position)
10. How is this subject dressed, adorned, or otherwise artificially
equipped? (Having possession)
Before discussing each of the categories in detail, we should clarify
one point concerning this division. The categories themselves are univocal.
This means that all those things which are in the category of substance
are equally substances, i.e., that substance means the same thing for each
of them. The same is true of quantify, quality, relation, and the others.
Yet these ten categories are themselves only analogously related to being.
Because being does not mean precisely the same tiring in reference to
each of them, they are spoken of as particular modes of being rather
than species of the genus being. “Genus” properly refers to a uni­
versal which can be divided into species through opposing differences
which add something to the genus. Thus animal is contracted to man
through the addition of rational, a note different from and determinative
of animal, and to brute through the addition of nonrational, similarly a
different and determinative note. But nothing can be added to being
which is different from being save nonbeing, which is, of course, nothing.
Hence there is no difference which could contract being as a genus to a
species inferior to it. The categories differ from one another, but the ven'
differences between them are, i.e., are beings. Hence being already in­
cludes the differences which distinguish its different modes and, accord­
ingly, already includes the different modes themselves. Being can be
considered a superior in reference to the ten categories, but not in the
way in which we have seen that a predicable is a superior. A univocal
superior, that is, a predicable, does not actually contain its inferiors;
it includes them only potentially. Being, on the other hand, is an
analogical superior or analogical whole. This means that being actually
includes its inferiors and that they are subjective parts in only a loose

62

THE LOGIC OF THE FIRST OPERATION

sense. Being is not a genus; it is at best only a quasi-genus. We cannot
here examine this extremely important point. However, we can note that
being, although analogically superior to substance and all its inferiors,
is not, like a univocal superior, related to these inferiors as a concept
which has greater extension and less comprehension. Being, in fact, is
both the most extended and, in a sense, the most comprehensive of
concepts. The law of the inverse ratio between comprehension and
extension applies strictly only to univocal concepts. Furthermore, we
have noted that being is not a genus. Consequently, the categories are,
as we have said above, the ultimate genera. Finally, even the differences
between the various modes of being are themselves being. This makes it
extremely difficult to define the categories in such fashion as clearly
to show how they do differ from one another as distinct modes of
being. This last observation leads us to the next section in this chapter
where we shall attempt to define the categories and give examples of each.

V.

The Ten Categories Considered Singly

1. Substance
Substance is defined as that nature which is apt to exist (i.e., calls for
existence) in itself and not in another.
Examples of substance are man, maple tree, horse, iron, water. These
are substances precisely because they are the types of natures which call
for existence in themselves, not in another. This does not mean they
exist by themselves, for each possesses the property of upholding accidents
which inhere in it as nonessential modifications. No substance — save
God, and God, as infinite being, cannot be categorized — exists without
some accidents. These accidents are beings which call for existence in
another as in a subject. Accordingly they are sometimes referred to as
beings-of-a-being, while substance is being simply speaking.
2. Quantity
Quantity is defined as that accidental modification of a subject which
causes it to have parts outside of parts.
Examples of quantity are six feet tall, sixty-four pounds, two quarts,
seventeen. Quantity is the first accident arising in material substance,
and it is that accident whereby the subject can be divided into constituent
parts and measured by number. Quantity is divided into continuous
quantity or extension and discrete quantity or number. Discrete quantity
is derived from continuous quantity by the separation of one part of

THE CATEGORIES

63

continuous quantity from another. Since it is the first accident arising in
material substance, quantity is an accident through which other accidents
inhere in material substance (c.g., color by way of surface).

3. Quality
Quality is defined as that accidental modification of a subject which
causes it to be determined in itself in a certain fashion.
Examples of quality are best understood if we first point out four
types or species of quality. These are (1) perfections of a substance
disposing it well or badly in its being (entitative qualities) or in its
operation (operational qualities) which are cither firmly founded (habits)
or are easily lost (dispositions); (2) abilities, i.e., potencies for operation,
including their deficiencies but not the lack of them, i.e., their privation;
Í 3) passive sensible characteristics which affect the sense powers; and
4) the qualitative termination of quantity known as figure or, if it
includes aesthetic proportions, form. Examples of the first species include
healthy, vicious, mathematician, even seeing, understanding and loving,
as we shall see later. Examples of the second are able to laugh, able to
sec. masculine, feeble-minded, and nearsighted. Examples of the third
arc red, sweet, pungent, and cold. Examples of the last are curved, tri­
angular, and oblong.

4. Relation
Relation is defined as that accidental modification of a subject which
consists simply in its reference to another.
For a fuller understanding of relation we should distinguish between
the relation itself (e.g., paternit)'), the subject of the relation (e.g.,
John), the term to which the subject is referred by the relation (e.g.,
John’s son), and the foundation of the relation (e.g., John's generative
act in begetting his son). With these distinctions in mind we can suggest
the following examples of relation: father, near, taller, teacher, equal.
5. Action
Action is defined as that accidental modification of a subject which
belongs to it precisely as the principle of a transitive operation.
Transitive operations differ from immanent operations in so far as
‘hey begin within an agent and pass from it to an extrinsic patient.
Immanent operations remain within the agent for its own perfection.
Pushing is an example of a transitive operation and seeing is an example
of an immanent operation. Immanent operations qualitatively modify
their subjects, whereas transitive operations modify their subjects in the

64

THE LOGIC OF THE FIRST OPERATION

category of action. Examples of action include striking, pulling, lighting,
injuring, helping, and tying.
6. Passion
Passion is defined as that accidental modification of a subject which
belongs to it precisely as the term of a transitive operation.
Examples of passion include being struck, being pulled, being hit, being
injured, being helped, and being tied.
7. When
When is defined as that accidental modification of a subject which
belongs to it precisely because it is measured by time.
Examples of when include today, in 1492, last month, and in the Middle
Ages.
8. Where
Where is defined as that accidental modification of a subject which
belongs to it precisely because it is measured by place.
Examples of where include in Massachusetts, at home, in the kitchen,
down South, and on the ball field.

9. Position
Position is defined as that accidental modification of a subject which
belongs to it precisely because of the disposition or arrangement of
its physical parts in a given place.
Examples of position include stooped over, standing erect, bent, sitting
down, and at ease.
10. Having possession
Having possession is defined as that accidental modification of a sub
ject which belongs to it precisely because it is clothed, armed, or other­
wise possessed of artificial trappings and equipment.
Having possession is perhaps the least easy of the categories to unde
stand. Strictly speaking, only man is subject to this accident. Other
subjects have it only under the care of men. Since man is a ration*
being, it is neither fitting nor necessary that he be substantially
equipped with all that he needs for survival and for the happy lifebecause he can devise by way of art the clothes, ornaments, and ano
he needs to lead the full life. When he actually possesses these extent*
trappings, man is accidentally perfected by way of having possess:-?!’
Examples of having possession are being dressed, wearing earrings, anntf
to the teeth, and shod.

THE CATEGORIES

65

Do not confuse the accident of having possession with a man’s
clothing or where with place or when with time. Having possession is an
accident belonging to a subject because of his clothing; where, because
of the place he is in; and when, because of the time. But in no instance
is the accident to be confused with the thing from which it is taken.
VI. The Categories and the Predicables

It is important to keep in mind the distinction between the categories
and the predicables. 'Ihc categories divide natures absolutely or in terms
of their comprehensive content. The predicables divide natures on the
basis of the universality referring them to their inferiors. Any univocally
universal concept, as long as it fits the requirements for categorical being,
can be classed both as a category' and as a predicable. For example, take
the proposition: Mary is sitting down. Here the predicate sitting down
is in the category of position and, as a predicable, is related to Mary
as a wholly separable accident. Or take: Fido is an organism. Here the
predicate organism is a substance prcdicably related to Fido as a remote
genus. In some cases predicates defy classification as categories because
they do not meet all the requirements for categorical being. These
predicates may, however, allow' for classification as predicables. As an
example, take the proposition, John is deaf. The predicate deaf is a
privation and consequently cannot be categorized. Yet deaf is prcdicably
related to John as a wholly separable accident. Similarly in the proposi­
tion: Genus is a logical relation, the predicate logical relation is not a
real being. Hence it cannot be put into any category. Yet it is prcdicably
related to its subject as its remote genus. In addition, we have seen that
complex natures are composed of a plurality of different natures. As such,
they cannot be categorized. However, a predicate representing a complex
nature may be used primarily to call attention to only one of the per­
fections it represents. The predicate of such a proposition can be classified
according to the category' of that perfection. Take Aristotle was a phi­
losopher as an example. Since philosopher here formally, i.e., primarily,
designates Aristotle as one possessed of philosophy and not as man,
philosopher is put in the category of quality. As a predicable, it is related
to Aristotle as a wholly separable accident.
It is particularly important, in view of the terminological identity,
to keep distinct the notions of accident as a category and as a predicable.
A nature is categorically an accident if it can exist only in another and"
not in itself. A nature is predicably an accident when it is related
noncssentially and contingently to its subject. Categorical accidents need

66

THE LOGIC OF THE FIRST OPERATION

not be prcdicably accidents. In Mary is able to talk, Running is an action,
and This figure is an isosceles triangle, the predicates are all categorically
accidents but arc prcdicably related to their subjects as specific property,
ultimate genus, and species respectively.
It may be worthwhile at this time to say something about the use of
the term essence. As a matter of fact, every' nature, substantial or acci­
dental, is an essence. It may seem that we have sometimes restricted
the use of “essence” to substantial natures, because the examples we have
used so far for the most part have involved substances, especially primary
substances, as subjects. Able to talk is nonessential to Mary, considered
as a primary' substance, precisely because it is outside of her essence as a
substance. This docs not mean that able to talk is not an essence in itself.
It is, of course, that kind of accidental essence classified as quality.
Similarly, virtuous is said nonessentially of John, but virtue is itself an
essence and is predicated essentially of John’s temperance. With this in
mind wc sec that substance is not the only category which can be divided
by essential differences into branches to form a tree-like logical schema.
Porphyrian Trees are theoretically possible for all ten categories, beginning
in each instance with the ultimate genus in question and descending
by way of essential differences to the singular instances in the category
in question.
VII. The Modes of Opposition

The categories are studied in logic in order that we may see how
different concepts are related one to another. Hence it is worthwhile
to note how some concepts are opposed to one another, for opposition
is a kind of relation. Accordingly the modes of opposition arc traditionally
considered immediately after the categories. Just as those things considered
as preliminaries to the study of the categories are known as anteprcdicamental, so the modes of opposition are spoken of as postpredicamental.
Two concepts arc opposed to one another when they arc so related that
the two cannot be realized in the same way at the same time in the same
subject. There are four different modes of opposition. In two of these,
one concept is positive and its opposite negative. In the other two, both
concepts are positive. The two negative modes of opposition are those ci
contradiction and privation; the two positive modes, those of contrarii
and relation.
The strictest form of opposition is that of contradiction. Contradictii
arises between two concepts when one is the pure and simple denial
of the other. This is the kind of opposition between an affirmation and

THE CATEGORIES

67

its denial, and it can be reduced to the opposition between being and
nonbeing. Contradictories arc so purely opposed that one or the other
is predicable of any subject whatsoever: if one of the contradictories can­
not be predicated of a subject, the other must. Thus, if a subject is not
white (whether or not it ever was or could be white), it must be non­
white. White and nonwhitc are contradictories. Thus whenever white
cannot be said of a subject (e.g., coal or grape juice), nonwhitc must
(for certainly if neither coal nor grape juice is white, both arc non­
white). Other examples of contradictories arc logical and nonlogical,
virtuous and nonvirtuous, higher and nonhighcr. The contradictory of
any term is easily formed simply by prefixing “non” to the term, 'rhe
resultant term expresses the pure denial of the first and is predicable of,
and only of, every subject of which the first is not predicable. Contradic­
tories can, of course, be expressed without use of the prefix “non,” just so
long as the negative term expresses the pure denial of its opposite. This is
the case with material and immaterial, for immaterial has the force of
nonmatcrial. ,
'Ihc second type of negative opposition is that of privation. This mode
of opposition exists between two concepts when one expresses a certain
positive perfection or formality and the other expresses the lack of this
perfection in a subject which could possess it. The difference between
contradiction and privation is immediately evident. Contradictories arc
predicable of all subjects in the sense that the one must be said of any
subject of which the other cannot. Concepts related by privation, how­
ever, arc predicable only of a certain type of subject. The privative
concept is predicable only of the subject which could possess its opposite.
Sight and blindness, sincerity and insincerity, legibility and illegibility
arc examples of concepts related by privation. Blindness is the lack of
sight in one who could see. Insincerity the lack of sincerity in one who
could be sincere. And illegibility is the lack of legibility in something
which could be legible. Thus blind, insincere, and illegible are said of
subjects when seeing, sincere, and legible cannot be said of them. But
they arc not said of all subjects which are non-seeing, non-sincere,
and non-legible. They arc said only of those subjects of which these
could be said because the subject could have sight, sincerity, and
legibility. Notice that wherever a concept has a privative its contradiction
can lie said of even- subject of which its privative can be said, but not
vice versa. For example, nonsinccre can be predicated of all the subjects
of which insincere can be predicated. But insincere cannot be predicated
of all the subjects of nonsincere. The reason for this is that the privative
concept says more than the contradiction. Like a contradictory concept, a

68

THE LOGIC OF THE FIRST OPERATION

privative-concept expresses the denial of its opposite. In addition, and
unlike the contradictory concept, it expresses the absence of its oppo­
site in a subject in which that opposite could be found.
The first type of positive opposition is that of contrariety. Contraries
are positive extremes in a given genus. Both of the concepts related by
contrariety are positive, each expressing an extreme in reference to the
other. Thus black and white arc contraries. They are opposed as opposite
extremes within the genus color. Contraries like concepts related by
privation arc predicable only of a certain type of subject. Colors belong
to substances precisely in so far as these substances are corporeal and
have a surface. Hence, both black and white can be said of, and only of,
bodies. Some contraries, such as black and white, are mediate, since they
admit of intermediary’ concepts (e.g., gray). Other contraries, such as
odd and even, which are said of whole numbers, admit of no inter­
mediaries and are called immediate contraries. In the case of immediate
contraries, as long as we stay with the appropriate subject, what one
contrary' cannot be said of, the other can be said of. This is not so with
mediate contraries. Thus, if a whole number is not odd it has to be even;
but a piece of paper may be neither black nor white, though it could be
either. Notice once again that the contradiction of a concept can be
said of any subject of which the contrary of this concept can be said,
but not vice versa. The contrary’ too says more than the contradiction.
It expresses not only the denial of its opposite but it affirms the positive
extreme of this opposite. It is sometimes difficult in practice to distinguish
a privative from a contrary. The difference, of course, lies in the fact
that a privative is negative and a contrary positive. However, some terms
sometimes express a positive notion and at other times a negative notion.
Bad can express cither the privation of good or a positive extreme, de­
pending on its use. Other pairs of concepts like merciful and merciless,
healthy and sick, and kind and unkind present a similar difficulty.
The second type of positive opposition is the opposition between cor­
relative notions which imply one another in different subjects. Tins is the
opposition of relation. Correlative notions arc defined in terms of the
relation between them; hence one always enters into the understanding of
the other and implies the other. They are opposites precisely because one
implies the other in a subject distinct from its own subject. What is
higher implies what is lower, for to be higher is always to be higher
than what is lower. But what is higher cannot in this same reference
be what is lower. Examples of relative concepts include parent and child,
husband and wife, taller and shorter, teacher and student, offensive and
defensive, half and double, upper and lower, and front and back. Here

THE CATEGORIES

69

again we note that where one of two correlatives can be said of a given
subject the contradiction of the other can also be said, but not vice versa,
llic relative expression also says more than the contradiction. In addition
to the denial of its opposite in its own subject it implies the existence
of its opposite in another subject.
Exercise VI

1. Define: predicament, univocal term, equivocal term, analogous term, sub­
stance, accident, real being, being-of-the-rcason, quantity, quality, relation
(as a category), action, passion, when, where, position, having possession,
primar}’ substance, secondary substance, continuous quantity, discrete
quantity, habits, dispositions, transitive operation, immanent operation,
opposition, contradiction, privation, contrariety, relation (as a mode of
opposition).
2. Precisely how do the categories differ from the predicables?
3. Classify the following terms as to univocity, equivocity, and analogy:
3.1 ............... “Fast” as signifying a runner and a color.
3.2 ............... “Slip” as signifying a trivial error and a case for a pillow.
3 3............... “Virtue” as signifying temperance and fortitude.
3.4 ............... “Right” as signifying the opposite of left andcorrect.
3.5 ............... “Quantity” as signifying continuous and discrete quantity.
3.6 ...............“Creator” as signifying God and an artist.
3.7 ............... “Head” as signifying part of the human body and the
leader of a group.
3.8 ............... “Water” as signifying a lake and an ocean.
3.9 ............... “Cabbage” as signifying a vegetable and money.
3.10 ............. “Lamb” as signifying a certain type of brute and Christ.
3.11 ............. “Science” as signifying arithmetic and geometry.
3.12 ............. “Medical” as signifying a physician and his surgical in­
struments.
3.13 ............. “Beautiful” as signifying God and a sunset.
3.14 ............. “Lark” as signifying a type of bird and a frolic.
3.15 ............. “Bright” as signifying a light and a student.
4. Natures as known exist in the intellect. Why is “nature known” not
synonymous with “being-of-the-reason”?

5. Keeping in mind the restrictions which make a being capable of being
categorized, give several examples of predicates which do not fit into any
category.
6. Explain the division of being into the ten categories.

7. Substance and the other categories arc true genera, while being is at best
a quasi-genus. Explain.
8. Check the following literary passages for as many examples of the ten
categories as can be found in them. Do the same thing with passages

70

THE LOCIC OF THE FIRST OPERATION

taken from textbooks and reference books for other courses, especially
those of a scientific nature.
8.1. This is a dreadful time, rendered the more dreadful by the gloom
of the weather and the country. I was never warm; my teeth chat­
tered in my head; I was troubled with a very sore throat, such as I
had on the isle; I had a painful stitch in my side, which never left
me; and when I slept in my wet bed, with the rain beating above
and the mud oozing below me, it was to live over again in fancy
the worst part of my adventures — to see the tower of Shaws lit
by lightning, Ransome carried below on the men’s backs, Shaun
dying on the roundhouse floor, a Colin Campbell grasping at the
bosom of his coat. From such broken slumbers, I would be aroused
in the gloaming; to sit up in the same puddle where I had slept,
and sup cold drammach; the rain driving sharp in my face or run­
ning down my back in icy tricklcts; the mist enfolding us like as
in a gloomy chamber — or, perhaps, if the wind blew, falling
suddenly apart and showing us the gulf of some dark valley where
the streams were crying aloud. (Taken from Kidnapped, by Robert
Louis Stevenson.)
8.2. On the various floors of this tower were the servants’ quarters, and
at the bottom a stable for horses. One room belonged to a woman
who often looked after the younger children of the master of the
house. At this particular moment '1'hornas was there, together with
the little sister whose name we do not know. Suddenly there was
a terrible thunderstorm and the tower was struck by lightning. 'Hie
little girl was killed, and also the horses in the stable below. The
mother in a state of panic rushed to the scene but found Thomas
quite safe and sound, and still fast asleep next to the nurse. (Taken
from St. Thomas Aquinas, by Angelus Walz, O.P., as translated by­
Sebastian Bullough, O.P. [Westminster, Md.: The Newman Press,
1951])
9. For each italicized predicate name the category' and the predicable
classification:
Category Predicable
9.1. Mary is able to laugh.
9.2. Mary is laughing.
9.3. Peter is sentient.
9.4. Peter is six feet tall.
9.5. Peter is rational.
11
9.6. Peter is stoop-shouldered.
1
9.7. Mary is studying her lessons.
9.8. Mary is combing her hair.
9.9. Mary is lying down.
9.10. This piece of paper is white.
9.11. Every organism is living.
9.12. Peter is dressed in a white shirt.
9.13. Peter is taller than John.
11
9.14. Peter is a mathematician.

:

1

the categories

Mathematics is a science.
(
Mary is looking at the sunset.
(
The sunset is being admired by Man-.
(
This tulip is yellow.
{
This tulip is a plant.
(
Mary’ is nearsighted.
(
Mary is my sister.
(
Paul was hit by a bus.
(
Paul was thinking about his future.
(
The missionaries were enjoyed by the
cannibals.
(
9.25. Your sister is in the next room.
(
926. Paul is courageous.
(
927. The will is an appetite.
(
9.28. Red is a color.
(
9.29. Red is striking.
(
9.30. Paul is armed to the teeth.
(
9.31. Every body is corruptible.
(
9.32. The student was worried about exams.
(
9.33. Every maple tree is a substance.
(
9.34. The highway was four lanes wide.
(
9.35. Paul was in Boston in February.
(
10. Why can the contradictory of a term be used wherever
privative or relative opposite might be used?
11. Name the type of opposition involved between:
11.1. ... ... Double — half
112. ... ... Good — bad
11.3. ... ... Black — white
11.4. ... ... Teacher — student
11.5. ... ... Before — after
11.6. ... • •. Logical — illogical
11.7. ... •.. Virtue — vice
11.8. ... •.. Intentional — nonintentional
11.9. ... ... 1 fearing — deafness
11.10. ... ... Sweet — bitter
11.11. ... • • • Intentional — unintentional
11.12. ... • • • Legible — illegible
11.13. ... • • • Healthy — unhealthy
1114. .. ... Sane—insane
HIS. ... • • • Better — worse
11.16. .
• • • Literate — illiterate
¡in.
... Love —hate
11.18.
11.19. .” ■.. Dry — wet
• • • Friendly — unfriendly
9 15
9'16
917
9M
9.19.
¿20.
9.21.
9 22.
9.23.
924.

)
)
)
)
)
)
)
)
)

(
(
(
(
(
(
(
(
(

)
)
)
)
)
)
)
)
)

) (
)
) (
)
) (
)
) (
)
)
(
)
) (
1
) (
)
) (
)
) (
)
) (
)
) (
)
) (
)
its contrary' or

CHAPTER

VII

Definition

I. Definition and Division

Simple apprehension admits of a more or less perfect grasp of its
object. This object, as we have seen, is fully understood only in terms of
its comprehension and extension. If there is only a vague, implicit knowl­
edge of an object’s comprehension and extension, then that object is
known only obscurely or confusedly. This would be the case were I to
know animal without knowing that its character as being sentient is
intelligibly distinct from its character as being an organism or without
knowing that man and brute are distinctly different immediate inferiors
of which animal can be predicated. The act of simple apprehension is
ordered to a much more perfect grasp of the objective concept than that
given in this obscure way. Simple apprehension seeks the clear and
distinct knowledge of objective concepts achieved by an explicit grasp
of their comprehensive notes and extensive parts. Here we must note,
however, that perfectly clear and distinct knowledge is difficult to achieve
for any concept and, at least in practice, is impossible for most concepts.
Yet we must try to realize such perfect knowledge as closely as possible,
for scientific discourse is possible only to the extent that such knowledge
is approached. The act whereby the intellect explicitly expresses the com­
prehension of a concept is the act of defining that concept. The act
whereby the intellect expresses explicitly the distribution of a concept
into its subjective parts is the act of logically dividing that concept. We
shall use the terms “definition” and “division” to signify not only the acts
of defining and dividing but also the objects known in each instance, as
well as the terms used to express these objects. Sentient organism is the
definition of animal; and man or brute is its logical division.
Definition and division arc both instruments for manifesting the object
of simple apprehension. Definition helps to disclose the meaning of an
object. From this point of view the act of defining involves a movement
72

DEFINITION

73

of the mind. This means that the mind truly advances when it defines.
However, from another point of view definition docs not imply an
advance in knowledge. The definition, as an object considered in itself,
is identically the same as the thing defined and, as such, must be strictly
coextensive with it. Thus, there is no movement from thing to thing
involved in the act of defining. Yet the definition does differ conceptually
from the defined, for it explicitly discloses the thought content which
was only implicitly in the defined prior to its definition. Thus, in defin­
ing, there is an advance in knowledge from the point of view of the
manner in which the object of simple apprehension is known. In the
strict sense, the term discourse applies to the movement of the mind in
the act of reasoning. However, in an analogical sense both the acts of
defining and of logically dividing can also be spoken of as discursive.
Definition and division both presuppose the doctrine of the predicables
and the categories. As we move through this chapter and the next we
shall see how much the acts of defining and dividing depend upon the
ability to distinguish between different predicables and different categories.
II. Nominal and Real Definitions

What we have said thus far about definition refers to the strictest type
of definition, namely, that which reveals the meaning of a nature by
making explicit reference to its comprehensive notes. Such definitions
arc rare, and in most instances we must be satisfied with less perfect,
but attainable, definitions. The essential definition, as the definition which
fully manifests the comprehension of the thing defined is called, is the
primary instance of definition; and other definitions arc more or less
perfect to the degree in which they approach or fall away from this ideal.
We shall begin to differentiate types of definition by distinguishing
between nominal and real definitions. The essential definition, as we shall
sec, is a type of real definition.
As the names would indicate, a nominal definition expresses what a
term stands for, and a real definition expresses what a nature is. This
distinction, apparently easy to grasp, is unfortunately not so simple as
it seems. Although wc can give some characteristic examples of nominal
and real definitions, there arc many instances when the nominal and real
definitions seem to be the same. Science of man would ordinarily be
suggested as an example of a nominal definition for “anthropolog}'.”
However, one might argue that this is as good a real definition of
anthropology' as possible. Rational animal is one of our clearest examples
of a real definition. Yet in a given case it could serve just as well as a

74

THE LOGIC OF THE FIRST OPERATION

nominal definition. Hence it is worthwhile to consider the distinction
between nominal and real definition more closely in order to clear up
possible difficulties.
Confronted for the first time with a new term, e.g., early in the logic
class with the term “simple apprehension” one asks, “What does this
term stand for?” The answer would be that it stands for the first opera­
tion of the intellect. This is a nominal definition proposed in answer to
a question asking for the meaning of a term. Now that the questioner
knows precisely what thing is signified by the term “simple apprehen­
sion,” he is ready to consider what this thing is in itself. He turns his
attention from the term to the nature signified and asks, “But what is
the nature of simple apprehension?” or “What is the first operation of
the intellect?” The answer to this must be a real definition more or less
perfect as it is closer or farther removed from a full and explicit dis­
closure of the comprehension of the nature concerned. The real defi­
nition of simple apprehension is the operation of the intellect by which
we grasp an object without affirming or denying anything about it.
Hence a definition is nominal when it expresses the meaning of a term,
and it is real when it manifests the nature signified by that term. 'Hie
importance of language for the communication of knowledge is un­
deniable. Thus nominal definitions, which clarify the meanings of words,
are highly important. But, as signs, words are important only in reference
to the concepts they signify. Thus real definitions, which show what
the nature is which these terms signify, are more important than nominal
definitions. Nominal definitions arc important in so far as they point
to objects known. One cannot attempt to find out what something is
(i.e., to discover, if possible, the constituents of its essence) until he
first knows what it is he is to concern himself with. Whenever a term is
so indeterminate as to leave this in doubt, a nominal definition is called
for. Just as terms are signs of natures known, so definitions of tenus
(nominal definitions) prepare the way for definitions of natures (real
definitions).
The search for nominal definition is a search for expressions which
point to the things signified by the terms we arc seeking to define.
Nominal definitions are formed very frequently simply by reflecting
upon the term to be defined. For example, the etymology of a term may
reveal its meaning. “Philosophy” comes from the Greek word “philos"
(lover or friend) and “sophia” (wisdom); hence it can be said to
mean the love of wisdom. Similarly “psychology” comes from “psyche"
(soul) and “logos” (discourse), and can be defined as discourse on, or sci­
ence of, the soul. Synonyms of a term may also be used to define it, as

DEFINITION

75

well as equivalent expressions from another language. Thus "nuptials”
can be defined as wedding ceremonies and “gendarme” as policeman.
The nominal definition must be better known than the term it defines
so that it will clarify the meaning of this term and point out precisely
what the thing is that the term signifies. Thus an expression offered as
(and suitably serviceable for) a nominal definition might also be a good
example of a real definition. One might ask what "man” stands for and
be satisfied with the definition rational animal. This could suffice, for
example, to indicate that "man” was used to stand for the species man
and not for a subclass of that species including only its masculine mem­
bers. This same definition, offered as an expression of what the nature
man is, is a highly acceptable real definition. Aristotle defines "soul” as
the first principle of life in the living being. In so far as this expresses
the meaning of the term "soul,” this is a nominal definition. It could
also be offered to express the nature of the soul, and thus it would be
a real definition. Etymological definitions and definitions by way of syno­
nyms and translation are rarely suitable as real definitions. But even here
tlie name may be so appropriate that its etymology adequately defines the
nature it signifies. Thus, as we have noted, "anthropology” can be nomi­
nally defined as the science of man, and this same expression can also be
used as a real definition of anthropology.
III. The Four Causes

Before considering the different types of real definition it will be neces­
sary to say something about causality and the four different kinds of
causes. Real definition is frequently definition by way of cause, and
some real definitions differ from one another because they express dif­
ferent causes. A cause is something from whose existence the existence
of another follows. Cause and effect are correlative notions. A cause is
some positive influence upon the being of something which follows
from it. 'That which follows from the cause is its effect. An effect is an
effect precisely in so far as it depends on its cause cither to be what
it is or to become what it becomes. Every corporeal thing depends for
its being upon causes which fit into four types, namely, material cause,
formal cause, efficient cause, and final cause.
Any corporeal thing is a certain kind of thing which can be changed
into some other kind of thing. A green apple is a determinate type of
apple which can be changed into a red apple through the process of
ripening. Any apple whatsoever is a certain kind of substance which can
be changed into man when eaten by a man and subjected to the process

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THE LOGIC OF THE FIRST OPERATION

of nutritional assimilation. 'Ihese examples differ, of course, in that the
first has reference to what a thing is and can be accidentally, whereas
the other refers to what it is and can be substantially. In both, how:.
ever, we note that something is dcterminatcly a certain kind of being
and can become a detcrminatcly different kind of being. A cause is
required within a being to make it precisely the kind of being it is. This
cause is the form of that being, its formal cause, The formal cause is
determinate and determining. What it determines is an indeterminate
and determinable substratum, which accounts for the ability of any
corporeal being to change. This substratum, subject to one form at a
given moment and with this form composing a certain kind of being,
is able to give up this form and receive another, with which it com­
poses a different kind of being. This indeterminate and determinable
substratum, which is truly the subject of change, is called matter or the
material cause. In so far as matter and form together constitute the
essence of the being of which they arc causes, they are spoken of
as intrinsic causes.
The intrinsic causes of a corporeal thing do not fully explain the
being of a corporeal thing. Matter, in itself indeterminate and lacking
form, is unable of itself to supply the form which determines it in any
given thing. There must be some agent, somehow extrinsic to both
matter and form, which brings about the union of matter and form in
any corporeal thing. This agent is called the efficient cause. But no agent
operates as an agent without sufficient reason. The reason why an agent
brings about the union of matter and form is the end or final cause. The
efficient and final causes are extrinsic causes. Matter exercises its causality
only in so far as it is actuated by a form. A form informs matter only
in so far as it is made present to the matter through the efficient
causality of an agent. An agent operates only in virtue of some final
cause as motive. Thus the extrinsic causes arc themselves related to the
intrinsic causes as cause to effect.
Briefly, the material cause is the “stuff” out of which a thing is made.
The formal cause is the design which is given the matter and deter­
mines the thing to be the kind of thing it is. The efficient cause is the
agent which produces the thing. The final cause is the purpose or end
of the thing. This does not mean that there must be only one material
cause, one formal cause, one efficient cause, and one final cause for a
given thing. The four causes are orders of causality, and any one cor­
poreal thing may be the effect of a plurality of causes in each order.
'Ihe doctrine of the four causes is of priman- importance in understand­
ing natural beings; however, it is perhaps most clearly illustrated for

DEFINITION

77

beginners in terms of artificial things. Thus a house is built out of
bricks and wood (material cause) into a particular type of building
(formal cause) by a corps of carpenters and bricklayers (efficient cause)
so that it might serve as a protection against the elements (final cause).
This brief discussion of the orders of causality is not intended as an
adequate philosophical exposition, yet it should serve as sufficient basis
for a division of real definition according to difference in the causes
involved.
IV. Types of Real Definition

The difference between real and nominal definitions is that the former
reveal the meaning of a nature expressed by a term, whereas the latter
simply point to what the term stands for. In the strict sense a nature
is fully revealed only by its intrinsic causes, which arc the causes which
constitute its essence. In the case of corporeal beings (both natural and
artificial) the intrinsic causes are the material and formal causes. Mathe­
matical entities and incorporeal substances are, for different reasons in
each case, essentially explained in terms of form alone. Whatever the
kind of nature to be defined, the most perfect type of real definition is
a definition in terms of intrinsic causality. This type of real definition
is spoken of as the essential definition.
Because of the extreme difficulty and, in many instances, the impos­
sibility of determining fully the intrinsic constitution of a nature, we
arc often unable to achieve essential definition. In such cases light may
still be shed indirectly upon the essence in question by an appeal to
something extrinsic to that essence, but necessarily connected with it. Two
possibilities, and only two, suggest themselves here. First, it is possible
to learn something about the nature by examining the extrinsic efficient
and final causes, since these arc causes in reference to the matter and
form which constitute the essence in question. Thus, we might define
a nature in terms of its extrinsic causes. Second, the attributes extrinsic
to the essence but necessarily caused by it can indirectly show what
it is. Thus, we might define a nature in terms of its properties. Defini­
tions manifesting a nature in terms of its extrinsic causes or in terms of
its properties are spoken of as nonessential definitions. A noncsscntial
definition will be either a nonessential definition by way of efficient
cause, a nonessential definition by way of final cause, or a nonessential
definition by way of property.
Our investigation into the nature and types of definition thus far has
yielded results which we can present schematically as follows:

78

THE LOGIC OF THE FIRST OPERATION

Nominal (an expression manifesting what a term stands for
by "pointing to" the thing it signifies)

Definition

Essential (by directly
revealing the essence
in terms of intrinsic
causes)
Real (a complex expres­
sion manifesting a nature
by explicitly disclosing
its comprehensive notes)

Through
final
cause

Nonessential
(by revealing
the essence
indirectly)

Through
efficient
cause
By way of
property

Wc have discussed the difference between nominal and real definitions
at some length. Now we shall look more closely at the types of real
definition.
Essential Definition
An essential definition manifests a nature by expressing its intrinsic
causes. These can be expressed as entitativcly, i.e., really, and not just
intelligibly distinct parts joined together in the composite they consti­
tute, which is, of course, the nature in question. (For immaterial entities
the form alone would be expressed, not as a part but as the whole of
the essence to be defined.) For example man can be essentially defined
as a being composed of a rational soul and a natural body potentially
having rational life within it. The parts of this definition are respectively
the form and the matter of human nature, which together constitute
the essence of man. But the better way of expressing an essential defini­
tion logically is to do so in terms of the proximate genus and the specific
difference of the defined. z\ definition in terms of proximate genus and
specific difference is an essential definition (i.e., a definition through
intrinsic causes), for the genus is taken from the material cause and
the difference from the formal cause. This second way of expressing
the essential definition is logically more strict than the first because
it is more suited to logical analysis and scientific discourse. It is easier

DEFINITION

79

to see how the definition expressed in this second way is logically related
to other intelligible objects than the definition expressed in the first way.

Definition by Way of Extrinsic Cause
Since both the final and the efficient causes of a thing exert a definite
influence in shaping its essence, it is possible to manifest the essence in
terms of final and/or efficient causes. Here, however, the manifestation
is indirect, whereas it is direct in the case of definition by intrinsic
causes. Thus, when an essential definition is impossible or when a
complementar}' definition is desired, definitions by way of extrinsic causes
may be called for.
Ñonessential definitions are naturally less perfect than essential defini­
tions; yet good nonessential definitions should approach essential defi­
nitions as closely as possible. Ordinarily it is difficult, and frequently
impossible, to determine the specific difference which is necessary for
an essential definition. If it is difficult to determine proximate genera,
it is usually possible to find the remote genera. The crucial problem in
attempting a nonessential definition is to determine an adequate sub­
stitute for a specific difference. Here we should recall why a specific
difference is so important for an essential definition. The reason is
that the specific difference pinpoints the definition so that it applies
only to the thing defined, distinguishing it from all other things. When
we cannot discover a specific difference and try to substitute for it some
extrinsic cause or causes, we are attempting definition by way of extrinsic
cause. To do this, the extrinsic cause or causes must contract the genus
precisely to the thing defined. Man can be defined as an animal destined
for eternal life. This is a good nonessential definition through final cause,
for animal is the proximate genus of man, whereas destined for eternal
life expresses a final cause which is limited as far as animals are con­
cerned to the species man. Certain things are characteristically defined
through final and/or efficient causes. For example, artificial things,
especially tools and other useful objects, can ordinarily be defined clearly
only in terms of the purpose for which they arc intended. And many
diseases arc frequently best defined in terms of the bacteria, germs,
or other agents which cause them.
Definition by Way of Property
Attributes flowing from an essence arc extrinsic to it; however, at
the same time they reveal something about that essence in so far as they
are caused by it. Thus these attributes can be used in nonessential defini­
tions. Since the definition and the thing defined must have the same

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THE LOGIC OF THE FIRST OPERATION

extension, only attributes related as properties to an essence can be used
to define it. A definition in terms of an accidental (rather than proper)
attribute would be either too broad or too narrow in extension for the
concept it was intended to define. Only an attribute that is necessarily
and exclusively an effect of the specific difference of the species can
substitute for that specific difference in a nonessential definition by way
of caused attribute. Tims, this type of nonessential definition is called
definition by way of property. Man can be defined in this way as an
animal endowed with speech, or as a tool-bearing animal, or as an artistic
animal. Triangle can be defined as a plane figure with three angles equal
to two right angles. Body can be defined as a corruptible substance.
We have seen that accidents cannot substitute for a missing specific
difference in a legitimate nonessential definition. This is true for any
individual accident. However, it is possible that a combination of acci­
dents might function as a property' for a given thing. So long as a num­
ber of accidents understood collectively belong necessarily and always to
every' member of a species and only to the members of that species,
this combination of accidents has the force of a property and can be
used in a definition by way of property. However, it is extremely difficult
to understand any combination of accidents well enough to see in their
combination the necessity of a strict property. Some have suggested
that man can be defined as a two-legged, featherlcss, nonaquatic ani­
mal. Two-legged distinguishes all actually existing men from all known
animals other than men, with the exception of birds. Featherless dis­
tinguishes men from all known birds except penguins. Nonaquatic dis­
tinguishes men even from penguins. This combination of accidents docs
sene to distinguish men from all other things in our experience. How­
ever, unless we can see a conceptual incompatibility between this com­
bination of accidents and nonman, and the lack of this combination of
accidents and man, then this definition falls short of being a legitimate
real definition. It can, however, surely serve as an acceptable nominal
definition of “man”; for it docs point among the things within our experi­
ence precisely to man, thus isolating man accurately as an object for
inquiry.
V. The Rules for Good Definition

Actually a set of rules for good definition is not necessary' for one
who understands the nature of definition. This is true because such
rules do no more than repeat, after the fashion of expressed norms, the
characteristics proper to definition. However, such rules arc useful for

DEFINITION

81

beginning students, as they remind them of the requirements of a good
definition. Since the primary instance of definition is essential definition,
the rules apply primarily to it. They apply proportionately to lesser
types of definition.
1. Definitions should include generic and differentiating elements. The
former show’s what it has in common with other things, and the latter
shows how it differs from them.
Clearly, in the strictest type of essential definition, the generic ele­
ment is the proximate genus and the differentiating clement is the
specific difference. Other definitions, with the exception sometimes of
the nominal definition, should contain a generic element as close to the
proximate genus as possible and a differentiating clement which can
substitute adequately for a missing specific difference. Neither analogical
concepts nor ultimate genera can be defined in terms of proximate genus
and specific difference, for neither has a genus or specific difference in
the strict sense. The best that can be done here is to approximate an
essential definition by wray of elements like, but not identical with,
proximate genus and specific difference. The definition of an accident
also presents a special case. Essentially an accidental form, e.g., the
ability to laugh, is purely a form without matter. An accident, however,
can be understood only in reference to the substance w'hich is its proper
subject. Hence the two elements in the definition of an accident will
express not only the form which is the accident but the matter which
is its proper subject, e.g., man as proper subject of ability to laugh.
2. The definition and the defined should be convertible (neither one
more or less extended than tlie other).
A definition is intended to clarify as far as is possible the thing it
defines. Thus, it must apply to every instance of what it defines and
only to what it defines. Anything less than this will confuse rather than
clarify. If triangle were defined as a three-sided plane figure with equal
sides one would think that only equilateral triangles, and not scalene
or isosceles triangles, were triangles. If triangle were defined as a rectilinear plane figure one would confuse triangles w'ith quadrilateral figures
and with evciy other type of polygon. The only adequate essential defini­
tion for triangle is three-sided plane figure. Every triangle is a three-sided
plane figure, and every three-sided plane figure is a triangle.
3. The definition should be clearer than the defined.
As noted, the purpose of a definition is to clarify the thing it defines.
It is obvious, therefore, that the definition must be clearer than what it
defines. This means that the definition ought to be as brief as possible.
It should not be in metaphorical terms. It should be in terms which

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THE LOGIC OF THE FIRST OPERATION

can be understood by those for whom the definition is intended. It
should be in positive rather than negative terms whenever this is possible.
This rule demands, especially, that the definition should not contain the
defined or any expression which is a derivative of the defined. Of course,
the thing defined must be known, indistinctly, prior to its definition and
before the parts of its definition are known precisely as such. But these
parts must be understood themselves before the definition as a whole
is understood. 'ITius, man is known before rational and animal arc known
as parts of its definition. But rational and animal must themselves be
understood before they can together manifest the signification of man.
4. The defined must be universal.
The ultimate intention of definition is to express distinctly the com­
prehension of a nature which is known in an act of simple apprehen­
sion, i.e., of a universal concept. Thus, definitions are possible only for
universals. Definitions are unchanging and eternal, because they express
the meaning of essences which have been abstracted from changing
individual situations. Individuals can be identified by descriptions involv­
ing the definitions of their species, but the ultimate difference pin­
pointing an individual will have to be some singular characteristic of
contingent and perhaps even momentary status.
Exercise VII

1. Define: definition, logical division, logical discourse, nominal definition,
real definition, cause, material cause, formal cause, efficient cause, final
cause, intrinsic cause, extrinsic cause, essential definition, nonessential
definition, definition by way of extrinsic cause, definition by way of
property.
2. In what sense is definition an instance of logical discourse, and in what
sense does it fall short of discourse strictly taken?
3. Explain: definition is to comprehension as logical division is to extension.
4. How do nominal definitions differ from real definitions? How do nomina,
definitions sene real definitions?
5. Define each of the four orders of causality, and give examples for each
order.
6. Explain the manner in which each cause is related to the others.
7. How does the essential definition differ from the nonessential? What
are the two different ways of expressing an essential definition? Which
is the stricter way from the point of view of logic? Why?
8. What are the two types of nonessential definition? Why are there these
two types, and only these? Subdivide these types where this is nccessan.
and illustrate each type of definition.
9. When can an accident (rather than a property) be used in a real defini­
tion? What are the strict limits to this use?

definition

83

10. List the rules for a good definition, and defend each rule by measuring
each one against the definition of definition itself.
11. Classify' each of the following definitions which are good, and explain
what is wrong with each of the bad ones. For each good real definition
point out the generic element and the differentiating element.
11.1. Man — a creature destined for eternal life.
11.2. Science — an organized body of knowledge.
11.3. Triangle — a plane figure bounded by three lines.
11.4. Osmosis — the passage of water through a membrane.
115. Second intention — a logical property belonging to an objective
concept precisely as known.
11.6. Edward — rational animal.
11.7. Zygote — fertilized egg.
11.8. Logic — the art of proceeding logically.
11.9. Definition — a principle of manifestation for the object of simple
apprehension.
11.10. Indian chief—top man on the totem pole.
11.11. Antibiotic — a chemical agent possessing bactericidal properties.
11.12. Human soul — principle of life.
11.13. Intelligence — the ability to solve problems.
11.14. Man — tool-bearing animal.
11.15. War —hell.
11.16. Quality'—that kind of a being apt to exist in another and not
in itself.
11.17. Bicycle — a light vehicle having two wheels, one behind the
other, a steering handle, a saddle scat or seats, and pedals by
which it is propelled.
11.18. Sign — that which signifies something beyond itself.
11.19. Substance — that kind of being apt to support accidents.
11.20. Man — creature composed of body and soul, created by God, to
give glory to God and share His happiness forever.
11.21. Organism — living body.
11.22. Body — corruptible substance.
11.23. Mathematics — the science of numbers.
11.24. Telescope — an optical instrument used to aid the eye in view­
ing distant objects.
11.25. Law — a reasonable ordinance, promulgated by the one in charge
of the community, for the sake of the common good.
11.26. Politics — a branch of ethics specifically distinct from other
branches of ethics.
11.27. Poker — a gambling game played with cards.
11.28. Gland — a cell or collection of cells which produces a deter­
minate product.
11.29. Accident—nonessential predicable.
11.30. Jazz — music which is improvised spontaneously.
12. Why is it so difficult to define the several categories?
13. Check this textbook for definitions. Classify and evaluate them chapter
by chapter. Do the same with other books. Prepare a list of faulty defini­
tions found in both technical and nontechnical works.

CHAPTER

VIII

Division

I. Logical Division

A definition, as we have seen, is a complex expression explicitly stating
the comprehension of a concept. Logical division, on the other hand,
looks to extension rather than to comprehension. It is thus a complex
expression which manifests the extension of a concept by pointing out
what subjects, i.e., inferiors, arc included within that extension. Just as
the comprehension of animal is made explicit in the definition sentient
organism, so the extension of animal is made explicit in the logical divi­
sion man or brute. The act of defining is an intellectual operation which
clearly and distinctly manifests the actual content, or meaning, of a
concept. Ilie act of logically dividing is an intellectual operation which
clearly and distinctly discloses the potential content of a concept. That
is, it shows how this concept can be realized in various types of subjects.
Each renders the relatively unknown — because only implicitly grasped
in an obscure and confused fashion — known. As such, each is a kind
of discourse, although we shall use the term “discourse” more strictly
later on in reference to the more complicated process of reasoning. Just
as essential definition answers the question, “What is this nature?" so
logical division answers the question, “In what different kinds of beings
can the nature be realized?”
Every' logical division involves three elements, lliese are: the logical
whole or totality divided, the subjective parts or dividing members, and
the principle of division or aspect of the whole in virtue of which it is
divided into parts. When animal is divided into man or brute, animal
is the totality divided; man and brute are the dividing members; and
specific differentiation is the principle of division.
For division as well as for definition the logician is concerned only
with the general theory. He defines division and divides and subdindes
it into the general types, and then defines those types. However, he
leaves the particular application of the theory of division for the proper
84

DIVISION

85

subjects of the special sciences to the scientists in question. The par­
ticular method of division best suited to the subject matter of a given
science can be determined only by one who is familiar with that
subject matter.
II. Types of Logical Division

Just as the primary instance of definition is essential definition, so the
priman- instance of logical division is essential division. This is the divi­
sion of a genus into its species through the addition of specific differences.
If the basis for logical division is something other than specific differences,
then the division is nonessential rather than essential. The division of
animal into man or brute is an essential division because it is made on
the basis of diverse specific differences. The divisions of animal into male
or female and of man into sane or insane arc nonessential divisions.
In all of these cases, the differences which arc the basis for the divisions
do not belong to the essence of the nature divided. They are not in the
same category' as that nature and the members into which the latter is
divided are not related to it as species to genus.
All the different nonessential aspects under which a universal concept
can be said to be realized in different types of beings can serve as the
basis for nonessential divisions. Traditionally the modes of nonessential
division have been classified under three headings: (1) the division of
substances by way of their accidents, (2) the division of accidents by
wav of the substances in which they inhere, and (3) the division of
accidents themselves by way of other accidents. Man is divided into
white, brown, red, or yellow according to the first mode. Able to repro­
duce one’s kind is divided into animal or plant according to the second.
What is cold is divided into liquid, gas, or solid according to the third.
All that is needed for a legitimate nonessential division is that the whole
must be divisible into parts on the basis of mutually exclusive or opposed
differences which do not belong to the essence of the whole. To illus­
trate the variety of possibilities, let us note several nonessential divi­
sions, each involving a different mode of opposition. The division of
disease into crippling or noncrippling is based on contradictory' opposi­
tion between differences. 1116 division of number into odd or even is
based on immediate contrariety. 'lire division of man into logical or
ilh"ical is based on privative opposition. Finally, the division of soldier
into superior or inferior rests on the opposition of relation.
Analogical concepts pose a problem for logical division just as they do
for definition. They cannot be defined as strictly as univocal concepts

86

THE LOGIC OF THE FIRST OPERATION

because they do not admit of genus and difference in any strict sense.
And they cannot be logically divided in the same way as univocal con­
cepts, namely, by the addition of opposed differences to which they are
in potency. The differences between the inferiors of an analogical whole
are actually present in the whole, although their presence together is not
distinctly known. Here the inferiors arc differentiated from one another
as differing modes of the analogical whole. Tin's is the case when being
is divided into substance and the nine accidents, as we saw previously in
our chapter on the categories.
Logical division can itself be divided into dichotomous division and
nondichotomous division. A dichotomy is a division in which two
dividing members exhaust the totality divided. A dichotomy is the
simplest possible division. A dichotomous division is the only kind of
division possible whenever there are but two subjective parts of the
whole from the point of view chosen for division. The division of
whole number into odd or even is a legitimate dichotomous division
which allows for no alternative, since the principle which distinguishes
some whole numbers as odd necessarily distinguishes all others as even,
'fhe division of triangle into equilateral or noncquilateral is legitimate,
but it is not the only division of triangle which can be made from the
point of view of the relations between its sides. Noncquilateral is a
negative notion which, positively expressed, would include both scalene
and isosceles. Under most circumstances the division of triangle into
equilateral, scalene, or isosceles would be preferred to the dichotomy
including equilateral and noncquilateral. However, if the intention of
the division is ultimately to focus attention on only one type of triangle,
namely, the equilateral, the dichotomous division may, because of its
simplicity, be more helpful. Suppose, for the moment, that in dividing
triangle one knew there were equilateral triangles and other kinds as
well. Suppose, however, that he did not know precisely what these others
were. In such a case, the dichotomy between equilateral vs. nonequilateral would be safe, whereas an attempt to divide triangle into equi­
lateral and more specific types might lead to error. The division of
triangle into equilateral or noncquilateral, as of logical division itself
into dichotomous or nondichotomous, is a division by way of contra­
diction. Division by way of contradiction is frequently helpful, and it
is usually safe. However it has one danger. The divisions of man into
white or nonwhite, Asiatic or non-Asiatic, rich or nonrich, are safe and
legitimate, although they have only slight value. However, the division
of man into risible or nonrisible is both useless and illegitimate, although
it might seem at first glance to be valid. The difficulty, of course, is that

DIVISION'

87

all men are risible. There can be no subjects in whom the second mem­
ber of this attempted dichotomous division can be realized. Nonrisiblc
simply cannot be a subjective part of man, and never, then, a member
of any logical division of man. This means that dichotomous division
by way of contradiction cannot be attempted if one of the members
is equal to the totality which is being divided.
III. Physical Division

Logical division is defined as the orderly distribution of a logical whole
into its subjective parts. We can extend the notion of division to include
both logical division and a second kind of division which we can call
physical division. Generally taken, division is defined simply as the distri­
bution of any whole into its parts. Logical division distributes the logical
whole into its subjective parts. Physical division divides an integral
whole into its composing parts. A logical whole necessarily involves sub­
jective parts, that is, parts which are included within its extension and
which together make up the whole of its extension. Man and brute
together make up the whole of the extension of animal. They are the
subjective parts of animal, and animal is a logical whole in reference to
them. Thus animal can be said of each of them. An integral whole is
actually the sum of its parts. Collectively they make it actually what
it is, and are its composing parts. The human body is an integral whole
actually composed of head, limbs, trunk, and extremities. No one of
them is the body, but together they compose the body. Unlike any
logical whole, every' integral whole actually and distinctly includes its
parts and is, in respect to its actual content, the sum of these parts.
Unlike any logical whole, an integral whole cannot be said of any one
of its parts. Every man is an animal, and every brute is an animal. But
no head is the body, nor is any limb or any other composing part of the
body able to be said to be the body.
There are many different kinds of integral wholes in the broad sense
in which we have used this term. We have seen that the distinct parts
of any body are composing parts of an integral whole. The entitative
parts of any nature are also composing parts of an integral whole. So too
are the generic and differentiating elements in a definition or the several
comprehensive notes of an objective concept. What is common to each
of these instances of the integral whole and its composing parts is this:
the whole involved is actually the sum of its parts and cannot be said
of any of them. There are as many examples of physical division as there
are integral wholes which can be divided into composing parts. The

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THE LOGIC OF THE FIRST OPERATION

division of the United States of America into fifty states is a physical
division. The division of human nature or of any individual man into
body and soul is a physical division. The division of the definition
rational animal into rational and animal, and of the concept man into
substance, material, living, sentient, and rational arc also physical divi­
sions. Notice that logical division requires a universal whole, whereas
the whole required in physical division may be either universal or singular.
IV. Rules for Good Division

The rules for good division arc but a reaffirmation of the nature of
division itself. They apply both to logical division and to physical
division.
1. The dividing members must be related to the totality divided as
parts to whole (i.c., in the case of logical division, as inferiors to
superior; and in the case of physical division, as composing parts to an
integral whole).
This rule is an immediate consequence of the definition of division
and needs no defense or explanation. This rule is violated by any divi­
sion in which one of the dividing members is coextensive with the
totality divided (e.g., man divided into risible or nonrisible).
2. The dividing members must adequately divide (i.c., exhaust) the
totality divided.
The purpose of division is to give a clear and distinct understanding
of a whole in terms of the parts which make it either potentially (in
logical division) or actually (in physical division) what it is. If the
division included only some of the parts comprising the whole this
would hinder rather than help in the understanding of the totality
divided. The division of predicable into species, genus, difference, or
property’ suffers this defect. Accident is required to complete the list
of predicables and render it adequate as a legitimate division. Even
universal concept involves an infinite multitude of singulars in its exten­
sion. Hence, there can be, strictly speaking, no division of a logical whole
into its singular inferiors. The attempt to list singulars under a logical
whole is an enumeration rather than a division properly understood.
3. The dividing members must be formally opposed to one another,
so that each part excludes the other.
A good division requires that the dividing members must be clearly
differentiated from one another. None of the parts should include another
cither in whole or in part nor should any of the parts be identical. Any
failure here renders the attempted division disorderly. To divide football

DIVISION

89

player into lineman, back, or tackle is confusing and illegitimate, for
lineman includes tackle as one of its types.
4. One and the same principle of division must be observed through­
out a division.
In a sense the principle or basis of division is the formal element in
division, for it makes the division to be precisely the kind of division
it is. Multiply principles of division and the result is chaos; for then
there is a division which is simultaneously more than one division.
Failure to observe this rule ordinarily breaks the previous rule. Even
when this is not the ease, the division is meaningless and confusing.
To divide people into men, women, or children is confusing or dis­
ordered, so long as it is based, simultaneously, on a difference in age and
on a difference in sex. To divide animal into risible or nonmoral yields
a set of dividing members which happen to exhaust the totality divided
while mutually excluding one another. Yet the division is not clear as it
stands, since one dividing member represents a part of the whole con­
sidered from one point of view and the other a part of the whole con­
sidered from an entirely different point of view. To make this division
meaningful it would be necessary to reduce risible to its source in man’s
rationality and nonmoral to its source in the brute’s nonrationality.
V.

Codivision and Subdivision

In many cases there is no reason why division must cease once a
totality has been divided into a set of parts. /\s a matter of fact, there
may be compelling reasons to divide each of the original dividing mem­
bers into its own parts and perhaps to continue this process through
several levels of division. To divide a whole into its parts and to divide
the parts of the original whole into their own parts is to subdivide. The
division of definition into nominal or real, coupled with the further divi­
sion of real definition into essential or noncssential, and finally of nonessential real definition into definition by way of extrinsic cause or
definition by way of property, is an example of legitimate subdivision.
Sometimes the whole divided into one set of parts according to one
principle of division can be divided into another set of parts from a
different point of view. To divide the same totality several times, each
time by a different principle of division, is to codivide. In the first chapter
of this book we divided logic into the logic of the first operation, the
logic of the second operation, and the logic of the third operation; and
we divided it also into formal logic and material logic. This is an example
of legitimate codivision.

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THE LOGIC OF THE FIRST OPERATION

The rules which apply generally for division apply, of course, for
subdivision and codivision. Each distinct division included within the
complex framework of either subdivision or codivision must measure up
to the four rules discussed in the preceding section. 'Hie more compli­
cated the process of subdivision or of codivision, the more carefully
must we attend to the rules. Increasing complexity brings with it a cor­
respondingly increasing possibility of disorder, and disorder is the death
of division.
VI. The Value of Definition and Division

Both definition and division have an importance proper to themselves.
Each aids in explaining the object grasped in the first operation of the
intellect. Definition clarifies the concept from the point of view of
comprehension, while logical division clarifies it from the point of view
of extension. It is a philosophical axiom that act is prior to potency,
in the sense that the actual is more excellent than the potential. Since
comprehension expresses what the concept is in act, while extension
expresses what it is in potency, definition is logically of greater importance
than division. Accordingly, it is not surprising to note that division is
of value not only in reference to itself, but also in so far as it is ordered
to definition. The ultimate perfection of the first operation of the intel­
lect comes in defining, and frequently definition can be achieved only
in conjunction with division. For example, the dividing members in an
essential division are differentiated from each other by opposed specific
differences and arc united in a common proximate genus. Essential divi­
sions, therefore, attain differing species. The process of dividing gives
us the elements needed for the essential definition of these species,
namely, proximate genus and specific differences. The same is true, in
proportionate fashion, for less strict instances of definition and division.
Definition is a process which demands for its integrity a high degree of
precision. Unless the defined and its definition are but different expres­
sions of identically the same thing, the definition in question is faulty.
Division is an extremely useful logical instrument for attaining the pre­
cision needed in definition. To illustrate: mathematics in the classical
sense can be defined as the speculative science of quantified being. In
attempting to understand this definition one might begin by noting that
of the ten ultimate types of being mathematics fits into the category' of
quality. Then it is significant to note that of the four species of quality
mathematics falls under the first as a habit. Habits are either entitative
or operational, and mathematics is of the second type. Operational habits

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91

arc either good or bad. Good habits are virtues, and mathematics is a
virtue. Virtues in turn are either moral or intellectual, depending on
whether they perfect the will or the intellect. Mathematics is an intel­
lectual virtue, and that special kind of intellectual virtue which we call
science. Science itself is either practical or speculative, depending on
whether its end is action or truth. Mathematics aims at truth, and so
is a speculative science. Speculative science, finally, is cither ordered to
a knowledge of physical being, quantified being, or being as such. Mathe­
matics is the second type, namely speculative science of quantified being.
Thus the definition of mathematics is arrived at, in this example at least,
after a fairly complicated set of divisions and subdivisions. The way to
definition need not always be as involved as this. The purpose of the
example is not to manifest an unvarying process. It simply illustrates
the connection between definition and division and the assistance that
division can render in the approach to definition. The illustration also
points up the fact that definitions are not always easily achieved. The
formation of a definition is usually the result only of a long and compli­
cated logical effort. Division is one of the chief instruments of the
intellect in its search for definitions.
As long as the intellect remains on the level of simple apprehension,
it is ordered to the act of defining. But the first operation of the intel­
lect is itself ordered to the second and to the third. Thus, definition is
not simply an end in itself. It is primarily of concern in so far as it
can be a principle of argumentation. We shall return later to the
question of definition in order to see how definition does function as
a principle of argumentation.
Unmistakable evidence of the importance of both definition and divi­
sion to science can be found in any scientific textbook, including this
one. Each chapter abounds in definitions and divisions of one type or
another. An examination of the two sections on definition and division in
this book reveals a variety of definitions and divisions. For example,
definition is logically divided and subdivided into its types, and each of
the dividing members in this logical subdivision is itself defined. In
addition, each of the dividing members of this logical subdivision is physi­
cally divided into its composing parts. Consider the case of definition
by way of property. We saw that this is a type of noncssential definition.
Xoncssential definition is a type of real definition. Real definition is a
type of definition. And definition is in a sense a type of logical dis­
course. We defined definition by way of property as a nonessential defini­
tion indirectly manifesting the essence of the defined in terms of a neces­
sary and exclusive attribute caused by that essence. Finally, we physically

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THE LOGIC OF THE FIRST OPERATION

divided it into the genus of the defined and one of the properties of the
defined. Division too has been defined and divided (both logically and
physically) with its dividing members themselves defined and sometimes
further divided. This is as it must be. There can be no systematic and
scientific discourse that dispenses with rigorous definitions and divisions.
Not that these represent the whole of any scientific treatise nor the end of
any scientific inquiry. But they do represent an indispensable part and a
beginning without which there can be no significant scientific conclusions.
Exercise VIII

1. Define: logical division, essential division, nonessential division, totalitydivided, dividing members, principle of division, dichotomous division,
physical division, logical whole, subjective parts, integral whole, com­
posing parts, codivision, subdivision.
2. Compare logical division to definition. Why is each spoken of as discur­
sive? What are limits to this manner of speaking?
3. Logically divide nonessential division into its types; explain the reason
for the division in terms of the principle used for making it.
4. What arc the modes of opposition which arc significant for the theory of
division?
5. What arc the advantages and possibly some of the disadvantages of
dichotomous division?
6. How precisely does a logical whole differ from an integral whole?

7. Define physical division and then illustrate the many ways in which
physical division can be realized by supplying examples of significantly
different physical divisions.
8. What docs it mean to say that for logical division the whole is in the
part, and for physical division the part is in the whole?
9. List the rules for a good division, and defend each rule by measuring it
against the definition of division itself.
10. Compare subdivision with codivision, and discuss the rules for a good
division in the context of each.
11. What is the value of division considered in itself? What is the value of
division as ordered to definition?
12. Classify' and evaluate the following attempts at division:
12.1. Predicable — species, genus, generic difference, specific difference,
property, and accident.
12.2. Science — philosophical science and experimental science.
12.3. Division — logical and physical.
12.4. Organism — plant, animal, and man.
12.5. Essential definition — proximate genus and specific difference.

DIVISION

93

Virtue — good habit, intellectual habit, and moral habit.
Man — artistic and nonartistic.
Student body — freshmen, sophomores, juniors, and seniors.
Student—freshman, sophomore, junior, and senior.
Snow — white and nonwhite.
Logic — logic of the first operation, logic of the second operation,
and formal logic.
12.12. Digestive tract — oral cavity, pharynx, esophagus, stomach, small
intestine, and large intestine.
12.13. Clothing — black and white.
12.14. Organism — living body, plant, and animal.
12.15. Predicate — essential predicable and nonessential predicable.
13. Logically divide, then subdivide and/or codivide the following:
13.1. Quality
13.2. Cognitive operation
13.3. Football player
13.4. Substance
13.5. Law
13.6. Society
13.7. Predicable
13.8. Logical discourse
13.9. Plane figure
13.10. Pleasure
14. Make a list of the divisions employed in this textbook, and in other
technical and nontechnical books. Classify and evaluate each.
15. There is a lengthy and logically rigorous passage in St. Thomas’ Com­
mentary on the De Anima of Aristotle in which several divisions are
proposed in preparation for an adequate definition of the soul. Study this
passage and pick out each division and each definition offered in it.
Classify and evaluate each one of them. Try to reproduce the passage in
outline form. 'Hie passage in question is Book II, Lesson 1, No. 213233 (Pirotta edition). It is available in English, translated by K. Foster
and S. Humphries in Aristotle’s De Anima (New Haven, Conn.: Yale
University Press, 1954), pp. 166-172.

12.6.
12.7.
12.8.
12.9.
12.10.
12.11.

part

n

The Logic of

the Second Operation

CHAPTER

IX

Judgment: The Second Operation

I. The Nature of the Second Operation of the Intellect

The first operation of the intellect bears upon quiddity or essence —
and nothing more. In simple apprehension we grasp natures without
affirming or denying anything about these natures, not even their
existence, actual or possible. Thus simple apprehension is a trun­
cated act of knowing. It gives us at best an incomplete and unsatisfying
view of reality. It can be called an analytic act which knows reality
only by “tearing it apart” and presenting but one aspect of it. Knowing is
completed only in an operation which “puts reality back together again”
by grasping essences in the only way in which they are, namely, as
exercising some type of existence. Tin's is the second operation of the
intellect, which we call judgment, whose expression is the proposition.
Judgment is the act of the intellect affirming or denying one thing
of anota, There arc basically two types of judgment. These are the
existential judgment and the attributive judgment. The existential judg­
ment is the less complex of the two. It is the intellectual act of know­
ing that something is (i.e., exists) or is not (i.e., does not exist). The
attributive judgment is more complex. It is the intellectual act of know­
ing that something exists or does not exist with certain determinations.
This man is (i.e., exists) and This man is not (i.e., does not exist) are
existential propositions, while This man is tall and This man is not tall
are attributive propositions. In the one case existence itself is affirmed
or denied of a subject. In the other case some quidditative determina­
tion is affirmed or denied of a subject. In both cases both quiddity
and existence arc involved: in the first something is thought to be, and
in the second something is thought to be something. In each case some­
thing 'which we shall call the logical subject) is thought to be somehow
modified or actualized by something else (the predicate). The predicate
is related to the subject as form to matter, because the predicate is
97

98

THE LOGIC OF THE SECOND OPERATION

understood to signify some sort of determination or perfection for
the subject.
Tire second operation of the intellect is frequently referred to as the
operation of composition or division. The reason for this is most clearly
seen in the case of the attributive judgment. In the affirmative attributive
judgment two intelligible objects arc united or composed by way of
affirmation. In the negative attributive judgment two intelligible objects
are separated or divided by way of denial. Man and able to talk are two
of the many distinct intelligible aspects existing together in subjects such
as Man', John, and Peter. They are distinct as intelligible aspects, and
each is known in a distinct act of simple apprehension. Each has its
own comprehension and is defined in its own way. In a word, they arc
distinctly different objects with distinctly different meanings. Yet, even
though man and able to talk are different intelligible objects they exist
together in the same thing, namely, this man, that man, and in fact every
man. As intelligible objects they are two, but as realized in existing
subjects they are one. To judge Every man is able to talk is to grasp
man and able to talk in their existential status, knowing as one in exist*
ence things which are two as objects of thought. This is to know that
man is identical in subject with the being which is able to talk. Man
and able to talk remain diverse in meaning, and the affirmative attribu­
tive judgment in no way intends to identify them in meaning. It
identifies them only in reference to existence by affirming them to be
two aspects of the same thing.
Judgment bears on existence as well as essence. The affirmation or
negation of the second operation is an intellectual commitment to the
way in which things are. This is clearly the case whenever the subject
about which something is affirmed or denied is an actually existing thing.
But Every man is able to talk would be true even if no men actually
existed, and Santa Claus wears a beard is true although Santa Claus
is only a fiction. It seems perhaps that we can make a judgment in
instances where the judgment need not or does not bear upon existence.
However, this is not the case. There is no affirmation or denial unless
there is some reference to the way things are, and there is no judgment
apart from affirmation or denial. The primary instance of existence is
the act of existing which makes a thing to be an actually existing subject
outside the mind —the kind of existence enjoyed by each one of u$
right now. However, this is not the only type of existence. Tilings can
exist in the mind, as objects known, as well as outside the mind in the
realm of real subjects. Two types of things can exist in the mind: first
of all, those things which also can exist outside the mind, and second.

JUDGMENT: THE SECOND OPERATION

99

those things which cannot exist outside the mind precisely because extra­
mental existence is contradictory to their natures. Those which can exist
outside the mind can be divided into those which actually exist extramentally and those which do not but could. We have called the first
actually real beings and the second possibly real beings. We have seen
that those things which can exist only as objects in the mind are called
beingsjof-thc-reason. The most significant type of judgment refers to
beings which exercise the act of real existence. But judgments about
possibly real beings and beings-of-the-reason are legitimate.
II. The Pre-eminence of the Second Operation

Judgment is the most important operation of the intellect. We have
seen that simple apprehension is a truncated act of knowing, which
is related to judgment as the imperfect to the relatively perfect. Reason­
ing presupposes prior judgments. Nevertheless reasoning is ordered to
judgment. Reasoning itself makes sense only in so far as it generates a
conclusion. This conclusion is itself grasped in an act of judging. Were
we able to compose and divide in every instance without the support of
premises, which are themselves compositions or divisions, we would
never reason. Man is not equipped intellectually so that he can do this,
lie is by nature a rational or reasoning being. When he docs reason he
reasons so that he may judge. Thus, the first and third operations are
each in their own way for the sake of the second. The second operation
is the intellectual operation par excellence.
III. Judgment and the Motive for Assent

A judgment is an active intellectual commitment in the face of evi­
dence deemed sufficient to warrant assent. Judgments can differ from one
another either on the basis of the firmness of the assent or on the basis
of the nature of the evidence for assent. As far as the firmness of assent
is concerned, some judgments arc certain and some are only probable. A
judgment is certain when the assent given to a proposition excludes the
fear that its opposite is true. This is the way in which I firmly assent to
the tmth that even’ man is rational. There is absolutely no reason to
suspect that some man is not rational. /X judgment is probable when
assent is given to a proposition with an awareness that its opposite might
possibly be true. Thus I may assent to the proposition that some runner
will run a mile in three minutes and fifty seconds. However, I assent to
this only with reservation, for I am aware of the possibility that no

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THE LOGIC OF THE SECOND OPERATION

runner may achieve this mark. We are speaking of the difference between
certain and probable judgments on the basis of a difference in firmness of
assent. Lest this be misunderstood, we should note that even in judging
with probability there is a sense in which a firm assent is given, but it is
a firm assent only to the probability of the proposition in question. I
am not sure that a runner will establish a record of three minutes and
fifty seconds for the mile run, but I am sure that it is a distinct
probability.
Judgments can also differ on the basis of the evidence which makes
them legitimate. Assent to a proposition may be given on the basis of
extrinsic or intrinsic evidence. Extrinsic evidence consists in the testimony
of some authority to the truth of a proposition. This is assent by way of
faith. Faith is natural or supernatural depending on whether the authority
in question is finite or divine. Faith yields certitude or probability depend­
ing upon the degree of the trustworthiness of tire authority involved.
Intrinsic evidence comes from the thing itself, not from some authority
who knows about it. Such evidence may be either immediate or mediate.
Immediate evidence is found in a direct sensory-intellectual experience
of an actually existing thing or in the very meanings of the terms involved
in a proposition. The first kind of immediately evident proposition is
factually evident, the second self-evident. Propositions are mediately evi­
dent when they are seen to follow by way of reasoning from either type
of immediate proposition. Tire certitude or probability of assent based
on intrinsic evidence depends on the force of the evidence in question.
Consider the following examples which illustrate the differences in
judgment considered in this section:

1. I know certainly on the basis of supernatural faith the fact that
there is a final judgment for each man — because God has re­
vealed this.
2. I know certainly on the basis of human faith that Tokyo is a
city' in Japan — because competent and honest geographers have
said so.
3. I know as probable, on the basis of human faith, that I will not
die within the year of a heart attack — since a competent physi­
cian has carefully examined me and assured me that my heart
is in a healthy condition.
4. I know with certitude and as immediately self-evident that man
is a rational animal — for rational animal is the very definition
of man.
5. I know certainly and immediately, but not as self-evident, that my

JUDGMENT: THE SECOND OPERATION

101

typewriter is gray — for I see it now as immediately present to
my sense of sight.
6. I know certainly as mediately true that every' triangle has three
angles equal to two right angles — for I see that this follows from
the necessary truths that every plane figure bounded by three
straight lines has three angles equal to two right angles and that
every’ triangle is a plane figure bounded by three straight lines.
7. I know as probable and mediately true the fact of limited evolu­
tion — for this is established in the light of very probable premises.
IV. Truth and Falsity

Truth is a conformity between what is known and what is. Falsity, on
the other hand, is a lack of this conformity. On the level of simple
apprehension the intellect has not yet committed itself to the way in
which things are, nor even to the fact that they are. Hence, there is, in
the strict sense, neither truth nor falsity on this level. It is absurd to
say “true” or “false” about objects such as man, green-eyed monster, and
sleek fast racing horse. On the level of judgment there is a commitment
(either by affirmation or denial) to the existential situation. This com­
mitment can be said to be true or false, according as it does or does not
measure up to things as they are. For example, Every' man is able to talk
is true because it expresses things as they really are, and Every man is a
green-eyed monster is false because it fails to conform to things as they
are. In the strict sense then, truth and falsity are found only in the
judgment, in the intellect composing or dividing.
V. The Proposition and Its Elements

The object of the second operation is interiorly expressed in a mental
proposition. Its verbal expression is an oral or written proposition. A
proposition is technically spoken of as a composite expression. But
a proposition is a special kind of composite expression. There are others,
for example, definitions and arguments.
Composite expressions arc basically divided into imperfect and per­
fect. An imperfect composite expression signifies an incomplete thought
which fails to satisfy the intellect. A perfect composite expression signifies
a relatively complete thought which does not leave the intellect essen­
tially in suspense. Rational animal by itself is an incomplete expression
which calls for something more to satisfy' the mind. The same is true
for a composite expression such as while the band played on. These are
imperfect composite expressions because they leave the intellect essen­

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THE LOGIC OF THE SECOND OPERATION

tially in suspense. On the other hand, there are several types of perfect
composite expressions. Deprecative statements (prayers), imperative state­
ments (commands), interrogative statements (questions), and vocative
statements (salutations and appeals) are all examples of perfect com­
posite expressions. None of these leaves the mind in suspense as does an
imperfect composite expression. Yet none of these is a proposition, for
a proposition is that special kind of composite expression which signifies
what the intellect knows in judgment. We have seen that the intellect
possesses truth or falsity in its second operation depending on its con­
formity to reality. But it makes no sense to speak of deprecative, impera­
tive, interrogative, and vocative statements as true or false. Come here!
does not leave the intellect in suspense, but it is neither true nor false.
All men arc rational not only does not leave the intellect in suspense
but it can also be said to be true. The proposition can thus be dis­
tinguished from other perfect composite expressions by being defined
as a composite expression which signifies the true or the false.
To judge is to affirm or deny one thing of something else. That
about which something is affirmed or denied is called the subject of
the proposition. What is affirmed or denied of the subject is its predi­
cate. In the verbal proposition the connection between the predicate and
its subject is most clearly signified by a copula which is some form of
the verb “to be.” The verb “to be” can be used as a copula distinct
from the predicate and the subject. Then both the predicate and the
subject are expressed basically as nouns (i.e., after the fashion of sub­
stances). The subject is always basically a noun, but sometimes the predi­
cate is expressed as a verb (i.e., a term which signifies in the mode of
existence and action) which signifies in one term both the predicate
and the copula. The difference between these two formulations of the
proposition is that the former has only a verb-copula while the latter
has a verb-predicate. “Some men are great artists” has a verb-copula
linking the two nouns which signify respectively the predicate and the
subject. “Some men dare to tempt God” has a verb-predicate which
simultaneously signifies the predicate and its link with the subject of
which it is said. From one point of view the elements into which the
proposition is resolved are the noun and the verb. From another point
of view they are the subject, predicate, and copula.
For the purposes of rational discourse it is necessary to distinguish
subjects and predicates clearly from one another and to recognize each
for what it is as a distinct object. Hence it is usually preferable to state
an attributive proposition with a verb-copula rather than a verb-predicate.
In this way the subject and predicate terms are clearly distinguished

JUDGMENT: THE SECOND OPERATION

103

from one another and from the copula which unites or separates them
in the proposition. In this type of expression it is clear just what the
subject-term is and just what the prcdicate-tcrm is. Propositions stated
in this manner are said to be in strictly logical form, and it is always
possible to state propositions in logical form even though they are
not originally given in this fashion.
Propositions which are not originally in strictly logical form can be
put in it by isolating the subject-term and the predicate-term, no matter
how complex these may be, and then by joining or separating these
through an affirmative or negative form of the verb “to be” in the
proper tense. Sometimes the logical subject is grammatically stated in the
place of the predicate, and the predicate in the place of the subject. In
such cases, of course, it is wise to invert the order of the terms as
verbally stated so the grammatical expression is as faithful as possible
to the logical expression. The literary quality of propositions in strictly
logical form is frequently very poor, and one would not ordinarily
speak and write in this mode. However, it is frequently wise for a
speaker or writer to recapitulate his assertions and arguments in strictly
logical form because they can be most easily logically evaluated in this
form. It is always wise for the student beginning his course in logic
to recast propositions in strictly logical form. His concern is with logical
structure and not literary excellence, and he cannot, especially at the
beginning, afford to obscure what is logical in his concern to write well.
Consider the following propositions. In each case the proposition is
originally stated in a form which is not strictly logical. The second expres­
sion of each proposition is in logical form, and it signifies exactly the
same thing as the first.

1. Water boils at 100° centigrade —
¡Water] is ¡something which boils at 100° centigrade.)
2. No elephant ever forgets —

|No elephant) is [something which ever forgets [
3. Husbands have wives —
[Husbands] are ¡things which have wives.)
4. Patriotic politicians put the good of their nation above the good
of their party —
[Patriotric politicians are ¡beings who put the good of their

nation above the good of their party.)

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THE LOGIC OF THE SECOND OPERATION

5. Fools walk in where angels fear to tread —
I Fools I are ¡beings who walk in where angels fear to tread?]

6. Pleasant remain the memories of a happy youth —
ic memories of a happy youth] are [things which remain
Exercise IX

1. Define: judgment, merely existential judgment, attributive judgment,
extrinsic evidence, intrinsic evidence, immediate proposition, factually
evident proposition, self-evident proposition, mediate proposition, truth,
falsity, certitude, probability, noun, verb, subject, predicate, composite
expression, perfect composite expression, imperfect composite expression,
proposition.

2. What is the difference between simple apprehension and judgment?
3. How can a predicate and a subject with diverse definitions be identified
in an affirmative proposition?

4. Pinocchio had a long nose. How can this be understood to be a proposi­
tion if the second operation is understood to bear upon existence as well
as essence? After all, Pinocchio is only fictional.
5. In what sense is the second operation more excellent even than the third
operation of the intellect?

6. Distinguish between the following propositions in terms of motive for
assent:
6.1. The whole must be greater than any one of its parts.
6.2. The Communists have made great progress in their attempts to
conquer the world.
6.3. Every man is a morally responsible agent.
6.4. Interplanetary travel will become commonplace by 1985.
6.5. It rained yesterday.
6.6. Joe Dimaggio is the greatest baseball player in the history of the
game.
6.7. I am standing up.
6.8. If I am standing up, I am not not-standing up.
6.9. No Communist can be trusted to be true to his word.
6.10. Every organism is a living body.
7. Defend the definition of the proposition as a composite expression which
signifies the true or the false.
8. What is the difference between a proposition with a verbcopula and a
proposition with a verb-predicate?
9. Wherever possible restate the following as propositions in strictly logi­
cal form:
9.1. John can run the mile in four seconds flat.

JUDGMENT: THE SECOND OPERATION

105

Some men who can be trusted in a tight situation cannot be
trusted when things are going smoothly.
9.3. Beware of the dog.
9.4. Happy is the man who owes no debts.
9.5. To the winner go the spoils.
9.6. Don’t give up the ship.
9.7. The third world war terrifies all who anticipate it.
9.8. All the world loves a lover.
9.9. No one can say 1 didn’t try.
9.10. Who can fail to admire a courageous man?

9.2.

CHAPTER

X

The Supposition of Terms

I. Signification and Supposition

We have seen that a term is a sign which stands for a quiddity or
meaning. The meaning for which the term stands is its comprehension
or signification. Since definition expresses the comprehension of con­
cepts or meanings, terms having a common definition have the same
signification, and the distinction between univocal, equivocal, and analo­
gous terms is made from this point of view. Univocal terms have identical
signification. Equivocal terms have wholly diverse significations. Analogous
terms have significations which are basically different but relationally
identical, 'lire importance of identity' in signification for logical dis­
course can be seen if we anticipate the third operation for the moment
and note the differences in the following attempts at argument:
1. Every triangle has three angles equal to two right angles.
But, every isosceles triangle is a triangle.
Therefore, every isosceles triangle has three angles equal to two
right angles.
2. Every triangle has three angles equal to two right angles.
But, John’s romantic situation is a triangle.
Therefore, John’s romantic situation has three angles equal to
two right angles.

The first argument is obviously good; the second obviously defective.
Yet both seem to involve three terms and the same pattern of arrange­
ment. However, in the first argument there is the same signification
attached to each term whenever it is used. In the second argument, on
the other hand, the same signification is not given to one term, namely
“triangle,” each time it is used. At best “triangle” in the second argu­
ment is only tenuously analogous. Because of this, the argument is
formally defective. Its conclusion fails to follow from its premises pre­
106

THE SUPPOSITION OF TERMS

p

107

cisely because the signification of “triangle” varies from premise to
premise.
Let us consider a third argument before moving on:
3. Triangle is more universal than isosceles triangle.
But, ever)- isosceles triangle is a triangle.
Therefore, every isosceles triangle is more universal than isosceles
triangle.

Something is obviously wrong with this argument. At first glance it
might seem that its defect is like that of the second argumentation. Yet
examination discloses that the terms have a unity of signification each
time they are used. The term “triangle” is defined in the same way each
time it is used. However there is some way in which “triangle” in the
second proposition differs from “triangle” in the first. The term has the
same signification in both propositions, but it is used differently in each.
In the first proposition “triangle” is used to stand for a concept which
is understood as enjoying an existence as an object for the intellect. In
the second proposition “triangle” is used to stand for a mathematically
real essence which is realized in every individual triangle. The term stands
for the same nature each time, but the mode of existence enjoyed by
this nature is different in the two cases. In short, the signification is the
same, but the term is nevertheless used differently. This difference is
sufficient to render this third argument defective. The use which a term
has over and above its signification is called its supposition. Terms have
signification even on the level of the first operation, but they have sup­
position only in propositions. Terms in isolation stand for meanings, or
else they arc not terms. However, terms in isolation cannot be used to
stand for one mode of existence rather than another because the first
operation of the intellect abstracts from all modes of existence. Wc define
signification as the meaning for which a term stands. We define supposi­
tion as the way in which a term is used in a proposition to stand for
what it signifies. Although supposition is a property of both subjects
and predicates, our concern is primarily with subjects. Predicates do not
vary in supposition nearly as much as do subjects. Predicates are usually
taken in what wc shall call real and personal supposition.
II. Material and Formal Supposition

Supposition is first of all divided into material and formal supposition.
When a term is used simply to stand for itself as a term and not for the
quiddity it is designed to signify, it is said to be taken with material

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THE LOCIC OF THE SECOND OPERATION

supposition. In “Triangle is a three-syllabled word,” “Man is easy to
pronounce,” and “Democracy is semantically loaded,” “triangle,” “man,”
and “democracy” all refer to themselves as words. They are used with
material supposition.
Whenever a term is used to stand for a quiddity and not for itself,
it is said to be taken with formal supposition. Terms in propositions are
ordinarily used with formal rather than material supposition. Formal sup­
position is divided into types, and we will consider its types before
examining illustrations of formal supposition.
III. The Types of Formal Supposition

Terms taken with formal supposition are used to stand for quiddities or
natures. If the nature for which a term stands is considered precisely as
an object existing in the intellect, the term has logical supposition. “Man”
in “Man is a universal concept” is taken with logical supposition.
“Universal concept” can be said of “man” only in so far as this term
stands for a concept which has existence in the mind. This is the
case for “triangle” in “Triangle is the proximate genus of isosceles figure,”
for “ability” in “z\bility is an abstract concept,” and for “animal” in
“Animal is more comprehensive and less extensive than organism.” On
the other hand, if the nature for which a term stands is regarded as
realizable in singular existente outside the intellect, that term has real
supposition. We will subdivide real supposition shortly and take up several
examples at that time. Here it is sufficient to note that “man” is taken with
real supposition in both “Man is rational” and “Some man is happy.”
A third type of formal supposition differs from both logical and real
supposition. At the same time, however, it seems to partake of both.
Hence we shall call it logico-real supposition and shall explain it by
reference to the two previous types. As noted, logical supposition results
when a term is used to signify a universal nature regarded precisely as
an intelligible object existing only in the mind and possessing, as known,
a positive unity. For example, in “Man is a species,” “species” is said
of “man” only in so far as the latter signifies a concept which has
positive unity and is regarded exclusively as an object of thought. Here
the term “man” has logical supposition. Real supposition, on the other
hand, results when a term is used to signify a nature regarded in its
capacity as being realizable in singular existente, that is, in subjects which
exist outside the mind. For example, in “Man is rational,” “rational” is
predicated of “man” is so far as it signifies a nature which can exist not
only as an object of thought but also in concrete things. Here the term

THE SUPPOSITION OF TERMS

109

“man” has real supposition. “Man,” taken in this sense, is open to
individual existents such as Mary and Peter, all of whom can be called
“rational.”
Now consider the proposition “Man is God’s finest earthly creature.”
Here, as in logical supposition, “man” refers to an object which possesses
a positive unity. But at the same time it refers to that object as exercising
real, not merely mental existence, for “Man is God’s finest earthly
creature” is true whether man is known or not. In this it differs from
logical supposition, which regards a concept precisely as known. “Man is
a species” is true only of the concept man as known by the mind. By
referring to concrete existence, then, the term “man” in this proposition
bears a likeness to real supposition. However, if a term is taken with
real supposition it is open to singular existents, and “man” in “Man
is God’s finest earthly creature” is taken with a positive unity which
precludes this. We cannot say of each man that he is God’s finest
earthly creature, as we might say of each that he is rational. In this,
then, the term “man” here differs from the same term taken with real
supposition. But since it docs partake of both logical and real supposition
in some aspects, we can say that it is here taken with logico-rcal
supposition.
IV. The Types of Real Supposition

A term is taken with real supposition when it signifies a nature or
quiddity which is understood as being capable of existence in singular
inferiors. But this nature can be regarded in two ways. First, it can be
regarded absolutely, that is, in itself, without explicit reference to the
individuals in which it is realized. Second, it can be looked at primarily
in its reference to those individuals. 'Ilius there are two basic types of
real supposition, spoken of as absolute and personal. Absolute real sup­
position results when the term is taken primarily for the universal nature
itself and only virtually for the individual existents which might par­
ticipate in that nature. Personal real supposition, on the other hand,
occurs when the term is taken primarily for those individual existents,
and through them for the universal nature in itself.
There is one important point to notice about personal and absolute
supposition. Terms taken with absolute supposition can be used only in
propositions with necessary predicates, whereas terms with personal sup­
position can be used in propositions with cither necessary or contingent
predicates. “Man” in “Every man is rational,” in “Every man is risible,”
and in “This man is tall” is taken each time with personal supposition.

no

THE LOGIC OF THE SECOND OPERATION

The difference between “man” in “Man is rational” (where “man” is
taken absolutely) and “Every man is rational” (where “man” is taken
personally) is that “man” in the first proposition signifies the nature
man directly and in the second it directly signifies the individual subjects
which possess the nature.man.
/
r
V. Universal, Particular, and Singular Terms

Terms taken with personal supposition signify natures or quiddities as
realized in singular inferiors. However, these terms can signify these
natures taken either in their full extension or with some limitation to
that extension. Terms used to stand for natures taken with some restric­
tion placed upon their full extension can signify a nature understood with
a determinate limitation placed upon their full extension or with an
indeterminate limitation. “Man” in “Every man is mortal” is used to
the fullest as far as extension is concerned. In this case the predicate is
identified with all the inferiors of the subject. When a term is used in
this way it is said to be distributed or universal. “Man” in “Some man
is happy” signifies the extension of the nature man as limited indeter­
minately. Clearly it is not used to the fullest, but just as clearly there
is no determinate indication as to the exact limits of the restriction
placed upon its extension. This is the particular or undistributed use
of the term. “Man” in “This man is tall” signifies a determinate limit
of the full extension of man. Once again there is clearly a restriction
placed upon the full extension of the term as used, but here there is
a determinate indication of the limits of that restriction. The term
“points” to the segment of its extension which is pertinent to the
proposition. Terms taken with a determinate limitation placed upon their
full extension arc said to be singular.
We shall discover that we must be able to classify both the subjects
and the predicates of propositions as universal, particular, and singular. We
shall refer to this as a classification of the extension of these terms. The
extension of the subject-term is usually indicated by some modifying
word such as “every,” “some,” or “this.” The extension of predicate­
terms is not so explicitly identified. However, as we shall see in the
next chapter, it is fairly simple to determine the extension of any predicate.
In regard to each of the types of personal supposition, there are a
few important points to note. In one sense all terms are universal
whether they are used in propositions or not, because they stand for
universal concepts. 'Phis is true even of those terms which are used
particularly and singularly in propositions. We have already defined the

THE SUPPOSITION OF TERMS

111

universal term as that which can be said of many. When we speak of
universal, particular, and singular terms here we are speaking of the use
of the universal terms. We can speak of a universal term as used uni­
versally (in its complete extension) or particularly (with an indeterminate
limitation to that extension) or singularly (with a determinate limitation
to that extension). Terms taken with absolute, real supposition do not,
strictly speaking, have any extension as used. However, they are virtually
universal and can be handled in logical discourse as though they were
universal. Whatever belongs intrinsically to a nature or is necessarily con­
sequent upon what belongs intrinsically to a nature must certainly belong
to all the subjects within the extension of that nature.
Students frequently find it difficult to understand what is meant by
the particular use of a term. Perhaps the reason is partially terminological.
We say that a term is particular when it is used with an indeterminate
limitation placed upon its full extension. Tin's is to say that it is used
with no particular segment of its extension in mind, using the term
“particular” in the colloquial sense. Particular subjects are characteristically
indicated by the word “some.” However, any word can be used so long
as it indicates an indeterminate limitation placed upon the full extension
of the term. “Certain men,” “several horses,” and “a few sailors” are
all particular terms. The shades of meaning which distinguish these
modifiers from one another are not of concern to the logician. Each is on
a par with the other as an indication of an indeterminate restriction upon
the extension of the term. For the same reason “some triangle” and
“some triangles” are logically equivalent; what is logically significant
in each case is only the indeterminate limitation of the extension of
“triangle.” Grammatical singularity and plurality neither add to nor
subtract from the status of the term in so far as it is used as particular in
logic. For the sake of logical discourse it makes no difference whether a
term used particularly refers to one or more subjects. The important point
is that it refers to some (i.e., at least one) extensive subject of a given
concept without determinately pointing to any definite one.
The singular use of a term in logic should not be confused with the
grammatical singular. A term is used singularly in logic if and only if
there is a determinate limit of the full extension of the term. 'Ilius both
“this man” and “these men,” and “that tree” and “those trees,” have
singular supposition. All universal terms modified by demonstratives are
taken in singular supposition. This is also true of proper names. Terms
taken with logico-rcal supposition do not, strictly speaking, have any
extension. However, they function in logical discourse as though they were
singular because they are understood as signifying natures which have a

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THE LOGIC OF THE SECOND OPERATION

positive unity. Terms with logical supposition can also be handled like
singulars for exactly the same reason.
VI. Collective and Divisive Terms

We have divided terms taken with personal supposition into universal,
particular, and singular. From another point of view they can be divided
into divisive and collective tenus. A term is used divisively when it
stands for its inferiors taken one by one. It is used collectively when it
stands for its inferiors as taken together, i.e., as a group. The term “man”
(like all terms) is ordinarily used divisively. This is the case whether
it is expressed grammatically in the singular or in the plural. “Every
man is subject to death” and “All men arc subject to death” arc identical
propositions. In each case subject to death is affirmed of all individual
men and of each one of them taken individually. There is no suggestion,
even in the second expression, that the predicate is to be understood as a
characteristic of all men taken together as one group. In both cases the
subject is taken divisively, although this is probably clearer in the first
expression than in the second. However, terms which are ordinarily used
divisively can be used collectively. When it is said that men may over­
populate the world, or that several horses ate all the hay in the barn, or
that the ball players filled the clubhouse, or that our soldiers won the
war, “man,” “horse,” “ball player,” and “soldier” are all used collectively.
Only as a group can men be thought to overpopulate the world. Only
as a group could the horses have eaten all the hay. It is impossible for
each of many men to overpopulate the world. And it is impossible for
each of several horses to have eaten all the hay. Similarly for “ball player"
and “soldier” as used above. Since terms which are ordinarily used
divisively can be taken collectively, we must be careful in our use of
terms. To confuse the collective and divisive use of terms can be logically
disastrous. Whatever can be said of a group of individuals need not be
said of each of the individuals in the group. An excellent team may have
a player or two who arc less than excellent. Whatever can be said uni­
versally of the individuals in a group need not be said of the group taken
as such. 'Though all the United States senators, divisively taken, represent
one state, the group as a group represents all fifty states.
A term like “man” can be realized in an individual (c.g., Peter or
Mary). A term like “army” is limited to a group of individuals (i.e., a
number of soldiers taken precisely as a body of men). Although the
difference between “man” and “army” might seem like the difference
between the divisive and the collective uses of a term, it is a difference

THE SUPPOSITION OF TERMS

113

in signification, not in supposition. It is by definition that man can be
said of individuals, and it is by definition that army can be said only of
a group of individuals. A question of definition, of course, is a question
of signification and not of supposition. As a matter of fact “army” can
be used both divisively and collectively. It is divisive in “Every army
needs a good commander in chief.” It is collective in “The armies of
the nation defended the nation’s many fronts.”
VII. Possible and Actual Existence

Terms taken with personal supposition can stand for natures under­
stood to exist either actually or potentially. Propositions with universal
subjects and necessary predicates may express only possible existence, and
ordinarily they arc proposed in this fashion. Such propositions are
equivalent to propositions with subjects taken with absolute supposition.
“Every man is capable of speech,” "Every animal is sentient,” and “Every
triangle is a polygon” are examples of propositions which ordinarily
express only possible existence for their subjects. Understood in this
way they are true so long as there is a necessary connection between their
predicates and their subjects even though no actual instance of their
subjects exists. Propositions with particular subjects and contingent
predicates may express the actual existence of the subject, and they are
frequently intended to do so. “Some trees arc tall,” “Some men are
Americans,” and “Some animals arc ferocious” arc examples of proposi­
tions which ordinarily express actual existence. When these propositions
are understood to express the actual existence of their subjects they arc
true only so long as their subjects do actually exist and have the determina­
tion which is signified by the predicate. However, propositions with
universal subjects can express actual existence. This is ordinarily the case
when they are used with contingent predicates, for example, "All men in
the second half of the twentieth century live under the threat of nuclear
devastation.” It can even be the case when they arc used with neces­
sary predicates. A proposition with a universal subject and a necessary
predicate can suppose the actual existence of the subject. For example,
"Every man (and there are men) is capable of speech.” On the other
hand, propositions with particular subjects and contingent predicates can
be understood sometimes to express merely possible existence. "Some trees
are tall” can be proposed to mean that there are trees actually in existence
and that at least some of these are tall. However, it might also be
proposed simply to signify that the nature of trees is such that there
is the possibility of their being tall. In the first case the subject-term

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THE LOGIC OF THE SECOND OPERATION

signifies the nature taken personally and as actually existing. In the
second case it signifies the nature taken personally but with no more than
possible existence intended. Propositions with a particular subject and
a necessary' predicate can be enunciated (e.g., “Some man is rational”)
and these too can express either the actual or the possible existence of
their subjects.
VIII. Nonsupposing Subject-Terms

We have noted that both subject-terms and predicate-terms have
supposition but that the problem of supposition arises primarily in
reference to the subject. The supposition of the subject of a proposition
is limited by its predicate and by the copula of the proposition. A
predicate may be such that it can be identified with a subject only if that
subject is understood to exercise a special kind of existence. For example,
the attribute of species can be predicated of a subject only in so far
as this subject is understood to exist as a universal object of simple appre­
hension. We can say a “Man is a species,” but we cannot say “John
is a species.” No singular subject could be legitimately used with this
predicate, for no singular subject could stand for something which enjoys
the kind of existence relevant to this predicate. The singular subject
used with this predicate simply lacks supposition. It can be described
as a nonsupposing subject. In a similar fashion any subject which docs
not meet the demands of the copula of the proposition is a nonsupposing
subject. The copula of a proposition which expresses actual existence
can be either in the present, past, or future tense. In each case the terms
in the proposition must stand for things enjoying the kind of existence
demanded by the copula. The subject of “Babe Ruth was a great batter”
stands for a man who existed in the past. Thus it meets the demands of
this copula, which is in the past tense. 'Ilie subject of “Babe Ruth is a
great batter” or of “Babe Ruth will be a great batter” does not measure
up to the requirements of the copula. “Babe Ruth” is a name which
can stand only for a figure in the past. It cannot stand for a subject
which measures up to demands of a copula in either the present or
future tense. When it is expressed with these copulas, it is a nonsupposing
subject. Every' proposition with a subject which ought to stand in a
certain way for something but cannot do so and be true to the demands
either of its predicate or its copula has a nonsupposing subject. It is
impossible that any affirmative proposition with a nonsupposing subject
be true. The predicate of an affirmative proposition is identified with
its subject by way of its copula. This identification cannot be realized

THE SUPPOSITION OF TERMS

115

so long as the subject as used in the proposition is not proportioned to
the predicate and the copula. “John is a species” must be false, since
“John” cannot stand for a subject which enjoys the kind of existence
demanded by the predicate “species.” “Napoleon will be a great general’'
must be false, since “Napoleon” cannot stand for a subject in the future
time demanded by the copula in the future tense.
IX. The Unity of Terms and Logical Discourse

At the beginning of this chapter we noted that an argument which is
apparently valid can be defective if any of the terms used in it changes
its signification or supposition. Now that we have distinguished between
the types of supposition let us return to a more detailed consideration
of our original point regarding the unity of terms in logical discourse.
This discussion can be meaningful to us, although we have not yet come
to the third operation of the mind. All that wc need to note here concern­
ing logical discourse is that it involves various terms which are used more
than once. That one and the same term be used more than once is
essential to valid argumentation. The terms must stand for the same
thing, both from the point of view of signification and supposition, if they
are to be said to be identical terms. And if they are not one and the
same, the attempted discourse is necessarily defective.

1. Signification
Univocal-terms stand for precisely the same nature and offer no diffi­
culty in discourse. Equivocal terms stand for absolutely different natures
and must be treated as completely diverse terms. Thus equivocation always
renders logical discourse defective. Analogous terms are basically diverse
in meaning, but they are relationally the same. Because they are not
absolutely diverse in meaning they can be used in valid discourse, but not
precisely in the way in which univocal terms can be used. We shall,
however, omit the discussion here in favor of a consideration later in a
metaphysics class. The problem of the use of analogous terms in argu­
mentation is a special problem of method in metaphysics. Metaphysics
is a science in which the significant terms are analogous.

2. Material supposition
A term taken with material supposition stands only for itself as a
word, and not for the nature ordinarily signified. Hence we cannot, in a
valid argument, use the same term once in its material supposition and
then in its formal supposition. If a term is taken once with material

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THE LOCIC OF THE SECOND OPERATION

supposition in a given discourse it must be taken materially each time it
appears in this discourse. I cannot argue that Tom (or Harry) is threelettered because Tom (or Harr}’) is a man and man is three-lettered.
Logical supposition
Whatever is said of a subject expressed by a term taken with logical
supposition is said of it exclusively in virtue of its existence in the mind
as an intelligible object. 'Ilius it is impossible to use a term once with
logical supposition in a given discourse and a second time with some
other supposition. I cannot argue that man is the definition of man
because rational animal is the definition of man and every man is a
rational animal.
3.

4. Logico-rcal supposition
Whatever is said of a subject taken with logico-rcal supposition is
said of it both in virtue of the real situation and in virtue of the positive
unity which it has only as an intelligible object existing in the mind.
Because of the first point a term cannot be used once with logico-rcal
supposition and a second time with logical supposition in the same
discourse. Because of the second point a term cannot be used once with
logico real and then with real supposition. Clearly, then, if a term is used
once with logico-real supposition, it must be used in the same way each
time it is used in the same discourse. I cannot conclude that Tom is
the campus ideal because the football player is the campus ideal and
Tom is a football player.
5. Real supposition
Real supposition can be coupled neither with material, logical, nor
logico-rcal supposition. Consequently, a term used with real supposition
in a given discourse must be so used ever}' time it appears in that
discourse.

6. Absolute and personal supposition
A term used absolutely once may, of course, be used absolutely a
second time in any given discourse. And a term used with personal
supposition once may be used a second time with personal supposition.
However, because terms taken with absolute supposition can be resolved
into universal terms taken personally (in propositions expressing possible
existence), it is sometimes possible for a term to be used with absolute
and with personal supposition in the same discourse. Thus, if I know

that man (absolutely taken) is rational, and that John is a man (person­
ally taken), I can legitimately conclude that John is rational.
z 7. Universal, particular, and singular terms
Discourse can be legitimate even though there is some difference in
the way in which one term is used in a given discourse so long as the
term is generally taken with personal supposition each time. There is
no reason why I cannot conclude that all men arc sentient from the fact
that all animals are sentient and all men are animals — even though,
as we shall see later on, animal is used both universally and particularly
in this argumentation. We shall take up this point at length in a later
chapter.
/8. Collective and divisive terms
We have already noted that a term taken collectively is so different
from that same term taken divisivcly that it cannot appear both collec­
tively and divisivcly in the same discourse. Thus I could not conclude
that John numbers over 10,000 because he is a Marquette student and
Marquette students number over 10,000.
Actual and possible existence
There is no difficulty in this respect so long as the same term is used
each time to stand for actual existence or each time to stand for pos­
sible existence. There is difficulty only when a term is used once to
signify something possibly existing and a second time to signify some­
thing actually existing. The general rule which applies in this case is
clear. It is legitimate to go from the actuality of something to its pos­
sibility, but not from its possibility to its actuality.

9.

Exercise X

1. Define: signification, univocity, equivocity, analogy, supposition, material
supposition, formal supposition, logical supposition, real supposition,
logico-real supposition, absolute supposition, personal supposition, uni­
versal term, particular term, singular term, collective term, divisive term.
2. Distinguish between the signification of a term and its supposition. Why
does a term have supposition only in a proposition? Can terms with
identical signification have different supposition, and vice versa? Illustrate.
3. What is the reason for our concern with the signification and supposition
of tenus? What is the importance of these characteristics of terms for
logical discourse?
4. Distinguish between material and formal supposition. Illustrate.

118

THE LOGIC OF THE SECOND OPERATION

5. Distinguish between logical, real, and logico-real supposition. Illustrate.
6. Distinguish between absolute and personal supposition. Illustrate.
7. Distinguish between the universal, particular, and singular use of a term.
Illustrate.
8. Discuss some of the terminological difficulties consequent upon the use
of the terms “universal,” “particular,” and “singular” in the context of
No. 7.
9. Distinguish between the collective and the divisive use of a term. Illustrate.

10. Distinguish between propositions expressing only possible existence and
propositions expressing actual existence. Illustrate.
11. Examine the following pairs of propositions very carefully. Check to
sec whether or not the term which is apparently used twice in the
propositions is used with sufficient unit}' of signification and supposition
to allow for valid discourse. When this is not the case explain the reason
why.
11.1. Army is a collective concept.
Armies demand expert organization.
11.2. Annies are different than they used to be.
An army of ants swarmed by.
11.3. Army is a concrete term.
Army is easy to spell.
11.4. The catcher is the most important man in baseball.
Eddie is a catcher.
11.5. Fire is defined as rapid oxidation.
The salesman who sold me my car was a ball of fire.
11.6. Ever}' fox has a bushy tail.
Field Marshal Rommel was a fox.
11.7. Man is a three-lettered word.
'I’om is a man.
11.8. Man is God’s finest earthly creature.
Man is a species of animal.
11.9. Man is a universal concept.
Every universal concept can be said of many inferiors.
11.10. Man is subject to death.
Some men are American.
11.11. Every square is a polygon.
Tin’s polygon is equilateral.
11.12. Square is inferior to polygon.
Ever}' square is equilateral.
11.13. The outfielders covered the whole outfield well.
Ted Williams was one of the outfielders.
11.14. Some dogs are collies.
The dog is man’s best friend.
11.15. Every one of Santa Claus’s helpers is an elf.
Some of Santa Claus’s helpers are elves.

THE SUPPOSITION OF TERMS

119

12. Evaluate the pairs of terms in the following argumentations from the point
of view of signification and supposition:
12.1. Since all prisoners hate long terms, and since calisthenics is a
long term, all prisoners hate calisthenics.
12.2. Since organism is superior to animal and man is an organism, it
follows that man is superior to animal.
12.3. Since the Senators are an august body, Senator Wiley is an
august body.
12.4. Since universal is concrete, man is concrete, because man is
universal.
12.5. God is inconsequential, because God is immaterial, and what is
immaterial is inconsequential.

CHAPTER

XI

The Categorical Proposition

I. The Difference Between the Categorical and
Compound Proposition

O

The proposition is the complex expression which signifies the object
known in the second operation of the intellect. In its fundamental form
it is composed of a subject-term, a predicate-term, and a copula (the
verb “to be”) which unites or separates these terms. This fundamental
type of proposition has one predicate which is affirmed or denied of
one subject. It is called a simple or categorical proposition. A more
complicated type of proposition can be constructed in which the elements
are propositions rather than terms and in which the copula is a con­
junction rather than the verb “to be.” This type of proposition is spoken
of as compound. Ever}' animal is sentient is a simple or categorical
proposition. Every plant is nonsentient is also a categorical proposition.
Each of them is resolved into elements which are terms, with a copula
which is the present tense of the verb “to be.” Every animal is sentient,
and every plant is nonsentient is a compound proposition immediately
resolved into elements which arc themselves categorical propositions and
a copula which is not the verb “to be” but the conjunction “and.”
Categorical propositions are defined in terms of one subject and one
predicate. In the case of John is happy there is obviously one predicate
and one subject. This is clearly a categorical proposition. 'Ilie same is
true of Mary is happy. John is happy, and Mar}' is happy is as obviously
a compound proposition. But what of John and Mary are happy? If this
means John is happy and Mary is happy, it is a compound proposition,
for happy is effectively said twice of two different subjects. But if instead
the meaning is John and Mary are a happy couple, then the proposition
is categorical, for the predicate is said of the two individuals only in so
far as they arc taken together as one pair. In this case there arc two real
subjects but only one logical subject, and one predicate. Accordingly this
proposition is categorical.
120

121

THE CATEGORICAL PROPOSITION

II. Simply Attributive and Modal Propositions

Not all predicates arc related to their subjects in the same way. Some
arc related necessarily to their subjects, others only contingently, some
impossibly, others possibly. Some categorical propositions express the
connection between subject and predicate without explicitly indicating
the mode of this connection (e.g., Every man is rational), llicse are
called simply attributive propositions. Others explicitly indicate the way
in which the subject and predicate are connected (e.g., Every man is
necessarily rational). These arc spoken of as 'modal propositions. The
expression of the way in which the subject and the predicate are joined
in the modal proposition is called the mode. The rest of the modal
proposition is called the dictum. A simply attributive proposition is true
if it measures up to the factual situation. A modal proposition is true
only_if both its dictum and its mode arc realized. If John is, as a matter
of fact, sitting down, the proposition John is sitting down is true. How­
ever, the proposition John is necessarily sitting down is false. The dictum
of this modal proposition measures up to the facts, but the mode is false
because John might not be sitting down. Propositions which express the
mode of necessity are directly denied by those expressing the mode of
contingency, and vice versa. Those which express impossibility are directly
denied by those expressing possibility, and vice versa. The following
examples should help to clarify the differences and relationships between
the different types of modal propositions. Some of the examples are true
propositions, and some of them, of course, arc false.

MODE OF NECESSITY
John is necessarily sitting down.
It is necessary that every man be
rational.
Ever)' man must be free.

MODE OF CONTINGENCY
John need not be sitting down.
z\ man might not be rational.

MODE OF IMPOSSIBILITY
No man can be unhappy.
It is impossible for John to be
here.
Angels cannot be weighed.

MODE OF POSSIBILITY
A man might be unhappy.
John could be here.

Not every man need be free.

Angels can be weighed.

/

122

the logic of the second operation

III. 'the Division of the Proposition by Way of Quality

The division of the categorical proposition into the affirmative and
the negative is spoken of as a division by way of quality. In an affirmative
proposition the predicate is said of its subject. In a negative proposition
the predicate is denied of its subject. A proposition is affirmative or
negative in quality on the basis of the affirmative or negative character of
its copula. In the affirmative proposition the copula unites the predicate
with the subject (e.g., Some man is happy). In the negative proposition
it separates the predicate from the subject (e.g., Some man is not happy).
Subject-terms and predicate-terms can themselves be either affirmatively or
negatively expressed, but this does not determine the quality of the propo­
sition. The following are all affirmative propositions:
Every man is rational.
Every brute is nonrational.
Some unhappy men arc unmarried.
Some baseball players who are not excellent hitters are of more
value to their team than those who are excellent hitters but
cannot field well.
Every A is B.
Some non-A is B.
Every non-A is non-B.
Some A is non-B.

The following, on the other hand, are all negative propositions:

No horses are nonsentient.
Many Americans are not rich.
Some non-Catholics do not fear a Catholic president.
Some of those below deck when the blast occurred were not in
the group unable to save themselves.
No A is B.
Some A is not non-B.
/Some non-A is not B.
/ No non-A is B.

>V
IV. The Division of the Proposition by Way of Quantity

Categorical propositions arc divided into the universal, the particular,
and the singular on the basis of the extension of their subjects. This is a
division of the proposition according to quantity. Propositions with fully
distributed or universal subjects are said to be universal in quantity.

THE CATEGORICAL PROPOSITION

123

Those with undistributed or particular subjects are particular in quantity.
Those with singular subjects are singular in quantity. A universal proposi­
tion cither affirms or denies its predicate of its subject with this subject
taken in all its extension (e.g., Every' man is rational, All triangles have
angles whose sum is equal to 180 degrees, and No organism is immaterial).
A particular proposition affirms or denies its predicate of an indeterminate
segment of the extension of the subject (e.g., Some triangle is isosceles,
Not all men are American, Several soldiers marched by, and Many terms
are difficult to define). A singular proposition affirms or denies its predicate
of a determinate segment of the extension of the subject (c.g., This pen is
mine, Those statements were false, Mary’s mother is ill, and John is a
student).
One or two additional words remain to be said on the quantity of
the proposition. First of all, the quantity of a proposition depends upon
the extension of its subject. What is to be done with a proposition
whose subject is indefinitely expressed in reference to extension? In such
a case it is necessary somehow to determine the extension of the subject
before the proposition can be quantitatively classified. In the proposition
Triangles arc isosceles it is clear from the nature of triangle that the
subject of the proposition is to be understood neither universally nor
singularly but particularly. In the case of Triangles arc three-sided, how­
ever, the subject should be considered as universal. Hence, Triangles are
isosceles is particular in quantity, and Triangles arc three-sided is uni­
versal. Second, the terms "particular” and "singular” could be misleading.
We must recall that technically "particular” refers to an indeterminate
limitation placed upon the full extension of a concept while "singular”
refers to a determinate limitation placed upon its full extension. The par­
ticular (technically speaking) is used when (colloquially speaking) no
one in particular is referred to. And the singular (speaking with logical
accuracy) might well be plural rather than singular (speaking in the
terms of the grammarian). We have pointed out before that the differ­
ence between the grammatical singular and the grammatical plural is of
little significance in logic. This is especially true in the case of the par­
ticular proposition. From the point of view of logical significance it makes
little difference whether we say "Some man is happy” or "Some men
are happy.” Logic is primarily ordered to science, and what is of scientific
importance in each statement is the indication that human nature is
open to happiness. In a similar fashion the fact that human nature is
open to the absence of happiness is expressed equally well in both "Some
man is not happy” and "Some men arc not happy.” Finally, in the strict
sense, only terms taken with personal supposition are divided into the

124

THE LOGIC OF THE SECOND OPERATION

universal, particular, and singular. Thus, propositions which have subjects
taken with supposition other than personal will not be classified strictly
as to quantity. They will, however, have a logical force similar to either
universal or singular propositions. Propositions with subjects taken with
logical or logico-rcal supposition function like singular propositions, while
those with subjects taken absolutely are virtually universal.

V. (jhe Extension of the Predicate
The extension of the subject of a proposition is ordinarily indicated
by some auxiliar}' term such as “ever}',” “some,” or “this.” However, this
is not the case with the predicate of a proposition. Predicates arc never
introduced by auxilian- terms indicating the extent to which they arc to
be taken in the proposition. Yet, it is necessary to know the extension
both of the subject and of the predicate of any proposition. As a matter
of fact it is as easy to determine the extension of the predicate as it is to
determine that of the subject. First of all, with rare exceptions, predicates
are not singular. Propositions which apparently have singular predicates
are ordinarily not stated in perfect logical form. These should be restated
so that the apparent predicate and subject are interchanged. For example,
One of the most important nations in the free world is the United Slates
in perfect logical form would be stated The United States is one of the
most important nations in the free world. Predicates, then, arc either par­
ticular or universal. They are particular in propositions affirmative in
quality and universal in propositions negative in quality. The reasons for
this arc clear, if we consider the nature of affirmative and negative propo­
sitions. The affirmative proposition signifies the identification of a predi­
cate with a subject. This identification is primarily from the point of view
of comprehension but correlatively, of course, from the point of view of
extension. To say Every' A is B is to assert that the total meaning of
B is realized in a subject which is A. That A is comprehensively more
than B is possible. But if A can be comprehensively richer than B, then
its extension may be less than that of B, for we have seen that concepts
richer in comprehension have less extension. Thus, from the formal point
of view A must be considered less extended than its predicate B and
consequently, from the point of view of extension, it is identified with
only some of B. 'Ibus the predicate of an affirmative proposition is fully
identified with its subject from the point of view of comprehension,
only partially as far as extension is concerned. It must, therefore, be
classified as a particular term. This is clearly the case with Every man is
animal. Since brutes as well as men are animal, man cannot exhaust the

THE CATEGORICAL PROPOSITION

125

extension of animal but can at best be considered as identified with some
of it. This is also the case with Every triangle is a plane figure and
Every' man is biped. But what of Every' man is a rational animal? As a
matter of fact the predicate and subject in this proposition arc coextensive
so that man does exhaust the extension of rational animal. This is true
because of the intelligible content of the predicate, but not because of
the formal structure of the affirmative proposition. The definition of a
species, the specific difference of a species, and a specific property of a
species are, as a matter of fact, coextensive with the species. Whenever
these are predicated of the species the predicate happens to be used
fully as far as extension is concerned. However, this is something acci­
dental to the nature of the affirmative proposition. The formal structure
of the affirmative proposition allows for a coextensive predicate but at
best guarantees only a particular predicate. It remains true, from the point
of view of formal logic, that the predicate of an affirmative proposition
must be classified as particular.
The negative proposition expresses the separation of a predicate from
a subject. From the point of view of comprehension the negative proposi­
tion signifies that the meaning of the predicate is not fully realized in the
subject. Since the meaning of the predicate is the sum of its compre­
hensive notes, this means that the negative proposition is an assertion
that not all the comprehensive notes of a predicate arc found in a subject.
But nothing falls within the extension of a concept unless it possesses
all the comprehensive notes of that concept. Thus the negative proposi­
tion separates its subject completely from the extension of its predicate.
To say No A is B is to assert that not all the comprehension of B is
found in any A and that no A is found within the extension of B. There
are no exceptions possible here even on the part of material or factual
condition. Without exception the predicate of a negative proposition is
universal.
A proposition can be so expressed that it stresses cither the comprehen­
sive or extensive relationship of predicate to subject. Since comprehension
and extension arc correlative it makes little difference which is stressed so
long as there is no intention to suppress the other. To say “Every man is
animal” is to stress the fact that the comprehensive notes of animal are
found in every' man. To say “Every- man is an animal” is to stress the
inclusion of all men within the extension of animal. Both express the same
truth, and either is acceptable. The same is true for “No horse is man”
and “No horse is a man.” The first points to the predicate comprehensively
taken and the second to the predicate extensively taken. Yet each signifies
the same truth, and either is an acceptable expression.

126

THE LOGIC OF THE SECOND OPERATION

VI. A, E, I, and O Propositions

Wc divided propositions from the point of view of quantity into uni­
versal, particular, and singular, and from the point of view of quality
into the affirmative and the negative. Combining these classifications we
find six types of propositions; universal affirmative, universal negative,
particular affirmative, particular negative, singular affirmative, and singular
negative. In order easily to identify these in the future we can assign
symbols to stand for the various types. Traditionally logicians have used
the letters A, E, I, and O to stand for the different propositions. A stands
for the universal affirmative, and I for the particular affirmative (from the
Latin word Affirmo). E stands for the universal negative, and O for the
particular negative (from nEgO). Because singular propositions, as we
shall see, are frequently used with a logical force similar to that of the
universal proposition, but not always and in even' way, singular affirmative
propositions can be spoken of as A1 and singular negatives as E1. The
following propositions illustrate the several types:
A — Every triangle is a plane figure.
z\ll who do not love their country are unworthy of its
protection.
Every non-A is B.
E — No good Christians are Communists.
None of the moral virtues are possessed by brutes.
No non-A is non-B.
I — Some men are American.
Several human faculties are cognitive.
Certain men arc mentally disturbed.
Some A is non-B.
O — Some men are not mathematicians.
Not every Communist is Russian.
All that glitters is not gold.
Some A is not B.
A1 — This nation is a democracy.
These triangles are equilateral.
John is president of his class.
This A is non-B.
E1 — This cow is not healthy.
Those men arc not students.
John is not wearing a hat.
That A is not B.

THE CATEGORICAL PROPOSITION

127

These examples do not exhaust the possible ways of expressing the
various types of categorical propositions grammatically. What is of
importance for the logic student is the logical fact of universal affirma­
tion, of universal negation, etc. These may be expressed in many ways
grammatically, and it is not the business of a logic text to list all the
grammatical possibilities. However, we might note that the universal
negative proposition is never rendered grammatically by the explicit use
of “every” with the subject and “not” with the copula. “Not” with
“every” not only diminishes the force of the copula by making for negative
quality, but it also diminishes the extension of the subject and renders
it particular. Not every A is B and Every' A is not B are equivalent to
Some A is not B. The universal negative expression is No A is B.
Exercise XI

1. Define: categorical (simple) proposition, simply attributive proposition,
modal proposition, mode, dictum, affirmative proposition, negative propo­
sition, universal proposition, particular proposition, singular proposition.
2. Contrast categorical and compound propositions in terms of their respec­
tive elements.
3. Distinguish between the dictum and the mode of a modal proposition
and explain what is demanded of each for the truth of the modal
proposition.
4. Supply original examples for each type of modal proposition. Then di­
rectly deny each with the appropriate modal proposition.
5. Explain how a proposition with either a negative subject or negative
predicate can be an affirmative proposition.
6. How do universal propositions differ from particular and singular proposi­
tions, and how do these latter differ from one another?
7. What does it mean to say predicates of affirmative propositions are used
fully in reference to comprehension and partially in reference to extension?
Explain why this is so in terms of the formal structure of the affirmative
proposition.
8. What does it mean to say predicates of negative propositions are used
partially in reference to comprehension and fully in reference to extension?
Explain why this is so in tenus of the formal structure of the negative
proposition.

9. What is the extension of the subject and of the predicate of each A
proposition? Of each E proposition? Of each I proposition? Of each O
proposition? Of each AJ proposition? Of each El proposition?
10. Indicate the extension of the subject in the parentheses following the
subject, the extension of the predicate in the parentheses following the

128

THE LOGIC OF THE SECOND OPERATION

predicate, and the classification of the proposition as a whole in the
parentheses following the proposition. Use the symbols U, P, S,\A, E,
I, OJA1, and E1.
£
10.1. No A (<a ) is non-B ( ).(•»*-)
10.2. Some triangles ( ) are isosceles ( i> ). (/ )
four-legged animals (u ) are human (« ). (£
103. No
.................................................................................
.. ”)
10.4. Not ever}' nonessential predicable ( P ) is an accident ( * ). ( )
10.5. No athletes who arc under six feet tall ( ) arc well suited to
the position of center on the modern basketball team ('-*•). (£ )
10.6. Some men who arc not happy in their work ( f ) arc unable to
do it well ( p ). (I )
10.7. No disloyal Americans (m ) are worthy of their privilege to vote
(“ )*
10.8. These numbers (4 ) arc divisible by two (P ). (/j )
10.9. All that glitters (p ) is not gold (<** ). ( O)
10.10. The Civil War ( a ) was fought in the nineteenth century
(J )• (*)
10.11. Flies ( p ) fly ( f ). ( I )
-^10.12. Ever}' non-A ( c
• p)
z10.B. Some A (,. ) is
• (¿»
10.14. This A ( u) is
10.15. A ( \ ) is B (/> , , A
10.16. Not any A (A ) is B (u ). (£)
11. Suggest examples of propositions with subjects taken with other than
personal supposition and discuss the “quantity” of each.

CHAPTER

XII

The Compound Proposition

I. The Nature of the Compound Proposition

A compound proposition is a single proposition composed of several
component propositions joined by a copula which is a conjunction rather
than a verb. The elements into which a categorical proposition are im­
mediately resolved are terms. Those into which a compound proposition
is immediately resolved are themselves propositions. In the least complex
compound propositions the elements are categorical propositions. In
more complex compound propositions the elements might themselves be
compound propositions. Thus The Communist world is militarily strong
is a categorical proposition immediately resolved into the subject-term
Communist world and predicate-term militarily strong. The Communist
world is militarily strong, and the free world is relatively weak is a com­
pound proposition immediately resolved into the two categorical proposi­
tions, The Communist world is militarily strong and The free world is
relatively weak. If the Communist world is militarily strong, and the free
world is relatively weak, then the free world is in danger of annihilation is
a more complex compound proposition immediately resolved into a com­
pound proposition and a categorical proposition. The compound com­
ponent is The Communist world is militarily strong and the free world
is relatively weak, while the categorical component is The free world
is in danger of annihilation. Note that the copula in the categorical
example is the present tense of the verb “to be.” In the two compound
examples the copula is respectively the conjunction “and” and the
conjunction “if . . . then.”
II. Conjunctive and Hypothetical Propositions

The elements of the compound proposition are propositions, ultimately
categorical propositions. Hence, the elements of the compound proposition
are logically divided after the fashion of the divisions of the categorical
proposition treated in the preceding chapter. Compound propositions
129

130

THE LOGIC OF THE SECOND OPERATION

themselves, however, have divisions proper to themselves. First of all,
some compound propositions express a definite commitment to the truth­
status of each of their elements. The proposition The Communist world
is militarily strong, and the free world is relatively weak asserts the truth
dctcrminatcly of both of its categorical components. On the other hand,
the proposition If the Communist world is militarily strong, then the
free world is in danger does not make a definite commitment to the
truth of cither of its elements. It docs, however, involve a commitment
to the nexus or link between these component parts. This commitment
requires one to admit that if the first is true the other must also be true.
The link or nexus between these two parts docs not depend upon their
truth, but it does exclude the possibility of the first clement being true
while the second is false. Those compound propositions which involve
a definite commitment to the truth-status of each element are conjunc­
tive propositions, whereas those which do not involve this commitment
are hypothetical propositions.1
III. The Copulative Proposition

The most elementary type of conjunctive proposition is the copulative
proposition. The copulative proposition simply asserts the truth of several
propositions joined by the conjunction “and” or its equivalent. The
copulative proposition expresses nothing more than the simultaneous
truth of several co-ordinate propositions. As such, a copulative proposition
is true when each of its elements is true. It is false if any one of its
elements is false, regardless of the truth value of the others.
Propositions which arc so constructed that they clearly involve several
co-ordinate propositions joined by “and” or its equivalent are said to
be openly copulative. This is the case with Every triangle has three sides,
and some triangles have equal sides. Propositions which do not clearly
and immediately reveal their commitment to the truth of several co­
ordinate propositions are said to be occultly copulative. Because occultly
copulative propositions can be expanded and expressed as openly com­
pound they arc called cxponiblc propositions. Only the good die young
is an occultly copulative proposition which can be expanded to The
good die young, and none who are not good die young. An cxponible
1 Many logicians call all compound propositions hypothetical. Others call only
conditional propositions hypothetical. Our use of the term “hypothetical" is more
restricted than the former and less restricted than the latter. Our use of the term
"conjunctive” is also different from the sense given it in many logic texts, where it is
used to signify the type oí proposition called “disjunctive” in this book. The term
"alternative” is used here to refer to propositions spoken of in most other texts as
disjunctive.

THE COMPOUND PROPOSITION

131

proposition is true, of course, only if each of the propositions which
appear in its expanded form is true. It is false if any one of them is false,
regardless of the truth of the others.
The preceding example of an occultly copulative proposition is called
an exclusive proposition because it identifies its predicate with its subject
to the exclusion of any other subject. A second type of occultly copulative
proposition joins a predicate to a subject which is understood to involve
exceptions. This is the exceptive proposition, A third type of occultly
copulative proposition is called the reduplicative proposition. The re­
duplicative proposition expresses the precise aspect of the subject in
virtue of which the predicate is cither affirmed or denied of it. The
following arc examples of cxponible propositions, together with the
openly copulative propositions to which they might be expanded and in
terms of which they must be understood for the purposes of verification:
EXCLUSIVE
God alone has the power
to create.

Only lovers can ade­
quately define love.

EXCEPTIVE
Everyone except Truman
expected Dewey to
win the presidency in
1948.
None of the American
League teams with
the exception of the
Yankees has been a
consistent pennant
winner.
REDUPLICATIVE
Every priest as a man
suffers temptation.

As a citizen you are en­
titled to vote.

God has the pow'er to create, and
nothing other than God has the power
to create.
Lovers can adequately define love, and
no one who is not a lover can ade­
quately define love.

All men other than Truman expected
Dewey to win the presidency in 1948,
and Truman did not expect Dewey to
win the presidency in 1948.
No American League team other than
the Yankees has been a consistent
pennant winner, and the Yankees are
an American League team, and the
Yankees have been a consistent pen­
nant winner.
Every priest is a man, and all men suffer
temptation, and every priest suffers
temptation precisely in virtue of being
a man.
You arc a citizen, and every citizen is
entitled to vote, and you are entitled
to vote precisely as a citizen.

132

THE LOGIC OF THE SECOND OPERATION

IV. Adversative and Causal Propositions

Copulative propositions assert the simultaneous truth of several co­
ordinate components. Other conjunctive propositions express not only
the simultaneous truth of their component propositions but the sub­
ordination of one component proposition to another. In an adversative
proposition the subordinate component is shown to be somehow opposed
to the main component. This opposition is verbally expressed by words
such as “although,” “despite,” and “whereas.” 7'ruman was elected in
1948, despite the fact that no pollster had predicted his victory' is an
example of an adversative proposition. It is true only if Truman was
elected in 1948 and if no pollster had predicted his victory and if there
is some suggestion of opposition between the latter truth and the former.
Other examples of adversative propositions include:
Although contraries seem like pure and simple opposites, they
might be false together.
Ilie weather turned out to be beautiful, despite an adverse
weather report in the morning.
John won the race, even though he lost his shoe twenty yards
from the finish.
A second type of conjunctive proposition which expresses the sub­
ordination of one element to another is the causal proposition. In the
causal proposition the subordinate component gives a reason for the truth
expressed in the main component. The subordination of the causal
component to the main component is expressed by words such as
“because” and “since.” Human beings are morally responsible agents,
because they are possessed of free will is a causal proposition. It is true
only if each of its elements is true and if the subordinate element is
truly related to the other after the fashion of cause to effect. Additional
examples of causal propositions arc:
Logical relations are not able to be categorized, because they are
beings-of-thc-reason.
This is a compound proposition, for it has a causal clause added
to a main clause.
No one understood the visitor, since he spoke only in French.
Adversative and causal propositions are conjunctive. Like copulative
propositions they assert determinately the truth of each of their elements.
But unlike copulative propositions they also assert the subordination of
one element to another. Thus more is required to verify adversative and
causal propositions than copulative. For the copulative proposition the
truth of each element suffices. For the adversative and causal propositions

THE COMPOUND PROPOSITION

IB

the truth of each component phis the realization of the adversative or
the causal nexus is necessary. 'Ilius, adversative and causal propositions
can be thought of as adding something to simply copulative propositions.
Each might be generally described as a copulative proposition phis.
V. The Conditional Proposition

As noted above, hypothetical propositions, unlike conjunctive proposi­
tions, do not assert determinately the truth of each of their elements
but only express a commitment to a nexus or link between them. The
example we used of a hypothetical proposition was If the Communist
world is militarily strong, the free world is in danger. Tins is a conditional
proposition. The conditional is not the only kind of hypothetical proposi­
tion. However, it is the most fundamental, and we shall treat of it
before the others. A simple conditional proposition is an assertion that
two propositions are so related that the truth of one of them (the ante­
cedent) necessitates the tnith of the other (the consequent). The com­
mitment of the mind in the simple conditional proposition is neither to
the truth of the antecedent nor to the tnith of the consequent. The
simple conditional commitment bears only on the nexus between the
components, namely, that the antecedent cannot be true without the
consequent being true as well. Thus the verification of the simple condi­
tional does not depend upon the truth of its elements. If the Communist
world is militarily strong, the free world is in danger is true not because
the Communist world is militarily strong (or is not militarily strong) nor
because the free world is in danger (or is not in danger). This proposition
is true because its antecedent and consequent are so related that if the
antecedent were true the consequent would necessarily be true. The
following examples illustrate the simple conditional:
If Marxist philosophy is true, the free world is doomed. (This
is verified because Marxist philosophy teaches the inevitable
universal success of Communism.)
If contingent beings exist, a necessary being exists. (This is verified
because only a necessary being can explain the existence of any
being whatsoever.)
In baseball, if a right hander is pitching, the odds favor a left­
handed batter. (This is verified because a ball thrown by a
right-handed pitcher curves in to a left-handed batter and is
thus better able to be seen by him than by a right-handed
batter. )

134

THE LOGIC OF THE SECOND OPERATION

If it is eleven o’clock, the mail has been delivered. (This is
verified as long as the habitual practice of the mailman is such
that he never delivers the mail later than eleven o’clock.)

These examples illustrate two things. First of all, they show that the
commitment involved in a conditional proposition bears essentially upon
the connection between antecedent and consequent. Second, they illus­
trate a variety of ways in which this connection can be guaranteed by the
reality of things. The verification of the simple conditional depends not
on the truth of its elements but entirely on the implicative nexus between
them. This nexus exists as long as the component propositions are so
related that the antecedent cannot be true without involving the truth of
the consequent as well. The scientific value of the conditional proposition
depends on the degree of necessity in the foundation for the nexus.
Thus far we have spoken only of simple conditional propositions, which
assert that an antecedent cannot be true without a consequent being true.
There is a stricter type of conditional proposition, the reciprocal condi­
tional proposition. The latter asserts as much as the former and, in
addition, affirms-that the consequent cannot be true without the ante­
cedent being true as well. In a simple conditional proposition we assert
that a given proposition implies a second proposition, meaning that the
first cannot be true without the second being true also. In a reciprocal
conditional we assert that a given proposition and only this implies
a second proposition. This means not only that the first cannot be true
without the second being true but also that the second cannot be true
without the first being true as well. Here too the truth of the proposition
as a whole does not depend upon the truth of its elements but solely on
the nexus of implication between them. The difference between the
simple and reciprocal conditional propositions is simply that in the latter
there is a stricter nexus or link between antecedent and consequent. A
simple conditional is true as long as the truth of its antecedent implies
the truth of its consequent, although its consequent can be true without
involving the truth of its antecedent. A reciprocal conditional is not
true under these circumstances. It is true only if both elements recipro­
cally imply one another. The following examples are illustrations of true
reciprocal conditional propositions:
If and only if man can know the intelligible, the human soul is
incorporeal.
If and only if God exists as a creator, contingent beings exist.
If and only if Oregon is north of California, San Francisco is
south of Portland.

THE COMPOUND PROPOSITION

135

If and only if each player on a team does his job well, the team
functions perfectly.
Clearly the existential situation which would guarantee the truth of a
reciprocal conditional would verify a similar simple conditional, but not
vice versa. Hence, if I can say If and only if a man is virtuous, he is truly
happy, I can surely propose If a man is virtuous, he is truly happy. How­
ever, I cannot assert If and only if a man runs, he moves just because If a
man runs, he moves is true. The reciprocal commitment includes the
simple commitment and more. Hence, we can move from the reciprocal
to the simple, but not necessarily from the simple to the reciprocal.
VI. The Alternative Proposition

The conditional proposition is only one kind of hypothetical proposi­
tion. r\ second type is the alternative, proposition. The object of the
commitment proper to the alternative proposition is a nexus or link
between component parts which requires that at least one of these com­
ponents lx; true. By this we do not mean simply that the alternative
proposition asserts that one of its elements is, perhaps quite independently
of the others, necessarily true. Rather we mean that the several elements
arc so related that they cannot all be false together, so that one, at
least, must be true.
The copula in the alternative proposition is signified by the conjunction
"or.” Generally speaking, this “or” indicates alternate propositions one
of which, in the light of the others, must be true. This is all that the
copula signifies in the less strict type of alternative proposition, called
the inclusive alternative proposition, so called because it includes the
possibility that more than one of the elements is true. The elements are
alternates only in the sense that, given the falsity of the others, one
must be true. The proposition is true only if its elements are so related
that, in the light of this relationship, at least one must be true. Here,
as with the conditional, the truth of the alternative proposition does not
depend on the truth of its elements taken in themselves but upon the
truth of the nexus between them. However in this case the nexus is not
one directly of implication, but rather of alternation. The following
examples illustrate true inclusive alternative propositions:

Either juvenile delinquency will be curbed or our streets will be
safe for no one.
Either the free world prepares to defend herself or the Com­
munist world will conquer her.

136

THE LOGIC OF THE SECOND OPERATION

A generic predicable is either a proximate genus, a remote genus,
or an ultimate genus.
Either a necessary being exists or no contingent beings exist.
These propositions arc true inclusive alternative propositions not simply
because one or more of the elements in each happens to be true but
because at least one must be true in the light of the others. These
propositions illustrate several points concerning alternative propositions.
First, they show that alternatives can involve two or more elements. We
shall be concerned chiefly with two-membered hypothetical, but we
should note that alternative propositions admit of any number of
elements and are logically manageable no matter how complex they
may be. Second, like the examples used to illustrate conditional proposi­
tions, these examples show that there is a variety of ways in which the
nexus between the elements of the proposition is realized. Here we must
also remember that these arc examples of inclusive alternative propositions.
Tins means that they assert merely that one clement at least must be true;
the others need not be false but may also be true. If this is not crystal­
cicar from the context then the propositions ought to signify the situation
explicitly. Expressed most clearly, then, these examples would be rendered:
Either juvenile delinquency will be curbed or our streets will be safe for
no one —or perhaps both, etc.
The more strict form of alternative proposition is the exclusive alter­
native. Like the inclusive alternative, the exclusive alternative asserts that
at least one of several alternate propositions must be true. Unlike the
inclusive, however, it excludes the possibility that more than one of the
alternatives can be true. It positively asserts that only one can be true.
'Hie elements are alternates in the strictest sense, for given the falsity of
the others one must be true, and given the truth of any one the others
must be false. Once again, of course, the truth of the proposition as a
whole does not depend upon the truth value of its elements taken in
themselves, but rather upon the nexus between them. This demands that
one and only one be true regardless of which one that is. The following
examples illustrate exclusive alternative propositions:

Either some being is finitely real or no being is able to be
categorized.
Either all men arc perfect or some men are not what they should
be.
Either no men are mentally unbalanced, or all men are mentally
unbalanced, or some men arc mentally, unbalanced while others
arc not.

THE COMPOUND PROPOSITION

137

Either the South won the Civil War, or the North won the Civil
War, or neither side won the Civil War.
These examples illustrate the same points illustrated by the examples
offered for the inclusive alternative. But these examples also show that
only one of the alternatives can be true for exclusive alternative proposi­
tions. Once again, if the context is not sufficiently clear to indicate they
arc to be taken as exclusive alternatives and not simply as inclusives,
this must be pointed out explicitly. Then the propositions would be
given as follows: Either some being is finitely real or no being is able to
be categorized — but not both, etc.
VII. The Disjunctive Proposition

The disjunctive proposition is the last of the hypothetical for us to
consider. The disjunctive proposition asserts, that its component proposi­
tions are so related that at least one of them must l>c false. The elements
of the disjunctive proposition are propositions whose mutual relationship
is signified by a copula which combines the negation “not" with the
conjunction “and.” TTîc disjunctive proposition thus appears to be merely
a denial of an original copulative proposition. 'litis, however, is misleading,
for the negation of a copulative proposition asserts no more than the
falsity’ of at least one of the components of the original copulative. The
disjunctive commitment, on the other hand, bears directly upon the
nexus between its component propositions and asserts that these com­
ponents are so related that in virtue of this relationship at least one of
them must be false. Though disjunctives with more than two elements
arc possible, ordinarily disjunctive propositions are of the two-membered
variety. The following examples illustrate the disjunctive proposition:
It can not be true that all predicates are superiors and that some
predicates arc singular.
Not all Americans are loyal so long as some arc Communist.
A predicate and its contradictory cannot both be affirmed of the
same subject.
It is impossible for a community to be well regulated and its
leaders unintelligent.
VIII. Reduction of Alternative and Disjunctive Propositions
to the Conditional

The conditional proposition is the most fundamental type of hvpothetical proposition. Any alternative or disjunctive proposition can be

138

THE LOGIC OF THE SECOND OPERATION

reduced to a corresponding conditional equivalent in meaning and basic
to it. Tire hallmark of the hypothetical proposition in general is the
logical nexus between its component propositions, in virtue of which the
truth value of one has a bearing on the truth values of the others. Even
hypothetical proposition, then, involves, at least implicitly, the relation of
antecedent to consequent. This relation is explicitly and directly expressed
in the conditional proposition, which is by definition a direct commitment
to a sequential connection between antecedent and consequent. This
relation is implied in both alternative and disjunctive propositions. Thus
all three — conditionals, alternatives, and the disjunctives — are hypo­
thetical propositions and, as wc shall see, all can be used as premises in
hypothetical argumentation. Yet conditionals remain the basic type of
hypothetical to which the others can be reduced.
The inclusive alternative and the disjunctive proposition are reduced
to simple conditionals. The stricter alternative is reduced to the stricter
conditional. Thus the exclusive alternative is reduced to the reciprocal
conditional. The following examples illustrate the reduction of alternative
and disjunctive propositions to equivalent conditional propositions:
INCLUSIVE ALTERNATIVE
Either juvenile delinquency will
be curbed or our streets will
be safe for no one.
Either the free world prepares
to defend herself or the Com­
munist world will conquer her.

SIMPLE CONDITIONAL
If juvenile delinquency is not
curbed, our streets will be safe
for no one.
If the free world does not pre­
pare to defend herself, the
Communist world will conquer
her.

EXCLUSIVE ALTERNATIVE
Either some being is finitely real
or no being is able to be cate­
gorized.
Either all men are perfect or
some men are not what they
should be.

RECIPROCAL CONDITIONAL
If and only if some being is
finitely real, some being is able
to be categorized.
If and only if all men are per­
fect, all men are what they
should be.

DISJUNCTIVE
It can not be true that all predi­
cates are superiors and that
some predicates are singular.
Not all Americans are loyal
while some are Communist.

SIMPLE CONDITIONAL
If all predicates are superiors, no
predicates are singular.
If all Americans are loyal, no
Americans are Communist.

THE COMPOUND PROPOSITION

139

IX. Categorical Propositions Expressed as Hypothetical

“If anything is a man, that thing is rational” apparently signifies a
simple conditional assertion. The same thing is true for “If anything is a
plant, it is not sentient.” However, these expressions can be taken as
equivalent to “Ever}' man is rational” and “No plant is sentient,” so long
as each of these categorical propositions is understood to intend only
possible existence. It makes little practical difference, at least on the level
of the second operation of the intellect, whether the expressions arc taken
as signifying the hypothetical or the universal categorical proposition, since
each reduces in meaning to the other. However, there is a difficulty which
arises on the level of the third operation if these propositions are
understood as hypothetical rather than categorical, since the rules for a
valid hypothetical syllogism do not apply without modification to a
syllogism based on one of these propositions taken as hypothetical. On
the other hand, if these propositions arc understood as categorical they
offer no difficulties as premises in a valid categorical syllogism. Hence it
seems preferable to take these propositions as categorical rather than hypo­
thetical, especially when they function as premises in argumentation.
What wc have said here of some apparently simple conditional expres­
sions applies as well to certain inclusive alternative and disjunctive
expressions. “Even' man is rational” is equivalent not only to “If any­
thing is a man, that thing is rational,” but also to “Either something is
not a man or it is rational” and “Nothing is both a man and not
rational.” And “No plant is sentient” is equivalent not only to “If
anything is a plant it is not sentient,” but also to “Either a thing is
not a plant or it is not sentient” and “Nothing is both a plant and
sentient.” The apparently reciprocal conditional expression “If and only
if a being is free, it is a moral agent” and the exclusive alternative expres­
sion “Either a being is free or it is not a moral agent” are both equivalent
to a copulative proposition composed of two universal categoricals with
inverted extremes, namely, “Even' free being is a moral agent, and every
moral agent is a free being.”
X. The Symbolic Representation of Compound Propositions

The constructs of formal logic, especially compound propositions, lend
themselves readily to symbolic expression. As long as they represent the
logical situation without distortion, symbols can frequently facilitate the
discussion of logical procedures. At this time we shall introduce the use
of symbols tor compound propositions. However, in order not to compli­

140

THE LOGIC OF THE SECOND OPERATION

cate things too much at this point, we shall limit ourselves somewhat
by symbolizing only the openly copulative proposition and the several
hypothetical propositions we have considered, namely, the simple con­
ditional, the reciprocal conditional, the inclusive alternative, the exclusive
alternative, and the disjunctive proposition. We shall also limit ourselves
for the most part to a consideration of two-membered compound proposi­
tions, though we have seen that multimembered copulatives, alternatives,
and even disjunctives are possible.
We have already used capital letters to stand for the subjects and
predicates of categorical propositions (e.g., Every A is B). We shall use
small letters now to stand for the elements of compound propositions.
Here, of course, the letters will stand for propositions and not for terms,
for the component parts of compound propositions are themselves
propositions. We shall use a slightly elongated dash (—) as the sign
of negation or contradiction, a dot (•) for the conjunction “and,” a
wedge (\Z) to stand for the copula in the inclusive alternative, an
inverted wedge (Z\) for the copula of the exclusive alternative, and an
arrow (-> ) to signify simple conditional sequence, with a two-headed
arrow (<->) to signify reciprocal conditional sequence. Where punctuation
is necessary, especially to insure the correct application of the sign of
negation, parentheses and brackets will be used. Finally, an identity in
meaning between two propositions will be indicated by the sign of
equivalence (=). Thus, the copulative proposition The Braves won the
National League pennant in 1958, and the Yankees won the American
League pennant that same year can be symbolized simply as a ■ b. The
simple conditional proposition If contingent beings exist, a necessary being
exists becomes symbolically a-» b. The reciprocal conditional If and only
if a man can know the intelligible, the human soul is incorporeal be­
comes a
b. Either juvenile delinquency will be curbed or our streets
will be safe for no one can be expressed a\/b because it is an inclusive
alternative, while the exclusive alternative Either some being is finitely
real or no being is able to be categorized is symbolically represented
a/\b. This leaves only the disjunctive to be symbolized. Here, how­
ever, there is some difficulty. Were the disjunctive simply the negation of
an original copulative we could symbolize it — (a • b). But this, as we have
seen, is not the case. However, we can symbolize the disjunctive by
modify ing the symbolic expression of the negation of the copulative. We
do this by adding an “n” to signify the kind of necessity involved in a
logically disjunctive proposition. —(a • b) is a copulative proposition which
simply denies the copulative a • b. But n — (a • b) is a disjunctive proposi­
tion which denies a • b precisely because it asserts the impossibility of a • b.

141

THE COMPOUND PROPOSITION

XI. The Truth-Functional Proposition and Its Truth Table

A compound proposition is said to be truth-functional when its truth as
a whole depends solely upon the truth values of its component parts.
This is the case with the copulative proposition a • h and with its
contradiction — (a-b). The various possible combinations of truth
and falsity for the elements of a truth-functional proposition can be
tabulated along with the corresponding truth value of the proposition
itself in a schematic arrangement called a truth table. The truth table
for the copulative proposition a-b and its contradiction —(a-b) is as
follows:
—(a-b)

a

b

a-b

T

T

T

F

T

F

F

T

F

T

F

T

F

F

F

T

Conditional, alternative, and disjunctive propositions express com­
mitments to a sequential nexus between their elements. Hence they
cannot be verified simply in terms of the truth values of their component
propositions. Thus they arc not truth-functional, and they’ cannot be
expressed on a truth table manifesting all the conditions under which
they will be detcrminatcly true or false. No combination of true or false
premises guarantees the truth of a ->b, a<->b, a\/b, a/\b, or n — (a • b).
The only purpose of a truth table for these would be to indicate the
situations in which they could not possibly be true. Although no
combination of true and false components will guarantee the truth of
a-»b, nevertheless, if a is true and b is false, a->b could not be true.
Similarly a/\b must be false if both a and b arc true. The same is
true for the other hypotheticals. Though no combination of truth values
for their components guarantees their truth, some combinations do neccs?
sítate falsity. TTiis is so because these combinations rule out the possibility
that the sequential nexus which the proposition asserts between its
components exists. Hence, truth tables cannot be used to verify or confirm
hypotheticals, but they can be used to show when hypotheticals cannot
be verified. With this understanding of copulative and hypothetical
propositions, the following truth table expresses all the conditions for the

142

THE LOGIC OF THE SECOND OPERATION

verification and disconfirmation of copulative propositions as well as those
which can be used to disconfirm impossible hypothetical propositions.’
a

b

a-b

a->b

a<->b

a\/b

a/\b

n — (o-b)

T

T

T

?

?

?

T

F

F

F

F

?

F
?

?

F

T

F

?

F

?

?

?

F

F

F

?

?

F

F

?

F

’Some logicians (both in the so-called Aristotelian tradition and in the school
of mathematical logic) understand the verbal expressions of the alternative and dis­
junctive propositions as signs of a truth-functional proposition. They understand the
inclusive alternative simply as an assertion that at least one of several component
propositions is true (note: that at least one is true, not that, in the light of the
others, one must be true). In similar fashion the exclusive alternative is understood
simply as an assertion that at least one, but only one, of several component propositions
is true. The disjunctive is taken as an assertion that at least one of several component
propositions is false. Some logicians (ordinarily only those in the school of mathe­
matical logic) take even the conditional expression as the sign of a truth-functional
proposition. They understand the simple conditional simply as an assertion that it is
not the case that the antecedent is true and the consequent is false. The reciprocal
conditional is understood as an assertion that it is not the case that cither component
is true while the other is false. To speak of alternative, disjunctive, and conditional
propositions in this sense is to use these terms equivocally. In recognition of this point,
at least in reference to the conditionals, the type of implication understood for a
conditional proposition taken as a truth-functional proposition is usually given a dif­
ferent name than the type of implication understood for a conditional taken as a
strict hypothetical proposition. Truth-functional implication is called material implica­
tion, and strict implication is called formal implication. Actually truth-functional alterna­
tive, disjunctive, and conditional propositions arc rcductivcly of the copulative type
(each one equivalently the contradiction of a more or less complex copulative proposi­
tion). Suppose we symbolize the truth-functional alternatives somewhat differently
than the alternatives taken strictly. Adding an "f" for "factual," the truth-functional
inclusive alternative can be symbolized a\¿/b. The truth-functional exclusive alterna­
tive will be symbolized as a/\b. a\'/b ■—(—«•—b), and a/\b a—(a • b) •
—(—a
b). Adding an "m” for “material,” the truth-functional simple conditional
can be symbolized as a
b; and the truth-functional reciprocal conditional will be
expressed as a*-^>b. a-^b«=—(a-—b), and a*^->b«—(a—b) •—(—a-b).
Finally, the truth-functional disjunctive is clearly nothing but the contradiction of an
original copulative proposition, e.g., —(a*b). Truth-functional conditionals, alterna­
tives and disjunctives, precisely as the copulatives, lend themselves completely to a
truth table for purposes of verification and disconfirmation. The truth table for each of
these is as follows:

143

THE COMPOUND PROPOSITION

XII. A Schematic Résumé

In the past two chapters we have divided the proposition into its types
through the use of logical division, codivision, and subdivision. Perhaps
the following schematic outline will help to summarize the work of these
two chapters:
Simply attributive
Categorical
(Simple)

<

kModal

(Universal

Affirmative
<

s Particular

(.Negative

(Singular

Propositions

Openly

Copulative

Conjunctive*
Copulative plus<

[Exclusive
.Occultly
¿Exceptive
Adversative [Reduplicative

Causal
.Compound <

Conditional iolTklmcal
I Reciprocal

Hypothetical < Alternative

Inclusive
Exclusive

.Disjunctive
Each of the following propositions would be true if it were taken truth-functionally,
but none of them is true if taken strictly as hypothetical: Inclusive alternative —
Either Washington is the capital of the U.S.A, or Milwaukee is the home of the
Yankees. Either Marquette University is in Wisconsin or the Civil War was fought
in the seventeenth century. Either seven is an odd number or Communism is a threat
to world peace. Exclusive alternative — Either Washington is the capital of the U.S.A.
or Milwaukee is the home of the Yankees — but not both. Either seven is an odd
number or the Civil War was fought in the nineteenth century — but not both.
Abraham Lincoln was either the sixteenth U. S. president or the first man to swim
the English Channel — but not both. Disjunctive — Washington is not both the
capital of the U.S.A. and the home of the Braves. It is not true that seven is an odd
number and that the Civil War was fought in the seventeenth century. St. Louis
is not both south of New Orleans and a former site of the World’s Fair. Simple con­
ditional — If Washington is the capita] of the U.S.A., then Milwaukee is the home of
the Braves. If Washington is not the capital of the U.S.A., then Milwaukee is the
home of the Braves. If Washington is not the capital of the U.S.A., then Milwaukee
is not the home of the Braves. Reciprocal conditional— If and only if Washington is
the capital of the U.S.A., then Milwaukee is the home of the Braves. If and only if
Washington is not the capital of the U.S.A., then Milwaukee is not the home of the
Braves. If and only if no man is rational, then every whole number is odd.

144

THE LOGIC OF THE SECOND OPERATION

Exercise XII

1. Define: compound proposition, conjunctive proposition, copulative propo­
sition, openly copulative proposition, occultly copulative (exponible)
proposition, exclusive proposition, exceptive proposition, reduplicative pro­
position, adversative proposition, causal proposition, hypothetical proposi­
tion, simple conditional proposition, reciprocal conditional proposition,
exclusive alternative proposition, inclusive alternative proposition, disjunc­
tive proposition, truth-functional proposition, truth table.
2. What is meant by “sequential nexus,” and how precisely does this serve
as a principle of differentiation between types of compound proposition?

3. In what way are causal and adversative propositions like hypothetical
propositions, and why, despite this, are they conjunctive rather than
hypothetical?

4. What is the meaning of “antecedent” and “consequent” as used in refer­
ence to the conditional proposition?
5. Explain the primacy of the conditional among the hypothetical, and
illustrate the reduction of alternative and disjunctive propositions to
conditional propositions.
6. Supply examples of each type of proposition so far considered, and evaluate
each example as true or false with an explanation for this evaluation.
7. Classify each of the following propositions and, where the intelligible
content is sufficiently familiar, indicate whether or not it is true. Give a
determinate explanation for the falsity of those which are not true. Make
a note of those which can be taken either as hypothetical or categorical.
7.1. Only the good die young.
7.2. A man who is afraid need not be a coward.
73. Either logic is necessary for science or a course in logic is a waste
of time.
7.4. No animal except man has ever given evidence of moral respon­
sibility.
7.5. “In Flanders Fields the poppies blow
Between the crosses, row on row,
That mark our place, and in the sky
The larks, still bravely singing, fly,
Scarce heard amid the guns below.”
(Taken from “In Flanders Fields” by John McCrae.)
7.6. Creatures are contingent beings, while the creator is absolutely
necessary.
7.7. Only men are rational.
7.8. A Catholic politician cannot be both a good Catholic and a good
politician.
7.9. The man who has not known love is an unfulfilled man.

THE COMPOUND PROPOSITION

145

7.10. So long as a number is odd it cannot be divided by two without
a remainder.
7.11. History cannot be counted on as an infallible index of the future.
7.12. Man is a corruptible being in so far as he is corporeal.
7.13. Logic is an art, since it is ordered to a work to be produced.
7.14. As long as, and only as long as, men have respect for God, the
human race will be capable of a peaceful existence.
7.15. Not all negative propositions have universal predicates.
7.16. Cheating is immoral, for it is at least a violation against the virtue
of truth-telling.
7.17. No nation can establish a manned station on the moon.
7.18. An argumentation is either valid or its conclusion docs not follow
from its premises.
7.19. A thing cannot be and not-be.
7.20. If and only if man is not free, the human race is not responsible
for the chaotic condition of the world.
7.21. Logic is an art and not a science.
7.22. The Braves won the National League pennant in 1958 despite the
fact that the Yankees won the American League pennant that same
year.
7.23. If a man is moving from place to place, he must be walking.
7.24. Haney Haddix is the only man ever to pitch twelve perfect innings
in one major league ball game.
7.25. Either the Communists were very sharp or the Western leaders very
naive in the policy-making conferences held at the conclusion of
World War II.
7.26. Nothing can be both a violation of human nature and something
morally good for any man.
7.27. Either Communism will be ultimately unsuccessful or the fret­
world will be destroyed.
7.28. All adult American citizens with the exception of the citizens of
the District of Columbia are eligible to vote for the office of presi­
dent of thç United States.
7.29. Either golf is a healthy sport or lots of Americans are wasting their
time.
7.30. A false modal proposition has a false mode or a false dictum.
7.31. Man is able to reproduce himself precisely in so far as he is an
animal.
7.32. Either sound travels in waves or in particles, but not both.
7.33. Since every triangle is three-sided, a scalene triangle has no two
sides equal to each other.
7.34. Causal propositions involve, as conjunctive, more than a commit­
ment to a nexus between elements.

146

THE LOGIC OF THE SECOND OPERATION

735. If and only if a subject of a proposition is a substance can its
predicate be a genus.
736. If the present cold war becomes hot, the human race might ex­
terminate itself.
737. If logic is a tool for the other sciences, then the other sciences
naturally follow logic in the order of learning.
738. No contingent thing is the full explanation of its own existence.
739. A triangle is cither scalene or equilateral, but not both.
7.40. All the predicables, with the exception of accident, arc related
essentially to their subjects.

8. What are the limits so far as the use of a truth table for hypotheticals
strictly taken is concerned?

9. Defend the charge that to speak of truth-functional hypothetical proposi­
tions is to use the term “hypothetical proposition” equivocally.
10. Indicate T, F, or ? in the parentheses preceding each proposition after
determining the truth value of the proposition to the extent to which this
can be done in the light of the data given.
10.1. (
10.2. (
103. (
10.4. (
10.5. (
10.6. (
10.7. (
10.8. (
10.9. (
10.10. (
10.11. (
10.12. (
10.13. (
10.14. (
10.15. (

) a • b, if a is true and b is true.
) a-5-»—b, if a is true and b is false.
) n — (a • b), if a is false and b is true.
) —a/i\b, if a is true and b is true.
) a<->b, if a is false and b is true.
) —(—a —b), if a is false and b is false.
) a\‘/b, if a is true and b is true.
) a/\—b, if a is false and b is false.
) —a-> b, if a is true and b is true.
) —a\/—b, if a is true and b is true.
) —a-^>—b, if a is true and b is true.
) —a • —b, if a is false and b is false.
) n — (a • b), if a is true and b is false.
) —a/i\—b, if a is true and b is true.
) a
b, if a is false and b is true.

CHAPTER

XIII

Relations Between Propositions

I. Propositional Relations

We have seen that terms can be logically related to one another in
different ways. For example, white and nonwhite are related by way of
contradiction, illegible is related to legible by way of privation, animal is
the proximate genus of man, etc. Propositions can also be related to one
another in different wap. We shall devote ourselves in this chapter to
an investigation into several of the most significant logical relations which
can be realized between propositions.
It is a property of propositions to be either true or false. The truth-status
of a proposition logically related to another is ordinarily an index of the
truth-status of that other proposition. For example, wc find that Every
man is potentially an artist and No man is potentially an artist arc re­
lated as contraries. We shall see that contraries cannot be true together.
Thus, from the truth of either of these two propositions we can move
immediately to the falsity of the other. Many logicians speak of the
immediate passage from the truth (or falsity) of one proposition to the
truth (or falsity) of a logically related proposition as an instance of im­
mediate inference. “Inference,” however, properly refers to the rational
movement of the intellect in its third operation; and, as we shall sec, to
move from the truth-status of one proposition to that of an immediately
related proposition is not to reason. Consequently, it is inaccurate to speak
here of an immediate inference. It is better to say simply that we are
concerned with propositions which are determinately related to one
another and which are, because they arc so related, indices for the truth­
status of each other.
Wc have discussed both categorical and compound propositions and
their subdivisions. In this chapter wc will limit ourselves to a considera­
tion of the relations between categorical propositions, and, to be even
more restrictive, only of relations between categorical propositions which
arc simply attributive rather than modal. Not that there are no logically
147

148

THE LOGIC OF THE SECOND OPERATION

significant relationships between modal propositions and between com­
pound propositions of one type or another. As a matter of fact, many
of the propositional relations we shall investigate within the context of
the simply attributive categorical proposition are also found between
modal propositions and between compound propositions. The reason for
restricting our consideration is simply to limit our investigations prudently
within the bounds indicated by the relatively short time available for a
semester’s study. Since simply attributive categorical propositions are the
fundamental type of proposition, we shall be able, despite the restrictions
in the breadth of our consideration, to acquire a solidly basic grounding
in the theory of propositional relations.
We have spoken of six types of categorical proposition, the A, E, I, O,
A’, and E' propositions. Each type admits of different propositional rela­
tions. For the most part the logical relations belonging to the universal
and particular propositions offer relatively little difficulty. However, the
relations belonging to singular propositions frequently involve difficulties
which make their investigation relatively troublesome. But the student
may be encouraged to learn that singular propositions are not nearly so
significant logically as the others. Logic is a tool pre eminently aimed at
science, and singular propositions are for the most part scientifically
irrelevant. Thus the beginning student can put aside the difficulties
involved in considering the logical relationships of singular propositions in
order to concentrate more on those belonging to universal and particular
propositions. With this in mind this chapter has been planned so that
the various propositional relations arc discussed first of all in reference
to A, E, I, and O propositions. A separate section of the chapter follows
in which all the propositional relations considered in the chapter are
discussed in the context of the A' and E1 propositions.II.
II. The Nature and Types of Opposition

Opposition is the logical relationship between propositions which affirm
and deny the same predicate of the same subject. Three different modes
of opposition can exist between propositions, namely, contradiction, con­
trariety, and subcontrariety.
1. Contradiction exists between two propositions with the same sub­
ject and predicate which purely and simply deny one another. Propositions
purely and simply deny one another when the truth of either excludes the
truth of the other while the falsity of either excludes the falsity of the
other. The contradictory of a given proposition, say, Every man is naturally
logical, is equivalent to an assertion that this proposition is not true,

RELATIONS BETWEEN PROPOSITIONS

149

that is, It is not true that every man is naturally logical. But this is
asserted in Some man is not naturally logical. Thus, Some man is not
naturally logical and Every man is naturally logical are related by way
of contradiction. They purely and simply deny one another: if one is
true the other is false, and if one is false the other must be true. Con­
tradictor}' opposition exists between both A and O propositions with
the same subject and predicate and E and I propositions. An example
of contradictory A and O propositions is: All men are white (A); Some
man is not white (O). An example of contradictory E and I propositions
is: No animals are rational (E); Some animal is rational (I). Another
example of A-O contradictories is: All men are rational (A); All men
are not rational (O). Although this O proposition may seem to be an
instance of an E proposition, it is, as we have seen, particular. As long as
one of these pairs of propositions is true the other is false and as long
as one is false the other is true.
2. Contrariety is the mode of opposition between two propositions with
the same subject and predicate which deny not only one another but
even less extended formulations of each other.
An A proposition with a given subject and predicate can be considered
a more extended formulation of an J proposition with the same subject
and predicate. The same thing, of course, is true for an E proposition
and a corresponding O. An A proposition, say, Even' criminal is neurotic,
is a denial of the corresponding E with the same subject and predicate,
namely, No criminal is neurotic. It is at the same time a denial of (in
fact, the contradiction of) the O proposition which represents a less
extended formulation of the E, namely, Some criminal is not neurotic.
Because the A in question is a denial of the corresponding E, and at the
same time a denial of the less extended formulation of that E, it is said
to be related to the E by way of contrariety. Just as the A proposition
is the contrary of the E, the E, in turn, is the contrary of the A, for the
E denies the A and also the I which is the less extended formulation
of that A.
At first glance contrariety might seem to be a stricter mode of opposi­
tion than contradiction. However, contrariety is less strict than contradic­
tion. Like contradictories, contraries cannot both be true, because they
deny one another. Unlike contradictories, however, they can both be
false, because they not only deny one another but also less extended
formulations of each other. This additional denial actually introduces a
weakness into the opposition of contrariety so that contraries need not
be opposed in falsity. No man is neurotic and Every man is neurotic arc
contrary propositions. Since they deny one another they cannot both be

150

THE LOCIC OF THE SECOND OPERATION

true. Yet both happen to be false, for some men are neurotic and some
men are not neurotic. It is, of course, not always the case that contraries
arc both false. Every animal is sentient and No animal is sentient are
contraries, and the first is true while the second is false. The rule
concerning contraries is simply this: since contraries cannot be true
together but may be false together, if one is true the other must be false,
but if one is false the other is questionable in the light of the first.
A person may know that the second proposition is true or false while the
first is false, but he will not know this on the evidence of the falsity of
its contrary. He will know it only on other grounds. However, if he
knows, to begin with, that the first proposition is true, then the fact
that the second is the contrary of the first is sufficient evidence for him
to assert its falsity. Although contraries might both be false, they can
never both be true.
Before proceeding from contrariety to the third type of opposition it
is worthwhile to suggest the following rule as a practical canon of effec­
tive debating: never propose a contrary when a contradiction will do. It is
true that one can show that his opponent’s position is false if he can
establish the contrary of his opponent’s position. Yet we must remember
that contrary positions may both be false. If one opposes a false position
with the equally false contrary of this position one does not destroy the
opposition and, in fact, one may seem to strengthen it. The surest and
simplest destruction of a position expressed either in an A or an E proposi­
tion is by establishing the contradictory O or I proposition. How much
simpler to refute the charge that no Americans arc loyal by presenting
in evidence some one loyal American than by attempting to establish the
loyalty of every American.
3. Subcontrariety arises between two propositions with the same subject
and predicate which do not deny one another but which do deny the
more extended formulation of the other.
Because I and O propositions each have a particular subject, an I
proposition does not deny the corresponding O; nor, of course, does the
O deny the I. However, the I docs deny (by way of contradiction) the
E, which is the more extended formulation of the O, and the O denies
the A, which is the more extended formulation of the I. Thus the O-I
propositions with the same subject and predicate are subcontrarics. Sub­
contrariety is a mode of opposition in only a tenuous sense, for sub­
contraries may both be true since they do not deny one another. Thus,
Some horses arc spotted and Some horses are not spotted are legitimately
subcontraries, and each is a true proposition. Of course, subcontraries
need not both be true. Some men are rational is true and, because all

RELATIONS BETWEEN PROPOSITIONS

151

men are rational, Some men are not rational is false. Whatever opposition
there is in subcontrariety is seen in reference to falsity, not truth. Though
subcon trades might both be true they can never both be false. Suppose
Some A is B is false, then Some A is not B must be true. The reason
for this can be shown in the following way. If Some A is B is false,
then its contradictory No A is B is true, since contradictories cannot be
true together. If, at the same time, Some A is not B were false, then,
its contradictor}' Every A is B would be true. But if that were the case
then No A is B and Ever}' A is B would both be true; but this is impos­
sible because they arc contraries. Ilius, if the I (Some A is B in our
example) is false, the corresponding O (Some A is not B) cannot be
false. A similar argument can be used to show that the I must be true
if the O is taken as false.
rIlie following examples of opposed propositions should help familiarize
the student with the modes of opposition:

1. Every democracy is a politically sound nation. (A)
Sonic democracy is not a politically sound nation. (O)
(Contradiction)

2. Some materialists arc Marxists. (I)
Some materialists are not Marxists. (O)
( Subcontrariety )
3. All that glitters is gold. (A)
All that glitters is not gold. (O)
(Contradiction)
4. No bcings-of-the-rcason can be categorized. (E)
Some bcings-of-thc-reason can be categorized. (I)
( Contradiction )
5. Some of the men who were not at the scene were unable to recall
the event. (I)
Some of the men who were not at the scene were not unable to
recall the event. (O)
(Subcontrariety)

6. None of the players was tired. (E)
All of the players were tired. (A)
(Contrariety)

7. All the troops crossed the Delaware. (A)
Some of the troops did not cross the Delaware. (O)
( Contradiction )

152

THE LOGIC OF THE SECOND OPERATION

8. No non-A is B. (E)
Some non-A is B. (I)
( Contradiction )
9. Some A is not non-B. (O)
Some A is non-B. (I)
(Subcontrariety)

10. Even- A is non-B. (A)
Every A is not non-B. (O)
( Contradiction )

III. Subalternation

Propositions related by way of opposition always involve the same
subjects and predicates and different quality, i.e., affirmation or negation.
Sometimes they involve different quantity, sometimes the same quantity.
What about propositions which have the same subject and predicate
and the same quality and which differ only in quantity? These proposi­
tions arc surely related to one another, but they are just as surely not
opposed to one another. We have already spoken of A and I (and E
and O) propositions with the same subject and predicate. We said
that the universal proposition was related to the particular as a more
extended formulation to a less extended formulation. The universal and
particular propositions are related to one another analogously as superiors
and inferiors. 'Hie universal proposition includes the corresponding par­
ticular proposition as a logical whole contains one of its subject parts.
This relationship is called subaltcrnation, and the superior proposition
is called the subaltcrnand, the inferior the subalternate. A predicate can
be said universally of a logical subject only if it can be said of even'
instance of that subject. Hence the subaltcrnand cannot be true unless
the subalternate is also true. But there is no guarantee that a predicate
can be said universally of a subject just because it can be said of some
instances of this subject. Thus the tnith of the subaltérnate leaves
undetermined the truth of the subaltemand. This implies, of course, that
the subaltemand must be false if the subaltérnate is false, but it allows
for the truth or falsity of the subaltérnate in the light of a false subalternand. Hence, if Every triangle is three-sided is true, Some triangle is
three-sided must also be true; and if Some men arc not rational is false,
No men are rational must also be false. But the falsity of No man is a
Communist leaves the truth-status of Some man is not a Communist

RELATIONS BETWEEN PROPOSITIONS

153

in question; and the truth of Some horse is spotted leaves the truth­
status of Every horse is spotted in question.
The following are examples of propositions related by subalternation:
1. Every Marine is taught to respect his rifle. (A)
Some Marine is taught to respect his rifle. (I)
( Superior [= subaltcrnand] to inferior [= subalternate] )

2. Some horses arc not spotted. (O)
No horses are spotted. (E)
(Inferior to superior)
3. No unintelligent man is fit to lead the nation. (E)
Not every unintelligent man is fit to lead the nation. (O)
(Superior to inferior)

4. Some non-A is non-B. (I)
Every non-/l is non-B. (A)
(Inferior to superior)
5. Every A is not B. (O)
No A is B. (E)
(Inferior to superior)
Before getting further into the discussion of propositional relations it
is necessary to insist upon a point which applies not only to the different
inodes of opposition but to any relationship between propositions. One
of the reasons why propositions are logically related to one another is
that they involve identical terms. Thus contraries have the same subject
and predicate while differing in quality but not in quantity. For terms
to be identical they must, as we have seen, involve generally the same
signification and supposition. It is understood throughout this chapter
whenever we speak of moving from the truth-status of one proposition to
that of a related proposition that there is no logically disruptive break in
the signification or the supposition of any given term from proposition
to proposition. Let us illustrate by way of reference to the relation of
subalternation. We have said it is legitimate to go from the truth of a
subaltemand to the truth of its subaltérnate, that is, from an A to its
corresponding I and from an E to its corresponding O proposition. This
would not be true if the A proposition expressed only the possible
existence of its subject and the I its actual existence. It can be true only
if the A as well as the I expresses actual existence or if the I as well as the
A expresses only possible existence. Furthermore, although an A may
frequently express only possible existence it can intend actual existence;

154

THE LOGIC OF THE SECOND OPERATION

and although an I frequently expresses actual existence, it may intend
only possible existence. The point here is that the terms must be the
same from proposition to proposition, and this means that they involve
the same signification and at least generally the same supposition.
IV. The Square of Opposition

Logicians have traditionally employed a visual aid, the “square of
opposition,” to assist in the handling of propositions involving opposition
or subalternation. 'Hie “square” is a rectangular figure whose corners
represent respectively A, E, I, and O propositions and whose sides and
diagonals as well represent the three inodes of opposition and subalterna­
ron. It serves as a graphic reminder that A and O (and E and I),
propositions with the same subject and predicate are related by way of
contradiction, that A and E propositions arc contraries of one another,
and that I and O propositions are subcontrarics of one another. Finally
it is a reminder that the A is related to the I and the E to the O as
die relationship of subalternation.
SU'
THE SQUARE OF OPPOSITION

'

(A) Every A is B.

(E) No A is B.

Contrariety

(I) Some A is B.

(O) Some A is not B.

It may be helpful to add a truth table as a companion piece to the
“square.” Given the truth value of one proposition on the “square,” the

155

RELATIONS BETWEEN PROPOSITIONS

truth table expresses graphically the truth values of each of the others.
The four horizontal columns in the table represent the A, E, I, and O
propositions. There are eight vertical columns in the table, each of which
records a truth value which is assumed for one of the four propositions,
along with the correlative truth values of the others which follow from
this. In each case the truth value which is assumed as a standard is
indicated by being encircled. Begin each time with the encircled T or F
proposition to ascertain the truth-status of the other propositions in the
same horizontal columns.
TRUTH TABLE FOR THE SQUARE OF OPPOSITION

(A)
(E)
(1)
(O)

Every
No A
Some
Some

A is B.
is B.
A is B.
A is not B.

0
F
T
F

F

?

SI
F

F
(T)
?

T

F
?
?

si

Si
?
?
T

?

SI
T
?

F
T
SI

T
F
T

T

SI

Neither the “square” nor its truth table arc suggested as mechanical
substitutes for an understanding of the relationships of opposition and
subaltcrnation and the logical rules which arc determined by these rela­
tionships. The “square” is recommended as an aid in seeing the concrete
realization of logical relationships in determinate propositions. The truth
table can be used as a check against the erroneous manipulation of
propositions related by way of opposition or subaltcrnation. Perhaps the
most profitable use of the truth table for the student consists first of all
in appreciating the principles according to which it is constructed and
then in repeated reconstructions of it, not from memory, but rather from
an understanding of the principles underlying it.
V. The Nature and Types of Conversion

Conversion is the logical relationship between propositions with inverted
or interchanged extremes which express cither totally or partially the
same truth. The act of converting is achieved by moving from an original
proposition (called the convertend) by inverting its subject and predicate
in a second proposition (called the converse) which expresses the same
truth as the original. The quality of the convcrtcnd and its converse must
always be the same. Some propositions can be legitimately converted
simply by interchanging their subjects and predicates while leaving the
quantity unchanged. This is called simple conversion. Other propositions
can be legitimately converted only if there is a change in the quantity

156

THE LOGIC or HIE SECOND OPERATION

of the original proposition. This is called accidental conversion. Some
.Americans are neurotic can be simply converted to Some neurotics are
American. Every Hoosier is an American cannot be simply converted to
Every American is a Hoosier, but it can be accidentally converted to
Some Americans are Hoosiers. To be legitimate a simple converse proposi­
tion must express totally the same truth expressed by the convertcnd.
To be legitimate an accidental converse proposition must express partially
the same truth expressed by the convertend. A converse expresses totally
the truth of the convertend whenever both of its terms have the same
extension that they had in the convertend. The converse only partially
expresses the truth of the convertcnd whenever cither of its terms has less
extension than it had in the convertend. (An attempted converse expresses
something not expressed by the convertcnd whenever any term in the
converse is more extended than the same term is in the convertend.)
Some neurotics arc American is a legitimate simple converse of Some
Americans arc neurotic because both terms have particular extension in
both the convertend and the converse. Every American is a Hoosier is an
illegitimate converse for Every Hoosier is an American because American
is taken with universal extension in the converse and with only particular
extension in the convertend. However, Some American is a Hoosier is a
legitimate accidental converse for Every Hoosier is an American. American
is taken with the same extension in both convertcnd and converse, while
Hoosier is taken with less extension in the converse than in the con­
vertcnd. If any of the terms in an attempted converse proposition should
have more extension than it had in the convertcnd, then the conversion
is illegitimate. That is why a proposition such as Some professors are not
scientists cannot be converted. In any attempted converse professor would
be the predicate of a negative proposition. As such, it would be universal
and therefore more extended than it was in the convertcnd, where it
was taken particularly.
The legitimate simple converse expresses exactly the same truth as its
convertend. Hence, if the convertend is true the converse is true, and
vice versa. If the convertcnd is false the converse is false, and vice versa.
The legitimate accidental converse expresses partially what its convertend
expresses. Since this relation is the same as that between subalternand
and subalternate propositions the rules that apply for subalternaron
apply for accidental conversion. Hence, a true convertend implies a
tme converse, but not vice versa; and a false converse implies a false con­
vertend, but not vice versa.
Conversion is of importance both on the level of the second operation
of the intellect and on the level of the third operation. But we should

157

REIATIONS BETWEEN PROPOSITIONS

note that the movement from a convertend to its legitimate converse is
not reasoning. The converse is a new proposition, but it does not
represent a new truth. The act of converting is an immediate passage
from a truth expressed in one way to that same truth expressed differently.
Tliis, it should be noted, is generally the case for the operations discussed
in this chapter. None of them is an instance of reasoning. The act of
reasoning is a mediate movement from previous knowledge to new
knowledge which is potentially in the previous knowledge. Still the
converse is a different expression of the same truth. It is logically worth­
while to be able to express a given truth in as many ways as possible
and especially to be able to recognize the same truth no matter how
it is presented. It is possible that, for a given person, one expression of a
truth might be more meaningful than another. When we study the
categorical syllogism we shall see another purpose for conversion. We
will find that syllogisms in one form can (and sometimes should) be
reduced to syllogisms in another form. This usually is done by converting
the propositions of the original syllogism.
VI. Legitimate Conversions

Since some propositions can be simply converted, some only accidentally
converted, and some not at all, we must determine exactly the number
and nature of the legitimate conversions. Since A, E, I, and O propositions
all theoretically admit two converses, the simple and the accidental, there
are eight different converses which we can test to see if they arc legitimate.
As we have seen, those which involve terms which have greater extension
than they had in the convcrtend must be rejected as illegitimate. The
results of the investigation into the legitimacy of the eight possible
converses can be expressed as follows:

(A)

Every A is B.

(E)

No A is B.

(1)

(O)

Some A is B.

Some A is not B.

Simple

Every Xis A*

Accidental

Some B is A*

Simple

No B isÂ-

Accidental

Some B is not A.

Simple

Some B is A.

Accidental

EveryJK is A.

Simple

Some Ji. is noh#C.

Accidental

NoB

158

THE LOGIC OF THE SECOND OPERATION

Whenever a term in the converse has the same extension that it had
in the convertend, a line is drawn over it. When it has less extension,
a line is drawn under it. When it has more extension it has been crossed
out. Any converse with a term which has been crossed out is illegitimate.
Those with lines over both terms are legitimate simple converses, as
they express the whole of the original truth. Legitimate converses with
any term underlined are accidental converses, as they express only a part
of the original truth. There are two legitimate simple converses, two
legitimate accidental converses. Only the E and the I propositions can
be simply converted. The E can also be accidentally converted, along
with the A. The O can be converted neither simply nor accidentally.
We have ruled out the possibility of the simple conversion of an A
proposition because its predicate, which is particular, would have to be
universal as subject of the converse. This is an indisputable point so
long as we arc considering the A proposition from the formal point of
view. Formally the A proposition can guarantee no more than the pos­
sibility of an accidental converse. However, in A propositions which
happen to have predicates which are coextensive with their subjects
simple conversion is legitimate. This exception is made strictly on the
basis of the matter of the proposition. The form of an A proposition
never guarantees a legitimate simple conversion. Thu$, A propositions
whose predicates arc related to their subjects as definition, specific differ­
ence, or specific property to species can be simply converted so long as
the conversion is made in virtue of the fact that the predicate happens
to be coextensive with its subject. In this way I can convert Every man is
a rational animal to Every' rational animal is a man and Every man is
capable of speech to Every thing capable of speech is a man. The predicate
of the first is the definition of the subject, and the predicate of the
second is a specific property of the subject.
The following examples of legitimate conversion should help to familiar­
ize the student with both simple and accidental conversion:
1. No triangles are four sided figures. (E)
No four-sided figures arc triangles. (E)
(Simple conversion)
2. Every logical relationship is a being-of-the-reason. (A)
Some being-of-the-reason is a logical relationship. (I)
( Accidental conversion )
3. No coward is able to be a true hero. (E)
Not everyone able to be a true hero is a coward. (O)
(Accidental conversion)

RELATIONS BETWEEN PROPOSITIONS

159

4. Some whole numbers are even. (I)
Some even things arc whole numbers. (Z)
(Simple conversion)
5. Every genus is an example of a predicable. (A)
Some example of a predicable is a genus. (I)
( Accidental conversion )
6. No scientist can afford to scorn logic. (E)
None who can afford to scorn logic arc scientists. (E)
(Simple conversion)
7. Bats have gnats. (A)
Some who have gnats are bats. (I)
( Accidental conversion )
8. No non-A is B. (E)
No B is non-A. (E)
(Simple conversion)
9. Some A is non-B. (I)
Some non-B is A. (Í)
(Simple conversion)
10. No A is non-B. (E)
Some non-B is not A. (O)
( Accidental conversion )
VII.

Obversion

Obversion is the logical relationship between propositions which ex­
press the same truth by affirming and denying contradictory predicates
of the same subject. Obversion, unlike conversion, is valid for all types
of propositions. An original proposition (called the obvertend) can be
obverted simply by changing its quality and contradicting its predicate,
'rhe result is a second proposition (called the obverse) which expresses
the same truth as the original. This is so because the two propositions
have opposite quality and predicates which are contradictorily opposed
to one another. Contradictory terms are related to one another as being
and non-being, or as affirmation and negation. Accordingly when one of
two contradictor}’ predicates is affirmed of a subject while the other is
denied of the same subject, the resulting propositions say exactly the
same thing, but, of course, in different ways. Since the obverse and its
obvertend express exactly the same truth, if either is true the other
is true, and if cither is false the other is false.
Strictly spcakingt the predicate of the obverse should be the contra­

160

THE LOGIC OF THE SECOND OPERATION

diction of the predicate of the obvertend. However, obversion is legitimate
as long as the predicates function as contradictories, even if, strictly
speaking, they arc not. This is the case with predicates which are
opposed as immediate contraries or with predicates of which one is the
privation of the other. Here, however, we must use these terms only
within the restricted area of their relevance. Strict obversion is based on
the fact that one term can be predicated of a subject if its contradictor)'
cannot. This is not absolutely true cither for immediate contraries or
for privatives. However, within the proper genus of things, the immediate
contrary of a term can be said of that of which the term itself cannot be
said, and vice versa. Thus, if a whole number is not odd, that whole
number is even. And if a whole number is even, that whole number
is not odd. Tn similar fashion, the privative of a term can be predi­
cated of that of which the term itself cannot, and vice versa, so long
as the terms are used with reference to the proper genus of things. If a
man is not able to see, that man is blind; and if a man is blind, that
man is not able to see. But wc could not say that Fido is even because
Fido is not odd, nor that Plymouth Rock is blind because Plymouth
Rock is not able to see. The reason is simply this: dogs and rocks are not
properly within the genera of things for which odd or even and able to see
or blind are relevant predicates.
Obversion is, in general, logically significant for the same reason that
conversion is important. It is worthwhile to be able to express the same
thing in more than one way. Obversion is especially suited to the elimina­
tion of excessive negatives in the expression of a truth. A proposition
overly laden with negative elements is often hard to grasp. Its obverse will
eliminate all or some of the negative elements and may help greatly to
clarify the meaning of the original proposition. For example, Some of
Poe’s short stories are not nonexciting is more clearly expressed as Some
of Poe’s short stories arc exciting. Similarly, None of the benefits of the
policy' are non transferable is more clearly expressed as All of the benefits
of the policy are transferable.

Examples of obversion include:
1. Every man is rational. (A)
No man is nonrational. (E)
2. Some solids have a density less than that of water. (I)
Some solids are not things which do not have a density less than
that of water. (O)
3. Some men are illiterate. (I)
Some men are not literate. (O)

RELATIONS BETWEEN PROPOSITIONS

161

4. Some politicians are not trustworthy. (O)
Some politicians are untrustworthy. (I)
5. Every horse is not spotted. (O)
Some horse is nonspotted. (I)
6. No animals are nonscnticnt. (E)
Every animal is sentient. (A)
7. Some A is non-B. (I)
Some A is not B. (O)
8. Every A is B. (A)
No A is non-B. (E)
9. Every A is not non-B. (O)
Some A is B. (Í)
10. No A is non-B. (E)
Every A is B. (A)
VIII.

Contraposition

Neither an A nor an O proposition admits of simple conversion.
However, logicians have discovered a substitute for simple conversion
for both the A and O propositions. It is a complex relationship spoken
of as contraposition. Contraposition is complex because it involves a
combination of obversion and conversion. An original A or O proposition
cannot be simply converted, but each can be obverted. The obverse of
an A is an E, and the obverse of an O is an I. The E and I propositions
can be simply converted, and their converses can be obverted once
again. The obverse of the simple converse of the obverse of an original
proposition is called the contrapositive of that original proposition. The
original proposition is called the contraponcnd, and the relationship be­
tween the two propositions is the complex logical relationship of con­
traposition. Neither the E nor the I proposition can be contraposed,
for in each case the obverse of the original is a proposition which defies
simple conversion. Since the contrapositive is the obverse of the simple
converse of the obverse of the contraponend, it expresses exactly the
same truth as the contraponend. Accordingly, if either of the proposi­
tions related by way of contraposition is true, the other must be
true. If either is false, the other must be false. Contraposition can
be effected simply by inverting the extremes of the contraponend
and by contradicting them, leaving their quantity and quality unchanged.
AH nurses are noncombatants can be contraposed by contradicting its
subject and its predicate and by using them in inverted order in an A

162

THE LOGIC OF THE SECOND OPERATION

proposition. The resulting proposition is Al! combatants arc nonnurscs.
In order to see that this is the legitimate contrapositive of AU nurses are
noncombatants, let us contrapose step by step:
All nurses are noncombatants. (A)
No nurses are combatants. (E)
No combatants are nurses. (E)

All combatants arc nonnurscs. (A)

(The original proposition)
(The obverse of the original)
(The simple converse of the ob­
verse of the original)
(The obverse of the simple con­
verse of the obverse of the
original )

Some A is not B is an O proposition. The O proposition whose
predicate is the contradictory of the subject of the original proposition
and whose subject is the contradictor}' of its predicate is Some non-B is
not non-A. Let us see that this is the legitimate contrapositive of Some
A is not B by contraposing step by step:
Some A is not B. (O)
Some A is non-B. (I)
Some non-B is A. (I)

Some non-B is not non-A. (O)

('rhe original proposition)
('Ilie obverse of the original
proposition )
(The simple converse of the ob­
verse of the original proposi­
tion)
(The obverse of the simple con­
verse of the obverse of the
original proposition)

Contraposition, as we have seen, is not a simple logical relationship. It
is complex, involving two obversions and one conversion. There is, prac­
tically speaking, no limit to the number of complex relationships which
can be set up between propositions. Contraposition is considered sepa­
rately and has been given a special name because it can be done very
simply and because the contrapositive serves as a convenient substitute for
a simple converse for propositions which can have no simple converse. It is
a helpful exercise for students to relate various propositions by way of
different complex relationships. For example, No B is non-A is the obverse
of the superior of the simple converse of Some A is B. No B is non-A
might also be considered the superior of the obverse of the simple con­
verse of Some A is B. In either case it can be seen that if No B is non-A
is true, Some A is B is true; and that if Some A is B is false, No B is
non-A must be false.

RELATIONS BETWEEN PROPOSITIONS

163

The following examples of contraposition will help to familiarize the
student with this complex logical relationship:
1. Some nonathletes are not healthy. (O)
Some who arc nonhealthy are not athletes. (O)
2. Not every non-Catholic is one who fears the Vatican. (O)
Some who do not fear the Vatican are not Catholic. (O)
3. Everything nonsentient is nonanimal. (A)
Every animal is sentient. (A)
4. Some A is not non-B. (O)
Some B is not non-A. (O)
5. Every non-A is B. (A)
Every non-B is A. (A)
IX.

Propositional Relations for the Singular Proposition

Singular propositions do not admit of opposition, subaltcmation, con­
version, and obversion as readily and as clearly as do universal and
particular propositions. Here we shall consider each of the propositional
relations investigated in this chapter in the context of the A' and E1
propositions. This consideration will reveal many difficulties not en­
countered with the A, E, I, and O propositions.

1. Contradiction
Here there is little difficulty. A' and E’ propositions do admit of
contradiction. As a matter of fact, the contradiction of the A‘ is the E‘
with the same subject and predicate, and, of course, the contradiction of
the E1 is the corresponding A'. This figure is not a triangle is the con­
tradiction of This figure is a triangle. John is a sick man is the contradic­
tion of John is not a sick man.

2. Contrariety
Singular propositions can never be contraries; for, as singular, they
cannot deny another proposition as well as a less extended formulation of
that other proposition.
3. Subcontrariety
Strictly speaking subcontrariety is a logical relation exclusively pos­
sessed by I and O propositions. Nevertheless there is a relationship be­
tween the E1 proposition and its corresponding I, and between the A'
and its corresponding O, which is similar to the relation of subcontrariety.
'Flic A proposition cannot properly be said to be a more extended
formulation of the A' in exactly the way in which it is a more extended

164

THE LOGIC OF THE SECOND OPERATION

formulation of the I. Given the A, the I is known immediately to be
true, but, given the A, the A' is known to be true only after the existence
of its singular subject is known. Nevertheless, once the existence of the
singular subject of the A' is presumed, the A1 can be seen to fall under
the A in a manner analogous to the way the 1 falls under it. The same,
of course, is true for the E and EJ propositions. The I is the subcontrary
of the O, and vice versa, precisely because each denies the universal
proposition under which the other falls, even though they do not deny
one another. Likewise the I does not deny the E‘, but it does deny the
E under which the E* analogously falls; and the O does not deny the A',
but it does deny the A under which the A' analogously falls. Thus the
I and the E1, as well as the O and the A', are related in a fashion
similar to the way in which subcontraries are related to one another.
Hence as long as the singular subject of the A1 proposition is presumed
to exist and to be under consideration, the falsity of the A1 implies the
truth of the corresponding O, and the falsity of the O implies the truth
of the corresponding A1. Nothing follows from the truth of either. For
the same reason, what has been said of the A' and the O can also be said
of the E1 and the I.
Thus, if it is false that some man is not a pure spirit, it must be true
that this man is a pure spirit, so long as this man is understood to exist
as a man. In similar fashion, if it is false that some Communist is a good
Christian, it must be true that this Communist is not a good Christian,
so long as this Communist is understood to exist as a Communist.
Further, if it is false that this fish is a salmon, it must be true that some
fish is not a salmon, and if it is false that this man is not a scientist, it
must be true that some man is a scientist.

4. Subalternation
The case of subalternation and the singular proposition is like that of
subcontrariety. In the strict sense singular propositions do not admit
of subalternation. Yet they do admit of a relationship similar to that of
subaltemation. As a matter of fact, they can be thought of as related
to the universal in a manner analogous to the way its inferior is related
to it. Similarly they are related to the particular in a manner analogous
to the way its superior is related to it. By subalternation the truth of the
subaltcrnand implies the truth of the subaltérnate, but not vice versa.
The A is related to the AJ, and the E to the E', in such fashion that
if the universal is true, the singular proposition must be true as long as
its singular subject is presumed to exist. Because of the relation lietween
the A and A1 and between the E and E1 propositions, if either of the

RELATIONS BETWEEN PROPOSITIONS

165

singular propositions is false, the corresponding universal is false also.
The A' is related to the I, and the E1 to the O, so that if the singular
proposition is true, the corresponding particular is true and if the par­
ticular proposition is false, any corresponding singular proposition must
also be false.
Thus, if it is true that every saint is a hero, it must be true that this
saint is a hero, so long as this saint is admitted to exist as a saint. And
if it is true that this saint is a hero, surely it is true that some saint is a
hero. Similarly, if it is true that no plant is sentient, it must be true
that this plant is not sentient, so long as this plant is admitted to exist
as a plant. And if this plant is not sentient, certainly some plant is not
sentient. Further, if it is false that some parrots can talk, it must be false
that this parrot can talk. And if it is false that this parrot can talk,
it must be false that every parrot can talk. If it is false that some plants
are not corruptible, it must be false that this plant is not corruptible.
And if it is false that this plant is not corruptible, it must be false that no
plant is corruptible.

5. Simple Conversion
Conversion is intended to restate the truth of a proposition differently.
When the original proposition is singular this can be done only if the
Singularity of the subject of the convertend is retained in the predicate
of the converse. 'Die legitimate converse of a singular proposition must,
then, have a singular predicate. This is an unusual logical situation. How­
ever, it can be tolerated. It is possible to convert both A1 and E‘ proposi­
tions simply, so long as tlie singularity of the subject of the convertend
is retained in the predicate of the converse. 'Plie A1 is converted to an I
proposition with a singular predicate. The E’ is converted to an E proposi­
tion with a singular predicate. Bertrand Russell is a well-known British
philosopher is converted to Some well-known British philosopher is
Bertrand Russell. John is not a hero is converted to No hero is John
(this individual John).

6. Accidental Conversion
As long as the singularity of the subject of the convertend is retained
in the predicate of the converse, an E1 proposition can be accidentally
as well as simply converted. John is not a hero is accidentally converted to
Some hero is not John.
7. Obversion
There is no difficulty in obverting A‘ and E1 propositions. They obvert
as easily as, and in exactly the same way as, universal and particular

166

THE LOGIC OF THE SECOND OPERATION

propositions. The cold war is a reality is obverted to The cold war is
not a nonreality. The cold war is not a fiction is obverted to The cold
war is a nonfiction.

8. Contraposition
Singular propositions admit of complex relations. For example, John is
not nonjust is the obverse of the contradiction of John is not just. How­
ever, because of the unusual status of the predicate for the converse of
a singular proposition, most complex relations which involve conversion
are of questionable value for singular propositions. This is the situation
as far as contraposition is concerned.
X.

Résumé

It may be helpful here at the end of this chapter to summarize these
rules in schematic fashion. Once again the student is warned against a
completely mechanical use of the schema.
Opposition
Contradiction

T<->F

Contrariety

T-»F

F->?

Subcontrariety

T->?

F->T

Subalternand to Subalternate

T->T

F->?

Subalternate to Subalternand

T->?

F-»F

Simple conversion

T<-»T

F«-»F

Accidental conversion

T-*T

F-»?

?<-T

F<-F

Obversion

T«->T

F<->F

Contraposition

iwr

F<-»F

Subalternation

Conversion

There are many helpful logical exercises which can be recommended
for the matter treated in this chapter. One good exercise is to construct
a number of different propositions and then attempt to make as many
logically related propositions as possible for each of these. The exercise

167

RELATIONS BETWEEN PROPOSITIONS

is completed by assuming first a true status, and then a false status, for
the original proposition and by attempting to ascertain the consequent
truth-status of each of its related propositions. As a practical hint the
student is reminded that it is usually prudent to state propositions in
strictly logical form before attempting to manipulate them. Suppose, for
example, Some Russians are Communist is taken for a start. This is an
I proposition, and it is already in strictly logical form. An I proposition
has a contradiction, a subcontrary, a superior, a simple converse, and an
obverse. If it is true, its contradiction is false, its subcontrary is question­
able, its superior is questionable, its simple converse is true, and its obverse
is true. If it is false, its contradiction is true, its subcontrary is true,
its superior is false, its simple converse is false, and its obverse is false.
Thus, beginning with Some Russians are Communist, the following exer­
cise is called for:

Original proposition — Some Russians arc Communist.
Contradiction
— No Russians are Communist.
Subcontrary
— Some Russians arc not Communist.
Superior
— Every Russian is Communist.
Simple Converse
— Some Communists arc Russians.
— Some Russians are not non-Communist.
Obverse

T
F
?
?
T

F
T
T
F
F

T F

Exercise XIII

1. Define: propositional relation, inference, opposition, contradiction, con­
trariety, subcontraricty, subaltcmation, subalternand (superior), subaltér­
nate (inferior), conversion, simple conversion, accidental conversion,
convertend, converse, obversion, obvertend, obverse, contraposition, contraponend, contrapositive.
2. Generally compare relations between propositions to relations between
terms.
3. Distinguish between a pure and simple denial and any other type of
denial.
4. Why is contradiction a stricter mode of opposition than contrariety is?
5. Why is subcontraricty an instance of opposition taken only in a wide
sense?
6. Why must contradictories have opposite truth values? Why can’t con­
traries be true together? Why can contraries be false together? Why can
subcontraries be true together? Why can’t subcontraries be false together?
7. Evaluate the following as a possible definition for contradiction: the
relation between propositions with the same subject and predicate which
differ in both quantity and quality.

168

THE LOGIC OF THE SECOND OPERATION

8. How does subalternation differ radically from any mode of opposition?
9. Why must a true subalternand necessitate a true subalternate, but not
vice versa? And why must a false subaltérnate necessitate a false subaltemard, but not vice versa?
10. How can the truth of an A proposition necessitate the truth of the cor­
responding I, since an A seems to demand only the possible existence of
its subject while the I seems to express the actual existence of its subject?
11. What is common to the legitimate accidental converse and legitimate
simple converse? How do they differ? How does this determine the way
in which convcrtend and converse are mutual indices of the other’s
truth value?
12. Can Ever}’ man is a rational animal be simply converted? Discuss.
13. How are obversion and simple conversion alike? How do they differ?
What advantage does obversion have over simple conversion? What ad­
vantage does simple conversion have over obversion?
14. When can something other than the contradiction of the predicate of
the obvertend be used as predicate of the obverse in legitimate obversion?
1 5. What, in general, is the significance of contraposition? What precise
function does contraposition have? Why does the contraponend have
exactly the same truth value as the contrapositive? Why can’t every propo­
sition be contraposed?
16. Discuss each of the propositional relations taken up so far in the context
of the singular proposition.
17. For each of the following propositions, (a) restate the proposition in
logical form if this is necessary; (b) determine the extension of its subject
and predicate; (c) classify it according to quantity and quality; (d) for­
mulate the several propositions related to it by way of opposition, sub­
alternation, conversion, obversion, and contraposition; and (e) suppose
the proposition as true (then as false) and indicate the corresponding
truth values of the related propositions:
17.1. All mothers love their children.
17.2. Some liquids are more dense than water.
17.3. Ever}' prudent athlete observes the rules of good health.
17.4. The fireman is an important man in the communit}’.
17.5. Man is sentient.
17.6. Everything worth doing is worth doing well.
17.7. Not every vital principle is an immortal soul.
17.8. Birds fly.
17.9. Some polygon is equiangular.
17.10. No polygon is a circle.
17.11. No king who does not respect his obligations to his people is
worthy of his crown.
17.12. Some students who do not study are taking the place in the school
rightfully belonging to others not fortunate enough to be in the
school.
17.13. Some men are just.

RELATIONS between propositions

169

17.14. All athletes who are not at least six feet tall are not unable to play
a good game of basketball.
17.15. Some propositions which are not openly copulative are not occultly
copulative.
17.16. All beings-of-the-reason which are not negations arc relations.
17.17. /Water boils at 100 degrees centigrade at sea level.
17.18. Tom’s handwriting is illegible.
x ¿
17.19. Not every man who speaks well of you to your face can be counted
upon as a friend who will support you when things arc not going
your way.
17.20. Every contradictory of the obverse of an E proposition is the sub­
contrary of an I proposition.
18. Indicate the truth value of the second proposition in the light of the truth
value of the first by marking T, F, or ? in the parentheses following the
second proposition in each pair of propositions. Then explain your answer
by stating the relationship the second bears to the first in the space fol­
lowing the parentheses.
18.1. Some
“
. (. .)_______________
A- ----is B. (T) Some _
B .is A.
18.2. No A is non-B. (F) Some non-B is not A. ( )
183. This A is B. (T) This A is not B. ( )
18.4. Not every non-A is B. (F) Not every non-B is A. ( I _
18.5. This non-A is B. (T) This non-A is not non-B. ( ) _
18.6. Every' A is B. (T) Some A is not B. ( ) __________
18.7. Some A is non- B. (F) Not every A is non-B. ( )
18.8. Every non-A is B. (T) Some B is non-A. ( ) _____
18.9. This A is non-B. (T) Some A is non-B. ( ) ________
18.10. No A is B. (F) Not every A is B. ( )
18.11. All A is B. (T) All non-B is non-A. ( )
18.12. Every A is non-B. (F) Some A is non-B. (
18.13. Not every A is B. (T) Some A is not B. ( )
18.14. Some non-A is B. (F) Every non-A is B. ( )
18.15. This A is B. (F) . B.
_ (. ). _____
Every A. is
18.16. Some A is B. (T) Some non-B is non-A. ( )
18.17. Every A is non-B. (F) No A is non-B. ( )
18.18. This A is not non-B. (T) No non-B is this A.
18.19. Some A is not non-B. (F) Some non-B is not
18.20. A is B. (T) B is A. ( ) _______________

19. State the proposition called for in each of the following. Then indicate
whether it is true, false, or questionable in the light of the original, taking
the original once as true and a.second time as false.
19.1. The obverse of the simple converse of the contradictory of the
contrary of the contradictory of Not every' A is B.
19.2. The superior of the simple converse of the subcontrary of the
obverse of Some propositions are hypothetical.
193. The contrary- of the obverse of the contradictory of the contra­
positive of Some liquid is not inflammable.
19.4. The accidental converse of the contradictory of the obverse of the
simple converse of the inferior of Every A is non-B.

170

THE LOGIC OF THE SECOND OPERATION

19.5.

The superior of the contradiction of the obverse of the contrary of
the contradiction of Some whole number is odd.

20. Criticize the following:
20.1. Since every law is reasonable, it follows that it is true that some
law is reasonable, that every nonreasonable thing is something
which is not a law, and that no law is something which is not
reasonable; and it follows that it is false that some law is not
reasonable, that some law is nonreasonable, and that everything
reasonable is a law.
20.2. Since it is true that every teacher is not a genius, it is true that
some teacher is a genius, and false that no teacher is a genius.
20.3. The following propositions can all be true together, but they can­
not all be false together: Some second intentions arc beings-of-thereason; some beings-of-the-reason arc not second intentions; every
second intention is a being-of-thc-reason; ever}- being-of-the-reason
is not a second intention.
20.4. It is not true that all Hibernians arc not drinkers, because it is
certainly true that several are.
20.5. Since it is true that some logic teachers are not idiots, it follows
that some are idiots. But if some are idiots, it must be false that
none of them are idiots. But this can’t be false unless it is true
that thej’ are all idiots. Hence, we can establish the fact that all
logic teachers arc idiots from the fact that some logic teachers
are not idiots.
20.6. Some noncombatants are nurses follows from the fact that no
nurses are combatants. For if it is true that no nurses are com­
batants, it is false that some nurses arc combatants, and false that
some nurses are not noncombatants, and true that some nurses
are noncombatants, and finally true that some noncombatants are
nurses.
20.7. Some husbands have wives, if all wives have husbands.
20.8. The obverse of the simple converse of the inferior of the contrary
of a true E proposition is a false O proposition.
20.9. The inferior of the contrary of the contradictory of the simple con­
verse of an I proposition is merely the simple converse of that I
proposition.
20.10. If it is true that some animal is not man, it is false that every
animal is man, and it is true that some nonanimal is not nonman,
and it is false that no animal is nonman, and it is false that some
animal is not nonman, and, finally, it is false that some man is not
animal.

PART in

The Logic of

the Third Operation

CHAPTER

XIV

Reasoning: The Third Operation of
the Intellect

I. The Nature of Reasoning

The intellect is made for truth. Tin’s, as we have seen, is grasped in
the second operation of the intellect, in judgment. Why then is a third
operation required? The reason is that the human intellect cannot know
all truth at once. In order to fulfill its potential for truth, the human
intellect must have recourse to a third operation which, paradoxically,
presupposes the second while being for the sake of the second. To under­
stand this, let us recall a few of the observations which were made earlier
with respect to judgment. There we saw that a judgment is true only if
motivated by something extramen tai which demands intellectual assent.
This motive is called evidence. We saw also that one type of evidence is
immediate evidence and that immediate evidence is divided into the
self-evident and the factually evident, with an example of a selfevident truth being The whole is greater than any of its parts and an
example of a factually evident truth being This typewriter is gray. It is,
however, obvious that we know many things which are neither self-evident
nor factually evident. Consequently, in order to assent to the truth of
these things a motive other than immediate evidence is required. One
possible motive for such assent is faith, which is based on evidence
extrinsic to the truth in question. But we arc not concerned here with
such extrinsic evidence as faith, however important it is in human life.
Hence we must search for another motive for assent, and we find this in
evidence intrinsic to the truth in question, but not immediately evident to
our intellect, cither by reason of being self-evident or factually evident.
Some propositions cannot be immediately seen to be true, but they can
be seen to follow as true from other propositions which are seen immedi­
ately to be true. To show' how such propositions do follow from proposi­
tions immediately evident is to reason. The operation of reasoning, then,
173

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THE LOGIC OF THE THIRD OPERATION

is a movement of the mind to a new truth seen in the light of older
truths. For example, consider the proposition Every man is subject to
death. As it stands this is neither self-evident nor factually evident. How­
ever, it is self-evident that every composite being is subject to death
(dissolution) and that every man is a composite being, and it is impos­
sible to consider these two truths together without seeing in them the
truth of Every man is subject to death. As a matter of fact, Every' man is
subject to death follows not only from the two self-evident propositions
given above; it follows also from the fact that this man is subject to
death (which is immediately evident when he is observed to die) and
this other man is subject to death, and this other man, and so on until
a sufficient enumeration of singulars has been achieved so that it becomes
evident that every man is subject to death. Here is a proposition, then,
which is neither self-evident nor factually evident. In itself, it is not
immediately evident, but it can be made evident in the light of both
self-evident and factually evident propositions.
Thus, if a man is to realize his full potential intellectually, he must
sometimes move from the second operation of the intellect to the third.
Yet he does so only so that he might find the evidence which moves
him to assent to a proposition. The third operation presupposes the
second; yet it is ordered to the second. Judgment remains the intellectual
operation par excellence. The necessity to reason is a sign of the intrinsic
weakness of the human intellect.
The conclusion in a reasoning process is said to follow from its
premises. Thus it depends on the premises. However, its truth is not
the same truth as any one of its premises. It is a new truth which exists
potentially in its premises, and it is known by anyone who sees how the
premises manifest the conclusion. It is potentially in the premises and it
is caused by them. Thus, when we speak of a conclusion following from
its premises, we do not mean cither that it is merely a novel statement
of an old truth (i.e., that it was in the premises actually all the time),
or that it is simply thought after something else is thought (i.e., that
perhaps it was in the premises not at all). The third operation of the
intellect is properly understood only as a true movement of the mind
from propositions already known to a new proposition potentially con­
tained in them.
II. The Argument and Its Elements

Mental and verbal arguments are related to reasoning in the same way
as mental and verbal terms are related to simple apprehension and mental

REASONING: THE THIRD OPERATION OF THE INTELLECT

175

and verbal propositions to judgment. 'Hie argument is the composite
expression which signifies the object of the third operation of the intel­
lect. Just as the categorical proposition is resolved into terms joined by
a copula, so the argument is resolved into propositions related to one
another in a certain way. In the act of reasoning the intellect moves
from previous knowledge to new knowledge. That part of the argument
which expresses the previous knowledge is called the antecedent. The
propositions making up the antecedent are called premises. That part of
the argument signifying the new knowledge is the consequent or conclu­
sion. Wc have used the terms “antecedent” and “consequent” before in
a similar but not identical fashion. We spoke of the antecedent and
consequent in the conditional proposition. The usage is the same in
this sense: neither antecedent can be true without its consequent being
true. The difference is clear, however. The antecedent in the argument
is composed of more than one proposition which together can be seen
to necessitate the consequent. The argument is more than one proposi­
tion, even one compound proposition; it is a composite expression
made up of several propositions which retain their identity as distinct
propositions and which together compose a logical entity which is
different from any conditional proposition. The conditional proposi­
tion is an assertion that of two component propositions the first cannot
be true without the second being true. The first of these two com­
ponent parts is the antecedent of the conditional, and, unlike the ante­
cedent of the argument, it need not be sufficient to yield the consequent
as a conclusion. It should be clear, then, that the terms “antecedent”
and “consequent” have their own distinctive meaning when they are
taken determinately in reference to the argument. The antecedent of
the argument is composed of the propositions which serve as premises
and, accordingly, it expresses the previous knowledge from which the
conclusion is generated. The consequent of the argument expresses the
conclusion which follows from the antecedent.
Reasoning is the third operation of the intellect, which presupposes
the second, which in turn presupposes the first. Similarly the argument
is resolved into propositions and these in turn into terms. 'Hie proxi­
mate matter of the argument is the number of propositions from which
it is constructed; its remote matter is the number of terms out of
which the propositions are ultimately constructed.
Any movement of the reason from previous knowledge to new knowl­
edge is, in a general sense, discursive. Thus we previously spoke of both
definition and division as instances of discourse in the sense that they
bring the mind from .a less perfect to a more perfect grasp of a concept.

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THE LOGIC OF THE THIRD OPERATION

The most perfect instance of discourse, however, is the argument, which
yields a conclusion which was virtually in the premises but is actually
different from cither of them. Definition and division result in a new
way of knowing an old object, whereas argument yields an entirely new
object for the intellect.
III. Validity and Truth

In an argument the conclusion follows from the premises. This means
that the premises are so interrelated that they manifest the conclusion,
and that the premises cannot be true without the conclusion being true.
The validity of an argument rests on the connection between its ante­
cedent and consequent. It follows that validity is a strict property of
argumentation. There is no argument which is not valid, because what
essentially constitutes an argument necessarily makes it valid. If there is
no validity, i.e., if the conclusion docs not follow from the premises, there
simply arc no premises and conclusion, no antecedent and consequent.
Strictly speaking, from the point of view of validity (which is the con­
cern of formal logic) there arc no bad arguments, no arguments in which
a conclusion docs not follow from the premises. Lacking validity', a
composite expression is not an argument at all, and it has neither con­
clusion nor premises. Nevertheless, it is customary to speak of invalid
argumentations and to note that in an invalid argumentation the con­
clusion does not follow from its premises. 'Ibis mode of speaking is
common enough so that we have adopted it here. Nevertheless, it is
acceptable to speak in this fashion only if a more strict usage of the
terms is recognized. To speak of an invalid argumentation is legitimate
only so long as it is understood that we speak of a composite expression
which is an attempted argument which falls short of being genuine
because the propositions proposed as premises do not in fact yield the
proposition proposed as a conclusion.
Validity and truth do not necessarily go .together. An argument can
be valid even though none (or only some) of its propositions arc true.
An attempted argument can be invalid even if each of its propositions is
true. Yet there is one connection between validity and truth which is
extremely important. In fact, if it were not for this connection there
would be no reason for perpetuating an interest in logic, or in any
science for that matter. The connection is this: given a valid argument,
a true antecedent necessitates a true consequent. The relation between
premises and conclusion is that of cause and effect. z\s long as a cause
exists (and exists as a cause), its effect exists. Since premises and con-

REASONING: THE THIRD OPERATION OF THE INTELLECT

177

elusions are propositions, the being of each is a being-true. Thus, so long
as the premises are true (and so long as they function as premises), the
conclusion must be true. If this were not the case, that is, if one could
begin with true premises and, arguing validly, still reach a false con­
clusion, there would be no purpose in reasoning, for one could never be
sure of his conclusion.
A true antecedent necessitates a true consequent. However, a false
antecedent does not necessitate a false consequent. Given an antecedent
with one (or all) of its propositions false, the conclusion which validly
follows from it may or may not be false. It is quite possible for it to be
true. Consider the following valid argumentations:
1. Every three-sided plane figure is rational.
But every brute is a three-sided plane figure.
Therefore, ever)' brute is rational.
2. Every three-sided plane figure is rational.
But every man is a three-sided plane figure.
Therefore, every man is rational.

Neither antecedent is true, while the consequent of the first is false and
the consequent of the second is true. Clearly a false antecedent can
generate a conclusion which is either true or false. Since this is the case,
we cannot say that false premises cause a conclusion to be cither false
or true. Some conclusions from false premises happen to be false, and
some happen to be true. We can only say that false premises, as premises,
cause a conclusion which would be true if these premises were true. But
as false, they can cause nothing, since to be false is, for a proposition,
not to be. The truth value of a conclusion following from false premises
will have nothing at all to do with the truth value of the premises.
Because a true antecedent in a valid argumentation necessarily yields
a true conclusion, it follows that a false conclusion can have come only
from a false antecedent. And because a false antecedent might happen
to generate a true conclusion, it is impossible to determine the truth­
status of an antecedent simply from the fact that its consequent is true.
The true consequent might have come from either a true or false ante­
cedent. Students are cautioned especially to remember that a true con­
sequent does not imply a true antecedent. It is quite possible to accept
a conclusion as true (c.g., the Communist conclusion that any exploita­
tion of the workingman must cease) without accepting its premises as
true (namely, the Marxist-Leninist theory' of economic evolution). That
day and night will succeed one another follows from the Ptolemaic
theory on astronomy, and also from the Copernican theory, which con­

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THE LOGIC OF THE THIRD OPERATION

tradicts that of Ptolemy. If one begins with premises which are known
to be true, he can be sure of a true conclusion. When one begins with
knowledge about the truth-status of a conclusion he can only say that
if the conclusion is false the premises themselves must be false.
IV. The Division of Argument

The meaning of argument should become clearer if we consider its
types. Besides, the several divisions will help to determine the content
and order of succeeding chapters in this part of our textbook.
Induction and Deduction
One of the most significant divisions of argument is into inductive
and deductive. Reasoning, as we have said, shows how a mediately
evident proposition is related to those immediately evident propositions
which manifest its truth. Generally speaking, resolution into self-evident
propositions is deductive, and resolution into factually evident proposi­
tions is inductive. There are, however, many exceptions to this. Only
the perfect form of deduction, which we shall call demonstration,
employs self-evident premises. Factually evident propositions can be em­
ployed as premises in deduction as well as induction. As a matter of
fact, both deduction and induction allow for propositions which are
not certainly evident at all, but which are only probable. Indeed, from
the point of view of formal logic, deduction and induction can l>c dis­
cussed independently of the truth-status of the propositions used in
argument. Nevertheless, the general point stands. Deductive argument
proceeds from the more universal to the less universal. A deductive con­
clusion is a less universal proposition which can be resolved into the
more universal premises which virtually contain it. And the self-evident
proposition is the prime example of the universal premise. However,
from the formal point of view, all that is required for deductive argu­
ment is a movement purely on the intelligible plane from a universal
premise to a less universal conclusion. The following argument is for­
mally deductive:
Every A is B.
But every C is A.
Therefore, every C is B.
Every C is B is a less universal proposition which falls under Every' A is
B. The reason why it does is that C is a subjective part of A. Thus, when
Every A is B is coupled with Every C is A, it becomes evident that

REASONING: THE THIRD OPERATION OF THE INTELLECT

179

Every C is B. This is a deductive movement of the mind. The argument
is called deduction or syllogism. Induction, on the other hand, proceeds
to the universal from the singulars of sense experience which represent
concretizations of the universal. Given a sufficient enumeration of singu­
lars, the intellect can move from the sensible plane to the universal
plane and form a proposition universally affirming the predicate found
in each of the singulars of the whole of which the singulars are sub­
jective parts. An inductive conclusion is a universal proposition which
can be resolved into a number of singular propositions sufficiently enu­
merated to manifest the universal conclusion. Thus, Every A is B might
be resolved back into This first A is B, and This second A is B, and
This third A is B, etc., etc.
Categorical and Hypothetical Argument
We know that arguments are composed of propositions. And we know
that some propositions are categorical and others hypothetical. Some
arguments are categorical because each of the propositions involved is
categorical. Others are hypothetical because the principal premise at least
is hypothetical. The first example below is a categorical syllogism. The
second is a hypothetical syllogism.
1. Every A is B.
But every C is A.
Therefore, every C is B.
2. a -> b
But a
b
Demonstrative, Dialectical, and Sophistical Argument
Some arguments yield a certainly true conclusion. These are called
demonstrations. Others give a probably true conclusion. These are dia­
lectical arguments. Others give only an apparently worthwhile conclusion.
These are fallacious or sophistical. Strictly speaking, we should divide
and then subdivide in order to make this a meaningful division. Argu­
ments arc either sound or defective. Defective arguments are, generally
speaking, fallacious or sophistical. This deficiency may be on the side of
form, or matter, or both. On the side of form, sound arguments must
be valid. On the side of matter, sound arguments will differ in so far as
one proceeds validly from absolutely certain premises to a conclusion
which cannot not-be true, while the other proceeds validly from premises
which can, at best, establish only a probable conclusion. The first type of
sound argument is demonstrative; the second, dialectical. Demonstration,

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THE LOGIC OF THE THIRD OPERATION

of course, is more perfect than dialectic. However, dialectical argument
is worthwhile. It is, in fact, highly significant, since in many areas the
human intellect is unable to reach demonstration and must be satisfied
with dialectic. Note that the difference between demonstration and
dialectic is material rather than formal. The first example below is demon­
strative; the second, dialectical; and the third, sophistical or fallacious:

1. Every rational animal is capable of speech.
But every man is a rational animal.
Therefore, every man is capable of speech.
2. Every mother loves her child.
But Marguerite is a mother.
Therefore, Marguerite loves her child.
3. Marquette students number over 10,000.
But John is a Marquette student.
Therefore, John numbers over 10,000.

Immediate and Mediate Inference

Some logicians distinguish between immediate and mediate arguments.
This is a false division since there are no immediate arguments or infer­
ences. The conclusion in an argument never comes from one premise
alone. There arc always at least two premises. The conclusion comes from
the principal premise, but only by way of another premise seen under
the principal premise. Alleged cases of inferences from one proposition
to another are really instances of restating a truth in another way. But this
is not reasoning. For example, to move from an I proposition to its
simple converse is a legitimate and immediate logical movement. How­
ever, the converse of the I is not a conclusion from the I. It is only the
original truth expressed in a new proposition. As far as argumentation
is concerned there are only mediate argumentations.
Exercise XIV

1. Define: reasoning (or inference), discourse, immediate proposition, medi­
ate proposition, argumentation, antecedent, consequent, validity, truth,
deduction( or syllogism), induction, categorical argumentation, hypotheti­
cal argumentation, demonstrative argumentation, dialectical argumenta­
tion, sophistical or fallacious argumentation.
2. Explain the reason why the third operation of the intellect is necessary
for intellectual perfection.
3. Explain the dependence of the third operation upon the second and its
simultaneous orientation to the second.

REASONING: THE THIRD OPERATION OF THE INTELLECT

181

4. In what sense can the acts of defining and dividing be spoken of, along
with the act of reasoning, as discursive?
5. In what sense is validity independent of truth, and in what sense are
they necessarily interrelated?
6. Supply examples of valid argumentations with true antecedent and true
consequent, false antecedent and false consequent, and false antecedent
and true consequent.
7. Evaluate the several divisions of the argumentation suggested in this
chapter by reference to the rules for a good division. Are these divisions
physical or logical? What is the precise principle of division in each case?

CHAPTER

XV

The Nature of the Categorical Syllogism

I. The Definition of the Categorical Syllogism

In an argument a consequent follows from an antecedent made up of
a given set of related propositions. The propositions in the antecedent arc
premises. The consequent is the conclusion. An argument with an ante­
cedent which is more universal than the conclusion is deductive. It is
called a syllogism. A syllogism which shows the connection between the
predicate and the subject of its conclusion by relating each of them to
a common third term in its antecedent is a categorical syllogism.
Consider the following proposition: Every man is a responsible agent.
This proposition is neither factually evident nor self-evident. Yet the
mind can be moved to assent to the truth of this proposition if its
subject-term (Every' man) and its predicate-term (responsible agent) are
joined to a common third term in antecedent propositions or premises.
Such antecedent propositions in the present instance could be: Every
intelligent being is a responsible agent and Every man is an intelligent
being. In these propositions the subject- and predicate-terms of our
original proposition arc joined to a common third term, in this instance
intelligent being. Expressing our premises and conclusion symbolically,
wc have: Every A is B, Every' C is A, and Every C is B. Here we see
that B belongs universally to C because we have previously seen that B
belongs universally to A and that A belongs universally to C. By linking
the terms of our original proposition (Every' man [C] and responsible
agent [B]) to a common third term (intelligent being [z\] ) in two pre­
vious propositions, we see that it must be true. If one of the terms of
our conclusion had been identified with the third term and if the other
term had been separated from it in the premises, then we would have
shown that those terms must be separated from each other in the
conclusion.
The distinctive feature of the categorical syllogism is its third term.
Tire antecedent shows that the predicate of the conclusion must either
182

THE NATURE OF THE CATEGORICAL SYLLOGISM

183

be affirmed or denied of its subject in the light of the way in which
each is logically related to the same third term. A categorical syllogism
is defined as a deductive argument which shows how the terms of its
conclusion arc related to one another by showing how each is related
to a common third term in its antecedent.
II. The Elements of the Categorical Syllogism

A categorical syllogism is so called because each of the three proposi­
tions of which it is composed is categorical. Two of these three proposi­
tions are contained in the antecedent of the syllogism as premises. The
third is the consequent or conclusion of the syllogism. The subject-term
and the predicate-term of the conclusion are known as the extremes of
the syllogism. Ordinarily the predicate-term of the conclusion is called
the major extreme or major term, and the subject-term the minor
extreme or minor term. The third term, which docs not appear in the
conclusion but which is found in each of the premises, is called the
middle term. The major term is related, either as subject or predicate,
to the middle term in what is called the major premise. The minor term
is related, either as subject or predicate, to the middle term in the
minor premise. The middle term is so called because it functions as the
principle through which the extremes are seen to be related to each other
cither by way of affirmation or denial. The major and minor terms are
given these names because the major is related to the minor as its logical
superior; at least it is normally apt to be affirmed or denied of the minor
term. The premises of a categorical syllogism are so related that one,
the major, is more universal than the other, the minor, and both shed
light on a conclusion which is less universal than the major premise. The
major term is related to the middle term in the major premise and the
minor term is related to the middle in the subordinate or minor premise.
The names for these terms are especially appropriate if we look at them
in the most perfect form of categorical syllogism. Here, as we shall see,
the major term is related to the others as the most extended of the three,
the minor as the least extended of the three, and the middle as less
extended than the major but more extended than the minor.
It is best to express the major premise first and the minor premise
second. This helps show that the minor premise should be taken as less
extended than the major premise and as subordinated to it. This point
can be further emphasized by introducing the minor premise with some
word such as the conjunction “but.” The conclusion is ordinarily last
and introduced by ap adverb such as “therefore” to indicate its status

184

THE LOGIC OF THE THIRD OPERATION

as a consequent. It is not necessary that categorical syllogisms be expressed
grammatically in precisely this fashion, and one should be prepared to
encounter them expressed in a number of different ways. However it is
ordinarily advisable, especially for students in elementary logic, to restate
syllogisms according to the plan just suggested when they are encoun­
tered in some other form. As long as the conclusion of a syllogism can
be discerned, it is easy to restate the syllogism grammatically in order to
express its true logical pattern as clearly as possible. The major and
minor terms are clear from the conclusion, and from them one can easily
locate the middle term and the major and minor premises. This is all
one need know in order to state the syllogism in strictly logical form.
The following example can be used to point out the elements com­
mon to everjf syllogism. Because it is an example of the first figure it
can be seen most perfectly to illustrate the reasons why the major, minor,
and middle terms have the names they have. It also serves to illustrate
how the minor premise is subordinated to the major, in conjunction with
which it yields a conclusion less universal than the major premise.
Every animal is an organism. — Major premise
M (middle
T
term)
But every man is an animal. — Minor premise
M
Consequent — Therefore, everyt man is an organism.
— Conclusion
t (minor
T (major
term)
term)

{

III. Direct and Indirect Conclusions

The major term of the categorical syllogism usually appears as the
predicate in the conclusion, where it is either affirmed or denied of the
minor term. The major term is normally used in this way because it
is normally suited or apt to be used in this way. Consider organism and
man. Organism is a remote genus of man; as such, it is calculated to be
said of man rather than vice versa. Thus we say Every man is an organism.
However, we can legitimately convert this to Some organism is a man.
Here the predicate is not a term naturally suited to be a predicate for
the subject to which it is related. Nevertheless, it can be used as the
predicate so long as the subject is taken particularly. Every man is an
organism is called a direct proposition, because the term used as the
predicate is naturally suited for this function. On the other hand, Some
organism is a man is an indirect proposition because the term normally

THE NATURE OF THE CATEGORICAL SYLLOGISM

185

suited to be used as the predicate is taken as the subject. Both of these
propositions follow as consequents from an antecedent including the
propositions Every animal is an organism and Every' man is an animal.
Every man is an organism is the more natural conclusion. We can call
it a direct conclusion. Some organism is a man is a less natural conclu­
sion. However, it is a legitimate indirect conclusion. A direct conclusion
differs from an indirect conclusion by using the major term as the
predicate, as it would ordinarily be used, whereas an indirect conclusion
uses the major term as the subject.
We shall see that there are three basically different patterns of syllo­
gistic procedure. 'Iliese are called the figures of the categorical syllogism.
Both direct and indirect conclusions are possible in each of the three
figures.
IV. The Figures of the Categorical Syllogism

We have spoken of the perfect form of the categorical syllogism,
while allowing for the possibility of less perfect forms. What are its
various forms, and why is one more perfect than the others? There arc
three basically different forms, called figures, of the categorical syllogism.
Each of these admits of variations of form within itself, known as the
moods of the syllogism. We shall speak first of the figures and then of
the moods.
The most important single principle of syllogistic procedure is the
middle term through which the major is united to or separated from the
minor term. The reason why there arc three basically different patterns
or figures of syllogistic procedure is this: there arc three, and only three,
basically different ways of relating the middle term to the major and
minor terms. The major term is always taken as more extended than
the minor. Understanding the major term in this way there arc three
basically different ways in which the middle term can be related to the
extremes. The middle can be logically less extended than the major and
more extended than the minor, or more extended than cither the major
and minor terms, or less extended than either the major or the minor
terms. These three possible relations of the middle term to the extremes
give us, respectively, the first, second, and third figures of the cate­
gorical syllogism, lliere is no fourth figure in which the middle term
would be more extended than the major and less extended than the
minor. This is an impossible combination, since the major is by defini­
tion taken as more extended than the minor. A fourth figure would
demand the impossible, namely, one term both less and more extended

186

THE LOGIC OF THE THIRD OPERATION

than another term or, in other words, a major term which is not a
major term. There is another way of expressing the three different ways
in which the middle term can be related to the extremes. In the first
figure the middle is affirmed or denied of the minor while the major is
affirmed or denied of the middle. In the second figure the middle is
affirmed or denied of both the major and minor. In the third figure
both the major and the minor are affirmed or denied of the middle. The
syllogistic patterns for each of the three figures can be diagramed as
follows:
Major premise
Minor premise

(1 )
M —T
t —M

(2)
T—M
t —M

(3)
M—T
M—t

Direct conclusion
Indirect conclusion

t—T
T—t

t—T
T —t

t—T
T —t

The following examples illustrate each of the three figures, with both
direct and indirect conclusions noted:
First figure — direct conclusion
Every rectangle is four-sided.
M
T
But every square is a rectangle.
t
M
Therefore, every square is four-sided.
t
T
First figure — indirect conclusion
Every rectangle is four-sided.
M
T
But every square is a rectangle.
t
M
Therefore, some four-sided thing is a square.
T
t
Second figure — direct conclusion
No plant is sentient.
T
M
But every animal is sentient.
t
M
Therefore, no animal is a plant.
t
T
Second figure — indirect conclusion
No plant is sentient
T
M

THE NATURE OF THE CATEGORICAL SYLLOGISM

187

But every animal is sentient.
t
M
Therefore, no plant is an animal.
T
t
Third figure — direct conclusion
Every good scientist has a sincere respect for truth.
M
T
But some good scientist is American.
M
t
Therefore, some American has a sincere respect for truth.
t
T
Third figure — indirect conclusion
Every good scientist has a sincere respect for truth.
M
T
But some good scientist is American.
M
t
Therefore, someone who has a sincere respect for truth is
T
American.
t

The first figure is said to be first because it is the most perfect form
of the categorical syllogism. It is the form of syllogistic procedure most
natural to the human reason. This is evident from experience. It can
also be seen in the fact that the basic principles of syllogistic procedure
can be applied directly only to the first figure. We shall say more of
this shortly. The first figure, as we have seen, is the only figure in which
the middle term is a middle in every respect, falling between the extremes
as far as extension is concerned. It is for this reason the only figure in
which the major is the most extended term while the minor is the least
extended. We shall see that the first figure is also the only figure in
which a universal affirmative conclusion is possible. For many reasons,
therefore, it is the most perfect form of categorical syllogism. The second
figure is put before the third because in it the middle term is more
extended than the other terms, whereas the middle is the least extended
of the terms in the third.
Some logicians hold that there is a fourth figure of the categorical
syllogism in which the middle term is the predicate of the major premise
and subject of the minor premise. Tin's argument would hold if the
figures of the categorical syllogism were determined according to the dif­
ferent ways in which the terms of a syllogism could be grammatically
arranged. There arc four different ways grammatically to express the cate­
gorical syllogism. But since there are only three ways in which the middle

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THE LOGIC OF THE THIRD OPERATION

term can be logically related to the other two terms, there can be only
three different logical figures. The fourth way of grammatically expressing
the syllogism ordinarily has the force of the first figure concluding in­
directly. Understood as such, it is an acceptable grammatical expression
of a legitimate logical procedure. It would be a more faithful expression
of this logical procedure, however, to transpose the premises, putting the
major premise first with the minor premise expressed along with and
under the major premise. For example:

Ever)’ man is an animal,
Every animal is an organism.
Some organism is a man. ■

■

Every animal is an organism.
Every man is an animal.
Some organism is a man.

V. The Moods of the Categorical Syllogism

The figures of the categorical syllogism depend upon the disposition
of the terms in the premises. 'Hie moods of the categorical syllogism
depend upon the disposition of the premises according to quantity and
quality. Each syllogism involves two premises. Each premise is cither
universal affirmative (A), universal negative (E), particular affirmative
(1), or particular negative (O). Singular premises (A* or EJ) do occur
as minor premises in the first and second figures. However, they’ present
no great difficulty' and do not constitute any distinct mood. For all
practical purposes they are handled as though they were universal
propositions. Disregarding the singular premises, we sec that there are
sixteen possible moods within each figure. The first figure can have a
major premise which is an A proposition coupled with an A, E, I, or
O minor. It allows also for an E, I, or O major as well, each with the
same possibilities for the minor premise. The same is true for the
second and third figures. 'The moods of the categorical syllogism are
identified ordinarily by describing them in terms of A, E, I, and O. The
description for a valid mood usually includes the conclusion as well as
the premises of the syllogism. For example:
1. (A) Every intellectual virtue is a quality.
M
T
(A) But every science is an intellectual virtue,
t
M
(I) Therefore, some quality is a science.
T
t

This is a first figure syllogism concluding indirectly with the mood
AAI.

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THE NATURE OF THE CATEGORICAL SYLLOGISM

2. (E) Nothing which fits into a category is a being-of-the-rcason.
T
M
(I) But some relation is a bcing-of-the-reason.
t
M
(O) Therefore, some relation does not fit into a category.
t
T

This is a second figure syllogism concluding directly with the mood
EIO.
3. (/)

Some triangle is ¡soreles.

(A) But every triangle is a figure possessed of three angles.
M

(/)

t

Therefore, some figure possessed of three angles is isosceles.
t
T

This is a third figure syllogism concluding directly with the mood 1A1.
Not every mood is valid. Certain moods are not valid in any of the
figures and most are invalid in one or another figure. Once we have
determined the rules for the validity of the categorical syllogism we shall
measure each of the possible moods against these rules and discover that
only a small fraction of them can yield a valid conclusion in any figure.
VI. The Basic Principles for Categorical Syllogism

An affirmative conclusion in a valid categorical syllogism follows from
its premises in virtue of the general principle of triple identity: two things
identified with the same third thing arc identified with one another. A
negative conclusion follows in virtue of the principle of the separating
third: two things one of which is identified with a third thing while the
other is separated from it are separated from one another. The prin­
ciples of triple identity and of the separating third cannot be called into
doubt, for their denials are patent absurdities. Syllogistic procedure is
sound precisely as grounded in these principles. Any syllogism is valid
only so long as it measures up to the requirements of either of these
principles. The principles arc expressed in terms of comprehension. They
arc best applied to the categorical syllogism by way of two principles
expressed in terms of extension which are more particularly proportioned
to the syllogism. The first of these is known as the dictum de omni
(literally, “what is said of all”): whatever is universally said of a logical
whole must be said of each of its subjective parts. The second is the
dictum de nullo (“what is said of none”): whatever is universally denied
of a logical whole must be denied of each of its subjective parts.

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THE LOGIC OF THE THIRD OPERATION

The dictum de omni and the dictum de nullo follow from the prin­
ciples of triple identity and of the separating third respectively. What­
ever is said of a logical whole must be said of its subjective parts because
the whole itself is said of each of these parts. Consequently, something
which is identified with a logical whole must be identified with its sub­
jective parts in accord with the demands of the principle of triple identity.
Consider the example: Every animal is an organism; but ever}' man is an
animal; therefore, every man is an organism. The comprehensive notes
of organism are seen to belong to any subject possessed of the compre­
hensive notes of man, because the comprehensive notes of organism
belong to any subject possessed of the comprehensive notes of animal,
which is the case with man. The comprehension of man and the compre­
hension of organism are each seen to be identical in subject with the
comprehension of animal and thus identical in subject with one another.
This is an evaluation of the syllogism directly in terms of the principle
of triple identity. It can just as legitimately, and more easily, be evaluated
directly in terms of the dictum de omni. Since organism is said of any
subject of which animal is said, it must be said of man, which is an
animal. The example is an instance of a valid categorical syllogism. It
is seen to be valid precisely because it measures up to the demands of
the principle of triple identity and the dictum de omni. We can con­
sider a second example: No rectangle is curvilinear; but every square is
a rectangle; therefore, no square is cun'ilincar. Tins syllogism is con­
structed in accord with the principles of the separating third and the
dictum de nullo. Comprehensively considered, curvilinear is seen as
separated from square, because curvilinear is separated from rectangle,
whereas rectangle and square are identified. From the point of view of
extension, curvilinear is denied of square precisely because it is universally
denied of rectangle, of which square is a subjective part.
The two examples of categorical syllogism we have used to illustrate
the force of the basic principles of the syllogism are both in the first
figure concluding directly. This is significant, for the basic principles
behind the syllogism are immediately applicable only to syllogisms in
the first figure concluding directly. This does not mean that second and
third figure syllogisms can never be valid, but it does mean that they are
valid only in so far as they can be reduced to syllogisms in the first figure.
The principles of triple identity and the separating third, as well as the
dictum de omni and the dictum de nullo, remain the ultimate canons
for the validity of any categorical syllogism. Each of the valid moods
in the second and third figures can be reduced to a valid mood in the
first figure. The second or third figure mood is seen to be valid because

THE NATURE OF THE CATEGORICAL SYLLOGISM

191

the mood of the first figure to which it is reduced measures up to the
demands of these basic principles. Hence, one method for evaluating
second and third figure syllogisms is to reduce the syllogism to a mood
of the first figure and then to measure the syllogism against the basic
principles. We shall soon discuss the reduction of second and third
figure syllogisms to syllogisms in the first figure. We shall see that the
reduction is not always easy and that it always requires care and precision.
Fortunately, there is a second method for evaluating second and third
figure syllogisms. Although the basic principles of the categorical syllogism
are directly applicable only to first figure syllogisms, we can deduce from
these principles a particular set of rules which can be applied to any
categorical syllogism in any figure. With this set of rules we need not
reduce a second or third figure syllogism to the first figure in order to
test its validity. We can do this simply by checking it against the par­
ticular rules for the categorical syllogism. Since these arc deduced from
the basic principles, they represent a true measure of the worth of the
syllogism in question. And, in so far as the basic principles remain directly
applicable only to the first figure, any valid mood in the second or third
figures is valid precisely because it is logically reducible to a valid mood
in the first figure. This is true even though we test it against the par­
ticular rules without bothering first to reduce it to the first figure.
VII. An Objection to the Categorical Syllogism

The dictum de omni requires that whatever is said universally of a
given logical whole be said of each of its subjective parts. Thus the
categorical syllogism in its affirmative form expresses a movement of the
intellect from a universal proposition in which a predicate is said of a
given universal subject to a conclusion in which this predicate is said
of a less universal subject related to the more universal subject as a
subjective part. The conclusion is reached because it is contained in the
more universal proposition from which it is generated. It is precisely at
this point that the categorical syllogism seems vulnerable to criticism.
If the conclusion of a categorical syllogism is already contained in its
major premise, then a syllogism does not seem to involve a movement of
the intellect. A categorical syllogism would seem to be no more an
instance of reasoning than the passage from the A or the E to the cor­
responding I or O by way of subalternation. Categorical syllogism would
seem to be guilty of the fallacy of begging the question, that is, of
presupposing the conclusion that it presumes to establish.
John Stuart Mill is-perhaps the best known author of the charge that

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THE LOGIC OF THE THIRD OPERATION

ever}' syllogism begs the question? He suggests the syllogism: AU men
are mortal; but Socrates is a man; therefore, Socrates is mortal. He argues
that no one can reasonably propose that all men arc mortal unless he
already knows that Socrates is mortal. Thus, it would seem, the syllogism
begs the question by presupposing its conclusion in its major premise.
Certainly if Socrates is a man, and if Socrates is not mortal, then it is
not true that all men are mortal. We can agree with Mill this far: All
men are mortal cannot be true unless it is also true that Socrates is
mortal. In fact it is because of this that we can conclude to the truth of
Socrates is mortal from the truth of All men are mortal. However, it is not
necessary to know that Socrates is mortal in order to know that all
men are mortal. All men are mortal expresses something other than a
factual report. Mortal is seen to belong to all men not because every
man has been observed and counted, but because human nature is
sufficiently understood so that it is seen to allow for the possibility
of death. Man, in abstraction from the individuating characteristics of
individual men, including Socrates, is understood to be such that any
subject which is a man is by that token also mortal. To be sure, the
proposition All men arc mortal results from an experience of individual
men. It is a universal proposition achieved by way of an induction from
singular instances of man and mortality. However, it is a proposition
which, in itself, is independent of any singular instance of which it is
a universal statement. It does not depend upon the fact that Socrates
is mortal. It is more than a report that all men are in fact mortal because
each has been observed to be. Thus All men are mortal does not pre­
suppose the proposition Socrates is mortal. Further, All men arc mortal
in conjunction with the proposition Socrates is a man yields the con­
clusion Socrates is mortal, lliis is a legitimate conclusion. It docs not
beg the question. All men are mortal is true independently of the fact
that Socrates is mortal and would be true even if Socrates did not exist.
It would not, of course, be true if Socrates were, per impossible, not
mortal. However, we say per impossible here precisely because All men
are mortal is true quite independently of Socrates, so that if Socrates
does exist as a man it is necessary that Socrates be mortal.
Not every proposition with the grammatical form of “All men are
mortal” can be a major premise in a categorical syllogism. Some represent
simply a factual report on an exhaustive examination of possible subjects.
Take the proposition All the students in the class passed the most recent
exam. This proposition can be made only if the examination grade of
1 See Mill’s Svstem of Logic, Bk. II, Chap. Ill (New York: Longmans, Green &
Co., 1930).

THE NATURE OF THE CATEGORICAL SYLLOGISM

193

each student in the class is observed and seen to be passing. This proposi­
tion is simply a report on the fact that this student, and that student,
and every other student in the class had a passing grade on the most
recent exam. There is no necessary connection between being a member
of this class and passing the most recent exam. It must be known that
each member of the class passed the exam before it can be known that
the entire group passed the exam. One could not conclude that John
passed the exam because John is a member of the class. The fact that
John passed the exam is presupposed to the proposed major premise and
is, in fact, partially what tliis proposition explicitly reports.
All men are mortal is independent of Socrates is mortal. It can be
called a universal proposition involving true universality. All the students
in this class passed the most recent exam is not independent of John
(i.e., this student in this class) passed the most recent exam. At best we
can call this a universal proposition which is enumcratively universal. The
middle term in a syllogism, however, functions as such only in so far as
it is a true universal, not an enumerative universal. The major, minor,
and middle terms may be one in subject (i.e., existentially identified in
the same tiling), yet they must be different as objective concepts (if not
in themselves, at least as conceived) or else the identification of both
the major and the minor in the middle causes no advance in knowledge.
The minor term in a syllogism whose middle term is only enumcratively
universal is not significantly different from the middle term itself. The
latter is simply the sum of the individuals of a given class of which
the minor is an instance. Thus the major cannot be communicated to the
minor by way of the middle. Inasmuch as the major term is said of the
middle it is already said of the minor.
VIII. The Expository “Syllogism44

The categorical syllogism generates a conclusion by linking the major
term to the minor term by way of the middle. The middle term is a
principle of communication for the major in reference to the minor. As
such, the middle must be a universal term so that it is impossible for a
singular term ever to function as a middle term in a categorical syllogism.
What, then, do we say of something like this?
Peter is a Canadian.
But Peter is a student in my class.
Therefore, some student in my class is a Canadian.
Tliis seems to be a syllogism, and the first and second propositions
cannot be true without the third being true as well. However, Peter docs

194

THE LOGIC OF THE THIRD OPERATION

not function as a middle term. There is nothing in Peter, as Peter, which
will allow us to relate student in my class to Canadian. Peter serves
rather as a concrete example of someone who is both a student in my
class and a Canadian. I point to Peter as a singular example which
establishes the proposition Some student in my class is a Canadian. The
function of Peter is simply to expose the truth that a certain student in
my class is a Canadian. Because Peter is not a middle term, this example
is not an example of a syllogism. However, because it has the appearance
of a syllogism, and because Peter does serve to expose the truth of what
appears to be a conclusion, it is said to be an example of an expository
“syllogism.”
The middle term in a categorical syllogism cannot be singular if it is to
function as a principle of communication. The major term cannot be
singular either. The major must be more universal than the minor.
However, the minor term can be singular, and in some cases is. It is
significant to note, however, that the categorical syllogism is primarily
ordered to conclusions concerning universal subjects. It allows for con­
clusions concerning singular subjects only in so far as the singulars fall
under the extension of the universals.
Exercise XV

1. Define: argumentation, syllogism, categorical syllogism, major term, minor
tenn, middle tenn, major premise, minor premise, direct conclusion,
indirect conclusion, figure, mood, true universality, enumerative universal­
ity, expository “syllogism."
2. Explain the meaning of categorical syllogism in terms of the elements
into which it can be both proximately and ultimately resolved. Define
each of these elements.
3. What is the difference between a syllogism concluding directly and a
syllogism concluding indirectly?
4. How are the figures of the categorical syllogism determined? Why have
some logicians spoken of a fourth figure? What makes a fourth figure a
logical absurdity?
5. How many possible moods of the categorical syllogism arc there? Explain.
6. What is the evidence for the truth of the basic principles for categorical
syllogism?
7. Explain the legitimacy of the following syllogisms in terms of the basic
principles for categorical syllogism:
7.1. Everj' polygon is rectilinear.
But every square is a polygon.
Therefore, every square is rectilinear.
7.2. No polygon is curvilinear.
But every square is a polygon.
Therefore, no square is curvilinear.

THE NATURE OF THE CATEGORICAL SYLLOGISM

195

8. How arc the basic principles for categorical syllogism applied to second
and third figure syllogisms?
9. How does the distinction between true universality and enumerative uni­
versality help to defend the syllogism against the charge of begging the
question?
10. Why is the expository "syllogism” not a syllogism in the strict sense, and
of what value is it, logically considered?

CHAPTER

XVI

The Rules for the Categorical Syllogism

I. The General Rules for Any Figure

The following rules are particular applications of the basic principles
of the syllogism. They are deduced from these basic principles and can
be applied to any categorical syllogism in any figure. If any one of them
is broken, the attempted syllogism is invalid. An attempted syllogism
might violate several of the rules at the same time. However, it is neces­
sary only to observe one infraction to know that the syllogism is invalid.
Thus a syllogism must conform to each of the rules in order to be valid.
Rule 1. There must be three terms and only three terms. These must
be so arranged that the middle term is found in one premise with the
major term and in the other with the minor term, while the major term
and minor term are found in the conclusion.
This rule follows from the very nature of the categorical syllogism as
determined by the basic principles of syllogistic discourse. It simply states
the essential pattern of syllogistic procedure.
I cannot argue that some animal is capable of speech because every man
is capable of speech and ever}' animal is mortal. Nor can I argue that a
certain musical instrument is colorless, odorless, tasteless, and weightless
because it is a triangle and because every triangle is colorless, odorless,
tasteless, and weightless. The first example obviously employs four terms
and is, therefore, not a syllogism. In the second example it looks as
though there arc only three terms, with each used twice, and in the
proper arrangement. But “triangle” is used with an analogy so close to
cquivocity that it does not stand for the same concept each time it is
used. Thus the second example also uses four terms. The principle of
triple identity demands that each of the three terms in the categorical
syllogism be taken generally with the same signification and supposition.
Anything less than this constitutes a violation of the ven’ nature of
syllogistic procedure just as surely as if four, five, or six completely diverse
196

THE RULES FOR THE CATEGORICAL SYLLOGISM

197

concepts expressed in completely diverse ways were employed in con­
structing the syllogism.
Other examples of syllogisms which arc faulty precisely as violating
this first rule include:
1. Every triangle is a rectilinear figure.
But no rectangle is a curvilinear figure.
Therefore, no triangle is a curvilinear figure.
(Four terms: [1] triangle, [2] rectilinear figure, [3] rectangle,
[4] curvilinear figure)
2. Americanism is'^opposed to Fascism)
But Fascism is\opposcd to Communism*
Therefore, Americanism is opposed to Communism.
(Four terms: [1] Americanism, [2] opposed to Fascism, [3] Fascism,
[4] opposed to Communism)

3. Substance is the ultimate genus of animal.
But every man is substance.
Therefore, every man is the ultimate genus of animal.
(Four terms: [1] substance taken with logical supposition, [2]
ultimate genus of animal, [3] man, [4] substance taken with per­
sonal supposition)
4. Marquette students number over 10,000.
But John is a Marquette student.
Therefore, John numbers over 10,000.
(Four terms: [1] Marquette students taken collectively, [2] num­
ber over 10,000, [3] John, [4] Marquette student taken divisively)

5. Foxes are valuable for their pelts.
But Rommel was a fox.
Therefore, Rommel was valuable for his pelt.
(Four terms: [1] foxes taken properly, [2] valuable for their pelts,
[3] Rommel, [4] fox taken metaphorically)

6. No A is B.
But some C is D.
Therefore, some C is not B.
(Four terms: [1] A, [2] B, Í3] C, [4] D)
7. Even’ A is non-B.
But some C is B.
Therefore, some C is non-A.
(Five terms: [I] A, [2] non-B, (3] C, [4] B, [5] non-A)

198

THE LOGIC OF THE THIRD OPERATION

Rule 2. No term in the conclusion can be universal if it is only par­
ticular in the premises.
This rule follows from the fact that the conclusion of the categorical
syllogism is caused by its premises. If any term were used more extensively
in the conclusion than it is used in the premises, the supposed effect
would be greater than its cause, and this is an absurdity.
I cannot argue that every Midwesterner is an American just because
every Badger is an American and every Badger is a Midwesterner. When
I attempt to argue in this way, Midwesterner is universal in the conclusion
and only particular in the minor premise. This would also be the case
were I to argue that no being is necessary because no creature is neces­
sary and every creature is a being. Here again the minor term is more
extended in the conclusion than in the minor premise. These syllogisms
suffer the fallacy of the illicit minor term. To argue that some horses
arc not four-legged because every horse is four-legged and some animals
arc not horses violates this same rule. In this case, however, it is the
major term which is used universally in the conclusion after being taken
only particularly in the major premise. This syllogism involves the fallacy
of the illicit major term. This is also the case with the syllogism which
attempts to show that every substance is a plant because every plant
is an organism and every organism is a substance.
Other examples of syllogisms which arc defective because of illicit
major terms or illicit minor terms include:
1. Every predicable is a being-of-the-reason.
But some logical relations are not predicables.
Therefore, some logical relations arc not bcings-of-tlie-rcason.
(Illicit major term)
2. No rectangle is a curvilinear figure.
But every rectangle is a rectilinear figure.
Therefore, no rectilinear figure is a curvilinear figure.
(Illicit minor term)
3. Not every substance is an organism.
But every plant is an organism.
Therefore, no substance is a plant.
( Illicit major term — in an indirect conclusion )
4. Every morally responsible agent is a free agent.
But some morally responsible agents arc not men of good will.
Therefore, some men of good will are not free agents.
(Illicit major term)

THE RULES FOR THE CATEGORICAL SYLLOGISM

199

5. Some laws are easily obeyed.
But no unreasonable ordinance is a law.
Therefore, some unreasonable ordinances arc not easily obeyed.
(Illicit major term)
6. No A is B.
But some C is A.
Therefore, no C is B.
(Illicit minor)
7. Every A is B.
But every A is C.
Therefore, every C is B.
(Illicit minor)
8. Every A is non-B.
But some A is not C.
Therefore, some C is not non-B.
(Illicit major)

9. Some A is B.
But no C is A.
Therefore, some C is not B.
(Illicit major)

Rule 3. The middle term must be universal at least once.
This rule follows from the principles of triple identity and the separating
third. A middle term which is particular both times might stand for
one segment of its possible extension the first time and for some other
segment of its possible extension the second time. Thus it cannot function
properly as a middle term, for it has the effect of being two different
terms, lire middle term serves its purpose precisely as one term simul­
taneously related to the predicate-term and subject-term of the conclusion.
It is by way of the middle that the predicate-term is communicated to
the subject. If the predicate were true only of some instance of the
middle while the subject was identified with only some (possibly other)
instance of it, then the predicate could not be communicated to the
subject by way of the middle. Under these circumstances the middle could
not be taken as one term simultaneously related to both. What we have
said about a middle intended as a principle of identification holds as
well for a middle intended as a principle of separation.
I cannot argue that all men are brutes because all brutes are sentient
and all men arc sentient, nor can I argue that all men are animals because

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THE LOGIC OF THE THIRD OPERATION

all animals are substances and all men are substances. Both conclusions
are equally invalid, although one is actually true and the other false.
To argue that some saints are neurotic because some saints see strange
things and neurotics see strange things is to argue invalidly. In each of
these instances the conclusion fails to follow from the premises precisely
because in none of these syllogisms is the middle term universal at least
once. Each can be said to suffer the fallacy of the unextended middle term.
Other examples of syllogisms defective because of unextended middle
terms include:
1. Every triangle is three-sided.
But some plane figures are three-sided.
Therefore, some plane figures are triangles.

2. Some men are intelligent men.
But some men arc blind men.
Therefore, some blind men are intelligent.
3. Not every substance is quantified.
But every body is a substance.
Therefore, not every body is quantified.

4. Every habit is a quality.
But every virtue is a quality.
Therefore, ever}' virtue is a liabit.
5. Every' even number is divisible by two without a remainder.
But some whole numbers are divisible by two without a remainder.
Therefore, some whole numbers are even numbers.

6. Some men are mathematicians.
But some men arc Americans.
Therefore, some Americans are mathematicians.
7. Not every A is B.
But some A is C.
Therefore, some C is B.

8. Every A is B.
But some B is not C.
Therefore, some C is not A.
9. Every A is non-B.
But some C is non-B.
Therefore, some C is A.

THE RULES FOR THE CATEGORICAL SYLLOGISM

201

Rule 4. If both premises arc affirmative, the conclusion cannot be
negative.
This rule follows immediately from the principle of triple identity,
which asserts an identity' between two things which are identified with the
same third thing. If one thing could be separated from another after each
was identified with the same third thing, the principle of triple identity
could be contradicted. This is impossible.
We cannot argue that no Communists are materialists because all
Communists are Marxists and all Marxists are materialists. This conclusion
would be absurd in the light of these premises. ITie contrary' of this
conclusion follows from these premises. There is little need to multiply
examples of this fallacy. To break this rule is so absurd that few are
ever tempted to do so and still fewer are fooled by those who do. For
this reason examples of this fallacy are relatively trivial.
Rule 5. Nothing follows from two negative premises.
This rule follows immediately from the principle of the separating
third. According to this principle two things must be separated from one
another if one is seen as identified with a third thing while the other is
separated from this same third thing. The third thing functions as a
principle for separating the other two only if it is positively identified
with one and separated from the other. If it were separated from both,
there would be no relationship set up between them and the third. In
this case it could function neither as a principle of identification nor of
separation.
It docs not follow that every Communist is a Marxist simply because
no Communist can be trusted and no Marxist can be trusted. Nor can
we conclude that kerosene is water because neither has a boiling point
of 10 degrees centigrade. We might also note that we could not conclude
that no Communist is a Marxist in our first example nor that kerosene
is not water in our second. When both premises are negative there is
no relationship between the extremes and the middle. No conclusion at
all is generated, and the supposed premises leave the supposed conclusion
decidedly open to either side of a contradiction.
The following examples suffer the fallacy of two negative premises:

/

1. No triangle is corporeal.
But some being is not corporeal.
Therefore, some being is not a triangle.

2 Communists do not favor the continuation of segregation in the
South.

202

THE LOGIC OF THE THIRD OPERATION

But Northern conservatives do not favor the continuation of
segregation in the South.
Therefore, Northern conservatives are Communists.
3. No A is B.
But no C is B.
Therefore, no C is A.
4. No A is B.
But not every C is A.
Therefore, some C is B.

Rule 6. If one premise is negative, the conclusion must be negative.
This rule follows immediately from the principle of the separating
third. A term positively related to one term and negatively related to
another manifests a negative relationship between these two. This rule is
the third rule concerned with the quality of the propositions in the
syllogism. Together these three rules demand that the two premises be
either both affirmative or one affirmative and one negative, and that the
conclusion be affirmative in the first case and negative in the second.
They rule out the possibility of two negative premises, of a negative
conclusion following two affirmative premises, and of an affirmative con­
clusion when one premise is negative.
In the light of the sixth rule it is impossible to argue that every man
is a horse because no horses arc endowed with speech and every man is
endowed with speech. Once again it is the contrary which follows. And
once again it is trivial to multiply examples to illustrate this fallacy.
Ordinarily a syllogism which breaks this rule is obviously invalid. Thus
it is of little value to multiply examples.
'Ihese are the six rules which govern the categorical syllogism. If any
one of them is violated, the syllogism in question is invalid. Any valid
categorical syllogism must measure up to all of them. The first rule merely
restates the basic pattern of syllogistic procedure, which lists the elements
required in a categorical syllogism and indicates generally the manner in
which they must be arranged. The second and third rules set the require­
ments for the three terms as far as their extension is concerned. The last
three rules state the requirements for the propositions used as premises
and conclusion from the point of view of quality. No other rules are
necessary, but two more can be added to the list. These additional
rules are: (7) Nothing follows from two particular premises, and (8)
If one premise is particular, the conclusion must be particular. These
are not independent rules. They do not follow directly from the two

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203

fundamental principles of the syllogism. They follow indirectly in so far
as their violation necessitates the violation of one or more of the six
rules already discussed.1 Thus it is quite possible to evaluate any cate­
gorical syllogism simply in the light of the six mies already given. We
add a seventh and an eighth to these six for the sake of convenience.
Sometimes it is easier to sec that the seventh or eighth is violated than
to see the accompanying violation of one of the fundamental six. The
seventh and eighth mies arc legitimate mies, but only in so far as they are
involved in the other rules.
Let us consider these additional rules in order to see why they pre­
suppose at least one of the basic six rules.
Rule 7. Nothing follows from two particular premises.
If both premises are particular, one of them must be an J proposition;
otherwise both would be O or negative propositions and thus break
Rule 5. An 1 proposition, however, has two particular terms. Consequently,
the other premise must be an O proposition; if it were another I, with
two particular terms, the middle term would be unextended and violate
Rule 3. The conclusion, therefore, must be negative, since one of the
premises is negative (Rule 6). But a negative conclusion has a universal
predicate. This means that one of the terms in the conclusion would
have greater extension than it had in the premises or that the middle
temí would be unextended (Rules 2 and 3), since the premises, which
arc O and I, allow for only one universal term. Thus two particular
premises ultimately require cither an unextended middle term or an illicit
major or minor term.
Each of the following examples violates this seventh mle as well as at
least one of the first six rules:

1. Some criminals are psychotic.
But some Americans are criminals.
Therefore, some Americans arc psychotic.
(Two particular premises, as well as an unextended middle term)
2.

Some plants arc green.
But some organisms arc not plants.
Therefore, some organisms arc not green.
(Two particular premises, as well as an illicit major term)

1 As a matter of fact Rule 4 cannot be violated without the violation of at least
one of the other five rules. Nevertheless, Rule 4 is given as a fundamental rule because
it follows directly from the principle of triple identity. Each of the other five of the
fundamental six rules can be broken independently of the others.

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THF. LOGIC OF THE THIRD OPERATION

3. Some A is B.
But some A is not C.
Therefore, some C is not B.
(Two particular premises, as well as an unextended middle term
and an illicit major term )

Rule 8. If one premise is particular, the conclusion must be particular.
A universal conclusion must be cither an E or an A proposition. If E,
both of its extremes must appear as universal in the premises, since both
are universal in the conclusion (Rule 2). Since the middle term must be
universal at least once (Rule 3), it follows that only one term can be
used particularly in the premises. Moreover, one of the premises must be
affirmative (Rule 5) and, as affirmative, must have a particular predicate.
According to our supposition, however, one of the premises is particular.
Its subject must also be particular. Thus we would have two terms used
particularly: one, as subject of the particular premise; the other, as
predicate of the affirmative premise. But, as we have noted, there is
room for only one particular use of a term in the premises if the con­
clusion is an E proposition. Consequently an E proposition cannot follow
from premises one of which is particular. With an A conclusion the
term used as subject of the conclusion is universal and must be used
universally in the premise in which it occurs (Rule 2). Since the middle
term must be used universally at least once (Rule 3) then there must
be at least two universal terms in the premises. Each premise must be
affirmative with an A conclusion (Rule 6). A particular premise would
have to be an I proposition with both terms particular. The other premise
would be an A proposition with a universal subject and a particular
predicate. This allows for only one universal use of a term when, as we
have seen, at least two arc necessary.
Each of the following examples violates this eighth rule as well as at
least one of the first six rules:
1. Some good athletes are not big.
But every good athlete is physically fit.
Therefore, no one physically fit is big.
(A universal conclusion with a particular premise, as well as an
illicit minor term)
2. Every man is endowed with speech.
But some two-legged animals arc not men.
Therefore, no two-legged animals are endowed with speech.
(A universal conclusion with a particular premise, as well as an
illicit major and an illicit minor)

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205

3. Every A is B.
But some C is B.
Therefore, every C is A.
(A universal conclusion with a particular premise, as well as an
unextended middle term and an illicit minor term )

Let us summarize the important points of this particular section. 'The
basic principles of categorical syllogism, namely, the principles of triple
identity and the separating third, and the dictum de omni and dictum de
nuHo, are immediately applicable only to syllogisms in the first figure.
Syllogisms in the other figures can be checked for validity in two ways:
first, by reducing them to the first figure and then measuring them directly
by the basic principles; second, by measuring them against a set of rules
which arc deduced from the basic principles and which are directly
applicable to syllogisms in any figure. Here our six fundamental rules
come into play, plus two additional rules which arc involved in the first
six and which can facilitate the evaluation of syllogisms when stated
as additional rules. The eight rules arc:
1. There must be three terms and only three terms. These must be
so arranged that the middle term is found in one premise with
the major term and in the other with tlic minor term, while the
major term and minor term are found in the conclusion.
2. No term in tlic conclusion can be universal, if that same term is
only particular in the premises.
3. The middle term must be universal at least once.
4. If both premises arc affirmative, the conclusion cannot be negative.
5. Nothing follows from two negative premises.
6. If one premise is negative, the conclusion must be negative.
7. Nothing follows from two particular premises.
8. If one premise is particular, the conclusion must be particular.
II. The Special Rules for Each Figure of the Categorical Syllogism

The eight rules just discussed apply to any categorical syllogism no
matter what its figure. Although there is no need to add to them, there
are special rules for each figure, and it may be helpful to use them in
evaluating syllogisms. These special rules for each figure arc not adequate
in themselves for an evaluation of a syllogism in that figure. They do not
dispense with the need for an evaluation in the light of the eight mies
already discussed. Rather each notes a requirement for a given figure
which cannot be ignored without breaking one of the eight mies.

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THF. LOGIC OF THF. THIRD OPERATION

Special Rules for the First Figure
For a direct conclusion —
1. The minor premise must be affirmative.
2. The major premise must be universal.
For an indirect conclusion —
3. If the minor premise is affirmative, the major must be universal.
4. If the major premise is affirmative, the conclusion must be par­
ticular.
5. If the conclusion is negative, the minor premise must be universal.
1. For a direct conclusion in the first figure the minor premise must
be affirmative. If the minor premise were negative, the conclusion would
be negative (Rule 6) and its predicate would be universal. Moreover, since
the universal predicate of the conclusion is the major term, a universal
major term would be required in the major premise (Rule 2). As predicate
the major term could be universal only if the major premise were negative.
But this would involve two negative premises and thus violate Rule 5.
2. For a direct conclusion in the first figure the major premise must be
universal. A particular major premise would have a particular subject,
in this case the middle term. The predicate of the minor premise, also
the middle term, would have to be universal or else the middle would
be twice particular (Rule 3). A universal predicate for the minor premise
would require a negative minor premise. But, as we have just seen, the
minor premise must be affirmative.
3. For an indirect conclusion, if the minor premise is affirmative, the
major premise must be universal. If the minor premise is affirmative,
its predicate, which is the middle term, is particular. If the middle term
is particular in the minor premise, it must be universal in the major
(Rule 3). As the subject of the major premise a universal middle term
would make for a universal major premise.
4. For an indirect conclusion, if the major premise is affirmative, the
conclusion must be particular. If the major premise is affirmative, the
major term, which is its predicate, is particular. The major term must,
then, be particular in the conclusion (Rule 2). As the subject of the
conclusion a particular major term would determine a particular conclusion.
5. For an indirect conclusion, if the conclusion is negative the minor
premise must be universal. If the conclusion is negative, its predicate,
the minor term, is universal. If the minor term is universal in the con­
clusion, it must be universal in the minor premise (Rule 2). As subject,
a universal minor term would make the minor premise universal.

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207

Special Rules for the Second Figure
For any conclusion —
1. One premise (and, of course, the conclusion) must be negative.
For a direct conclusion —
2. The major premise must be universal.
For an indirect conclusion —
3. The minor premise must be universal.
1. For any conclusion in the second figure one premise (and, of course,
the conclusion) must be negative. In the second figure the middle term
is a predicate in both premises. Consequently, one premise must be nega­
tive; otherwise the middle term will not be universal at least once
(Rule 3).
2. For a direct conclusion in the second figure the major premise must
be universal. If the major premise were particular, its subject, the major
term, would be particular. But the major term, as predicate of a negative
conclusion, must be universal in the conclusion. TTie major term cannot
be universal in the conclusion and only particular in the major premise
(Rule 2).
3. For an indirect conclusion in the second figure the minor premise
must be universal. If the minor premise were particular, its subject, the
minor term, would be particular. But the minor term must be universal
in the conclusion because it is the predicate of the conclusion, which
must, as we saw above, be a negative conclusion. The minor term cannot
be universal in the conclusion and only particular in the minor premise
(Rule 2).
Special Rules for the Third Figure
For a direct conclusion —
1. The minor premise must be affirmative.
For an indirect conclusion —
2. The major premise must be affirmative.
For any conclusion —
3. The conclusion must be particular.
1. For a direct conclusion in the third figure the minor premise must
be affirmative. If the minor premise were negative, the conclusion would
have to be negative (Rule 6). If the conclusion were negative, its predicate,
the major term, would be universal. If the major term were universal in
the conclusion, it would have to be universal in the major premise (Rule
2). As predicate of the major premise the major term could be universal
only if the major premise were negative. But this would involve two
negative premises and rule out validity (Rule 5).

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THE LOGIC OF THE THIRD OPERATION

2. For an indirect conclusion in the third figure the major premise
must be affirmative. If the major premise were negative, the conclusion
would have to be negative (Rule 6). If the conclusion were negative, its
predicate, the minor term, would be universal. If the minor term were
universal in the conclusion, it would have to be universal in the minor
premise (Rule 2). As predicate of the minor premise the minor term
could be universal only if the minor premise were negative. But this
would involve two negative premises and rule out validity (Rule 5).
3. For any conclusion in the third figure the conclusion must be par­
ticular. If the minor premise leading to a direct conclusion must be
affirmative, as we have just seen, its predicate, the minor term, must be
particular. If the minor term is particular in the minor premise, it must
be particular in the conclusion (Rule 2). As subject of the conclusion a
particular minor term constitutes a particular conclusion. The same is
true in reference to the major term in a syllogism leading to an indirect
conclusion.
III. The Valid Moods of the Categorical Syllogism

A student who knows the six (or eight) rules for any categorical
syllogism can confidently and capably evaluate any categorical syllogism.
In this section we shall determine which of the possible moods of the
categorical syllogism yield valid conclusions. The student should not
memorize the list of these moods with the idea that he can thereby
mechanically eliminate any syllogism whose mood is not valid. Rather
the student should apply the rules he has learned to each possible mood
to test its validity. By doing this he will draw up a list of the valid moods
of the categorical syllogism.
First of all, what are the various possibilities? A mood is determined
by the disposition of the premises in terms of quantity and quality.
Consequently there arc sixteen possible moods for each figure, since
both major and minor premises can be either A, E, I, or O. Each of
the four different major premises admits of four different minor premises
so that the total number of combinations of premises is sixteen. Each
mood, at least theoretically, can also have either an A, E, I, or O con­
clusion. Multiplying the sixteen possible combinations of premises by the
four possible kinds of conclusions we find that there are sixty-four
combinations possible. Since there are three figures, each with the pos­
sibility of both direct and indirect conclusions, the total number of
combinations is 384 (4 [major premises] X 4 [minor premises] X 4 [con­
clusions] X 3 [figures] X 2 [types of conclusions in each figure]). As a

THE RULES FOR THE CATEGORICAL SYLLOGISM

209

matter of fact only a small fraction of this total number is valid. Of
the 128 possible combinations in each figure there are only four distinctly
different valid moods leading to a direct conclusion in the first figure and
only five leading to an indirect conclusion, four of each in the second
figure, and six of each in the third figure. Thus, out of 384 possible
combinations of propositions there arc only 29 valid moods of the cate­
gorical syllogism.
We begin with the 384 possible combinations of major premise,
minor premise, and conclusion. Let us sec how many survive an evaluation
in the light of the general rules for the propositions in the categorical
syllogism. First of all, wc can eliminate any combination which violates
Rule 5 by including two negative premises. This eliminates the com­
binations with EE, EO, OE, and OO premises. Since these four sets of
premises admit theoretically of any one of four conclusions and can be
found in each of three figures in reference to both direct and indirect
conclusions, we have succeeded in eliminating 96 of our 384 possibilities
simply by the application of Rule 5. Next let us look among the remain­
ing combinations for those which violate Rule 7 by including two
particular premises. This group includes all those combinations with II,
10, and OI premises (OO already eliminated by way of Rule 5). The
total number in this group is 72. Now let us eliminate all those combina­
tions which violate Rule 4 by concluding negatively from two affirmative
premises. This group includes AA, Al, and IA premises when they are
coupled with either an E or an O conclusion (II already eliminated by
Rule 4). The total number of combinations in this group is 36. Some
of the remaining combinations violate Rule 6 by concluding affirmatively
even though one premise is negative. These include the AE, AO, EA,
El, IE, and OA combinations (the IO and OI combinations already
eliminated by Rule 7) when coupled with an A or an I conclusion. The
total number here is 72. Finally, we can apply Rule 8 to eliminate from
what remains all those with universal conclusions and one particular
premise. This includes the AIA, AOE, EIE, IAA, IEE, and OAE com­
binations (OEE, OEA, OAA, AIE, AOA, EIA, EOA, EOE, IAE, and
FEA all having been eliminated through the use of other rules). The
total in this group numbers 36. Thus, through the application of the
general rules for categorical syllogism, we have ruled out as invalid all but
72 of the 384 possible combinations wc started with.
These 72 remaining combinations include AAA, AAI, AEE, AEO, All,
AOO, EAE, EAO, EIO, I Al, IEO, and OAO considered in reference to
both direct and indirect conclusions in each of the three figures. If we
apply the special rules for the different figures of the categorical syllogism

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210

to these combinations, we find it necessary to eliminate all but 35. For
example, Rule 4 rules AAA out for an indirect conclusion in the first
figure; Rule 1 rules AAA out for any conclusion in the second figure; and
Rule 3 rules AAA out for any conclusion in the third figure. The legitimate
combinations for each of the figures can be listed in this fashion:

First Figure

Second Figure

Third Figure

Note that six pairs have been boxed together. For each boxed pair the
conclusion in one of the combinations is related to the other conclusion
as a subaltérnate. The premises are able to generate the universal con­
clusion and a fortiori the particular, since the particular is contained in

THE RULES FOR THE CATEGORICAL SYLLOGISM

211

the universal. Consequently, the combination with the particular conclu­
sion is not a distinct mood. It is only a variation of the mood which
is proportioned to the universal conclusion. Out of the original 384
possibly valid combinations, only 35 survive an evaluation in the light of
the rules — and these 35 represent but 29 distinctly different valid moods,
14 leading to direct conclusions in the three figures and 15 leading to
indirect conclusions.
Most logicians do not count the indirect moods of the second and third
figures, considering them as variant expressions of the direct moods of
these figures. Thus, they give only 19 distinctly different valid moods,
14 leading to direct conclusions in the three figures and 5 leading to an
indirect conclusion in the first figure. There is justification for this
position, since each of the indirect conclusions in the second and third
figures can be reached directly if the premises are transposed. This
transposition of premises involves no difficulties at first glance, since, to
begin with, the middle term is in the same position in both premises.
However, it does demand that what was originally the major term now
be considered as the minor, and vice versa. In some cases this does not
pose any difficulty, for in some instances neither extreme is in itself
more apt than the other to be used as major term. Consider the following
syllogism, for example:
Every genus is said of many differing in species.
But no species is said of many differing in species.
Therefore, no genus is a species.
This appears to be a second figure syllogism concluding indirectly, with
genus as major term and species as minor term. We can legitimately
consider it as such, or wc can transpose its premises, consider species as
major term and genus as minor term, and count it as a second figure
syllogism concluding directly. There is nothing to rule out this second
interpretation, for certainly neither genus nor species has any logical
priority over the other. One is as apt to be taken as major term as the
other, for each is equally (i.e., on the same level) a subjective part of
the same logical whole. We arc as free to consider one the major term
as the other. However, this does not mean that we can dispense with the
indirect moods in the second figure simply because a syllogism such as
this one need not be looked upon as having an indirect mood. First of all,
let us take the syllogism in question to be the equivalent of this second
figure syllogism concluding directly:
No species is said of many differing in species.
But every genus is said of many differing in species.
Therefore, no genus is a species.

212

THE LOGIC OF THE THIRD OPERATION’

'rhe rules of syllogistic procedure allow this conclusion and, with these
very same premises looked upon in exactly this same way, a second
conclusion: No species is a genus. This second conclusion, in the light
of these premises, is an indirect conclusion. Tims, no matter whether wc
take the premises as originally given (in which case the original con­
clusion is indirect) or transpose them (in which case the simple converse
of the original conclusion is a legitimate indirect conclusion), we find
ourselves faced with the actuality of an indirect conclusion in the second
figure. The same thing can be shown to be true for the third figure.
The case for the independent consideration of indirect moods in the
second and third figures is even stronger if we take an example with
extremes which are not indifferent to consideration as to major and minor
status. For example:
Every body is corruptible.
But somebody is not a plant.
Therefore, some corruptible thing is not a plant.

Tins appears to be a third figure syllogism concluding indirectly. It is
possible to transpose its premises and thus reduce it to a third figure
syllogism concluding directly. Plant would then be considered to be
the major term and corruptible the minor term. However, it is clear that
corruptible, since it is an attribute of plant, is apt to be said of or
denied of plant rather than the other way round. The conclusion of
this syllogism concluding directly would be inappropriately an indirect
proposition. It is more accurate to take this syllogism as one which con­
cludes indirectly than to consider it the equivalent of a syllogism con­
cluding directly. And what is true of this example in the third figure is
true of others, both in the second and third figures. For these reasons
wc have chosen to consider the valid indirect moods not only of the
first figure but of the second and third as well.
We have treated the indirect moods in all three figures. However, it
is only fair to point out that these are most significant in the first figure,
and especially for two of the five legitimate indirect moods of the first
figure. We have seen that the premises yielding valid indirect conclusions
in the second and third figures can be looked upon as giving these same
conclusions directly once the premises are transposed? This maneuver
is not effective in the first figure. Once the premises in the first figure are
transposed and the predicate of the conclusion is considered to be the
2 However, we have also seen that this way of looking upon indirect conclusions in
the second and third figures involves some difficulties, and that it is more accurate to
admit a real distinction between direct and indirect conclusions in all figures.

THE RULES FOR THE CATEGORICAL SYLLOGISM

213

major term, then the major term occupies a place in the conclusion and
in the premises which would indicate that it is both greater in extension
and less in extension than the minor term. This is a logical absurdity.
However, the (indirect) conclusions in AAI, All, and EAE can be
looked upon as being generated directly from their premises in the sense
that each is the converse of a direct conclusion. This is not the case
with AEO and IEO. The conclusions in these two indirect moods of
the first figure can be considered as generated from these premises only
indirectly. Neither can be seen to be generated directly from transposed
premises (as in the case of the valid indirect moods of the second and
third figures) nor as a converse of a direct conclusion (as in the case of
the other valid indirect moods in the first figure). 'Hiere is no way of
viewing these except as distinctly independent moods concluding indirectly
in the first figure. This is the reason why the indirect moods in the first
figure have not been ignored (though, as we have seen, they have some­
times been mistaken for moods of a fourth figure), while the indirect
moods of the second and third figures have usually been passed over as
insignificant.
IV. The Reduction of the Syllogism to the First Figure

We have seen that the first figure of the categorical syllogism is the
most perfect form of syllogistic procedure. The two basic principles of
syllogistic discourse directly apply only to the first figure concluding
directly. 'Hie first figure remains the only independent figure. 'Hie other
figures are legitimate inasmuch as each can be reduced to the first. Wc
can, of course, evaluate these figures directly by applying the eight rules
which flow from the two basic principles without reducing them to a
valid mood in the first figure. Still it is important to know how to
reduce the moods of the second and third figures to a valid mood in
the first. Only when this is done can we see clearly and immediately how
the two basic principles of syllogistic reasoning are realized in any syl­
logism other than a syllogism in the first figure concluding directly.
'Hiere arc two ways of doing this, direct and indirect. Direct reduction
operates by converting and/or transposing the propositions of the original
syllogism. Indirect reduction appeals to the principle of contradiction to
show the impossibility of supposing that the mood in question is invalid.
In direct reduction a syllogism on a par with the original is constructed
in the first figure concluding directly. In indirect reduction a first figure
syllogism concluding directly is constructed to show the absurdity of
holding that the original syllogism is invalid.

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THE LOGIC OF THE THIRD OPERATION

Many reductions require relatively little effort. For example, any
second figure syllogism with a direct conclusion whose major premise can
be converted simply is directly reduced to a legitimate mood of the first
figure by so converting it. The same thing is true of any third figure
syllogism concluding directly as far as its minor premise is concerned.
To illustrate:

No B is A.
No A is B.
->
But every C is B.
But every C is B.
Therefore, no C is A.
Therefore, no C is A.
(Second figure concluding directly with an E major)

Every B is A.
Every B is A.
But some C is B.
But some B is C.
Therefore, some C is A.
Therefore, some C is A.
(Third figure concluding direct y with an 1 minor)
Another relatively simple reduction is possible in the case of several of
the valid moods of the first figure concluding indirectly. Whenever a
conclusion of this kind is simply convertible, the syllogism can be
reduced to a mood of the first figure concluding directly merely by making
the simple conversion of its conclusion. Thus:
No B is A.
No B is A.
But every C is B.
But ever}' C is B.
Therefore, no C is A.
Therefore, no A is C.
(First figure concluding indirect y with an E conclusion)

Other direct reductions are more complicated. For example, an IAI
syllogism concluding directly in the third figure can be directly reduced
to a first figure concluding directly only through the simple conversion
of both its major premise and its conclusion along with the transposition
of its premises. Thus:

Some B is A.
But ever}' B is C.
Therefore, some C is A.
(Third figure concluding directly
sitions )

Every B is C.
But some A is B.
Therefore, some A is C.
with IAl combination of propo­

The AAI mood in the third figure concluding indirectly admits of
two direct reductions. It can be reduced either by converting its major
premise accidentally and transposing its premises or by converting its

THE RULES FOR THE CATEGORICAL SYLLOGISM

215

minor premise accidentally and converting its conclusion simply. To
illustrate:

Every B is A.
But every B is C.
Therefore, some A is C.

Every B is C.
But some A is B.
Therefore, some A is C.

(Third figure concluding indirectly with AAI propositions)
Every B is A.
But every B is C.
Therefore, some A is C.

Every B is A.
But some C is B.
Therefore, some C is A.

All the valid moods which are not already in the first figure concluding
directly can be indirectly reduced to it. Four of them can be reduced
only through the method of indirect reduction. Indirect reduction is
achieved in this way: first, contradict the conclusion of the original
syllogism; then, use this as a premise in a first figure syllogism concluding
directly along with the appropriate premise of the original syllogism to
establish the denial of the other premise. The original syllogism is valid
if the contradiction of its conclusion coupled with one of its original
premises contradicts its other premise. A syllogism is valid precisely
because its premises cannot be true if its conclusion is false. We can
show that this is the status of any valid syllogism by assuming the
opposite situation and then showing its absurdity. If we assume that the
conclusion of any syllogism is false while its premises are true, then we
assume the truth of the contradiction of that conclusion. If this contradic­
tion is true, then it cannot be coupled with one of the premises we have
assumed as true to establish the falsity of the other premise, also assumed
as true. Whenever the contradiction of the conclusion and one of the
premises do establish the falsity of the other premise, it is clear that we
have an impossible situation. Thus the original conclusion could not
have been false. It must have been true, and the original syllogism must
have been valid.
We have seen how an EAE syllogism in the second figure concluding
directly can be reduced through the simple conversion of its major
premise. This same syllogism can be indirectly reduced to the first figure
by contradicting its conclusion and using this as a premise along with
its major premise to establish the contradiction of its minor premise:

No A is B.
But every C is B.
Therefore, no C is A.

No A is B.
But some C is A.
Therefore, some C is not B.

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THE LOGIC OF THE THIRD OPERATION

An example of a valid syllogism in an imperfect mood which can
be reduced to a perfect mood only by way of indirect reduction is the
AOO syllogism in the second figure concluding directly. This reduction
takes place as follows:
Every A is B.
But some C is not B.
Therefore, some C is not A.

Every A is B.
But every C is A.
Therefore, every C is B.

It is not necessary here to illustrate each reduction. However, it is a
worthwhile exercise for the student to reduce directly each imperfect
mood to a mood in the first figure concluding directly and, where this
is impossible, to do so by an indirect reduction. The steps to be taken
for each reduction are as follows:

Rr»t figure
concluding
indirectly

AAI — simply convert the conclusion.
AEO — accidentally convert the major, simply convert the minor,
and transpose the premises.
All — simply convert the conclusion.
EAE — simply convert the conclusion.
IEO—simply convert both premises and transpose the premises.

Second figure
concluding
directly

AEE — simply convert the minor, simply convert the conclusion, and
transpose the premises.
AOO — indirect reduction only.
EAE— simply convert the major.
EIO— simply convert the major.

Second figure
concluding
indirectly

AEE—simply convert the minor and transpose the premises.
EAE — simply convert the major and simply convert the conclusion.
IEO — simply convert the minor and transpose the premises.
OAO—indirect reduction only.

Third figure
concluding
directly

simply convert the minor.
accidentally convert the minor.
simply convert the minor.
simply convert the major, simply convert the conclusion, and
transpose the promises.
OAO—indirect reduction only.
AAI — accidentally convert the minor.

Third figure
concluding
indirectly

AEO — accidentally convert the major and transpose the premises.
All — simply convert the minor and simply convert the conclusion.
AOO — indirect reduction only.
IAI — simply convert the major and transpose the premises.
IEO—simply convert the major and transpose the premises.
AAI —■ accidentally convert the major and transpose the premises
or accidentally convert the minor and simply convert the
conclusion.

All —
EAO —
EIO —
IAI —

THE RULES FOR THE CATEGORICAL SYLLOGISM

217

Most logic textbooks suggest a timeworn set of mnemonic verses in
Latin hexameter to help the student to recall the valid moods of the
categorical syllogism and the method of reducing the various imperfect
moods to perfect moods. Because Latin hexameters are, unfortunately, of
little value as an aid to study for the modem student and because these
traditional verses omit reference at least to some of the indirect moods,
we shall spend little time on them. However, it may be at least of
historical importance to note the names given the valid moods leading
to direct conclusions and to note how the spelling of each name reveals
the pattern of the mood and, in the case of the second and third figures,
the method of reduction to the first figure. The names given the valid
moods in the first figure are:

Barbara
Celarent

Darii
Ferio

Those in the second figure are called:
Cesare
Canicstrcs

Festino
Baroco

In the third figure the valid moods arc known as:

Darapti
Disamis
Datisi

Felapton
Bocardo
Ferison

In each instance the vowels represent in order the major premise, minor
premise, and conclusion. Thus, Barbara indicates an AAA mood, and
Bocardo an OAO mood. The initial letter for the valid moods in the
second and third figures indicates the mood in the first figure to which
it must be reduced. Thus, Caesare is reduced to Celarent, Datisi to
Darii, and Ferison to Ferio. An s (for simplex) indicates that the
proposition for which the preceding vowel stands must be simply con­
verted in reduction. A p (for per accidens) indicates that the preceding
vowel stands for a proposition which must be accidentally converted. An
m (for mutatio) in the name indicates that the premises must be
transposed. Finally a c (for contradictio) indicates that the only pos­
sible reduction is an indirect reduction by way of contradiction. It should
be of interest to the student to check the accuracy of these names and,
in the event they can be helpful, to use them as a mnemonic aid in
recalling the valid moods of the categorical syllogism.

218

THE LOGIC OF THE THIRD OPERATION

Exercise XVI

1. Differentiate between the basic principles for the categorical syllogism,
the general rules for the categorical syllogism, and the special rules for
the different figures of the categorical syllogism.
2. State each of the eight general rules for the categorical syllogism, and
defend each one in the light of the basic principles.
3. State each of the special rules for each figure, and defend each one in
the light of the general rules.
4. Explain why it is that there are 384 possible combinations of propositions
for the categorical syllogism. Show how all but 35 of these are rejected
as invalid. Explain why it is that only 29 of these 35 are considered to
be distinct moods.
5. Some logicians speak of 19 valid moods in four figures. Others speak of
19 valid moods in three figures. We have spoken of 29 valid moods in
three figures. What can be said for and/or against each position?
6. What is the significance of the reduction of second and third figure syl­
logisms and first figure syllogisms concluding indirectly to first figure
syllogisms concluding directly?
7. How does indirect reduction differ from direct reduction?
8. Construct an example for each valid mood in the second and third figure
and in the first figure concluding indirectly and reduce each one directly
and/or indirectly to a mood of the first figure concluding directly.
9. Try to construct mnemonic names for the valid indirect moods in the
three figures, using those for the direct moods as models.
10. Classify each of the following syllogisms by indicating its figure and by
pointing out whether its conclusion is direct or indirect. Then evaluate
it as valid or invalid, and, if it is invalid, explain what is wrong with it.
Reduce each valid syllogism to a valid mood of the first figure concluding
directly:
10.1. No crime is perfect.
* But every crime is an immorality.
^-Therefore, no immorality is perfect.
10.2. "Every virtue is worth having.
But some habits are not virtues.
'Ilierefore, some habits are not worth having.
10.3. No creature is necessary.
But no creature is God.
Therefore, God is necessary’.
10.4. Every horse is a substance.
But every' horse is a brute animal.
Therefore, somcjubstance is a brute animal.
10.5. No man is a hippopotamus.
But some man is virtuous.
Therefore, some hippopotamus is not virtuous.

THE RULES FOR THE CATEGORICAL SYLLOGISM

219

The philosophy teachers teach all the logic classes in this school.
But Professor Allen is a philosophy teacher.
Therefore, Professor Allen teaches all the logic classes in this school.
10.7. No polygons are circles.
But some figures are not circles.
Therefore, some figures are not polygons.
10.8. Every definition is a composite expression.
But ever}' definition is convertible with its subject.
Therefore, every composite expression is convertible with its subject.
10.9. No news is good news.
But the end of a war is news.
Therefore, the end of a war is not good news.
10.10. Every 300 hitter is a help to his baseball team.
But Jamie is not a 300 hitter.
Therefore, Jamie is not a help to his baseball team.
10.11. 'Every polygon is rectilinear.
But every polygon is a plane figure.
Therefore, some plane figure is rectilinear.
10.12. Every virtue is a quality.
But every ability is a quality.
Therefore, every ability is a virtue.
10.13. Some liquid is inflammable.
But some liquid is kerosene.
Therefore, some kerosene is inflammable.
10.14. Every moral agent is free.
But no plant is free.
Therefore, no moral agent is a plant.

10.6.

11. Supply whatever direct and indirect conclusions are legitimate for the
following premises. Classify each completed syllogism according to figure
and mood:
11.1. Every A is B.
But no A is C.
Therefore,
¿
11.2. Every A is B.
But no C is B.
Therefore,
.■
113. Some A is B.
But every A is C.
11.4.
11.5.
11.6.

But some C is B.
Therefore,
¿
No A is B.
Therefore,
Some A is B.
But no C is A.
Therefore,

THE LOGIC OF THE THIRD OPERATION

220

Some A is not non-B.
«
But every C is A.
Therefore,
‘
\J
11.8. Every A is non-B.
.
But no C is B.
U
Therefore,
11.9. Some A is not B.
| .
But every A is C.
Q
Therefore,
11.10. Some A is not B.
.
But no A is C.
* x.
Therefore,
12. Restate each of the following syllogisms in strictly logical form; then
classify’ each as to figure and mood, and evaluate each in the light of the
rules:
12.1. Since you must believe what you can plainly see, you must believe
me, because you can plainly sec me.
12.2. Labor unions should be abolished since all unjust organizations
should be abolished, and some labor unions are unjust organ­
izations.
12.3. John is a humble man, because he knows his own weaknesses, and
every humble man knows his own weaknesses.
12.4. No circle is a polygon, because no circle is quadrilateral, and some
polygon is quadrilateral.
12.5. No prudent politician will accept expensive gifts, and no one who
accepts expensive gifts can escape suspicion and criticism. Hence,
some prudent politicians escape suspicion and criticism.
12.6. Some men are not patriotic, because some men are not loyal
Americans, and loyal Americans are patriotic.
12.7. Capital punishment is forbidden by the Bible, because to inflict
capital punishment is to kill, and one of the commandments of
God recorded in the Bible forbids killing.
12.8. Some scientists are not rigorously logical, for all scientists are men,
and not all men are rigorously logical.
12.9. Because it is a book, Lady Chatterley’s Lover is one of the greatest
influences for spiritual good in the history of man; for books arc
most certainly one of the greatest influences for spiritual good in
the history of man.
12.10. Some drugs are habit-forming, and not all drugs are of medicinal
value. Hence, not all habit-forming substances are of medicinal
value.

11.7.

13. Restate in strictly logical form the syllogisms in the following passage
from St. Thomas Aquinas, classifying each as to figure and mood, and
evaluating each in the light of the rules for syllogistic procedure:
“From this, moreover, it is also clear that riches are not the highest
good for man.
“Indeed, riches are only desired for the sake of something else; the}'
provide no good of themselves but only when we use them, either for

THE RULES FOR THE CATEGORICAL SYLLOGISM

221

the maintenance of the body or some such use. Now, that which is the
highest good is desired for its own sake and not for the sake of something
else. Therefore, riches are not the highest good for man.
"Again, man’s highest good cannot lie in the possession or keeping of
things that chiefly benefit man through being spent. Now, riches are
chiefly valuable because they can be expended, for this is their use. So,
the possession of riches cannot be the highest good for man.
"Besides, an act of virtue is praiseworthy in so far as it comes closer
to felicity. Now, acts of liberality and magnificence, which have to do with
money, are more praiseworthy in a situation in which money is spent than
in one in which it is saved. So, it is from this fact that the names of
these virtues are derived. Therefore, the felicity of man docs not consist
in the possession of riches.
"Moreover, that object in whose attainment man’s highest good lies
must be better than man. But man is better than riches, for they are but
things subordinated to man’s use. Therefore, the highest good for man
does not lie in riches.
"Furthermore, man’s highest good is not subject to fortune, for things
subject to fortune come about independently of rational effort. But it
must be through reason that man will achieve his proper end. Of course,
fortune occupies an important place in the attainment of riches. Therefore,
human felicity is not founded on riches.
"Again, this becomes evident in the fact that riches are lost in an
involuntary manner, and also that the}' may accrue to eval men who must
fail to achieve the highest good and also that riches are unstable — and
for other reasons of this kind which may be gathered from the preceding
arguments.”
(St. Thomas Aquinas, Contra Gentiles, Bk. Ill, Chap. 30. Translation
by V. Bourke, On the Truth of the Catholic Faith, Bk. Ill, Pt. I
[Garden City, N. Y.: Image Books, 1956], pp. 116-117.) Reprinted
with permission of Doubleday and Co.

14. Look for examples of syllogistic argumentation in the textbooks and
reference works used in this course and in other courses. Restate each
syllogism in strictly logical form, and classify and evaluate each of them.

CHAPTER

XVII

The Hypothetical Syllogism

I. The Nature and Types of the Hypothetical Syllogism

In our investigation into the division of the proposition we spent a
good deal of time and effort on the hypothetical proposition. One reason
for this is the fact that hypothetical propositions can be employed as
major premises in a distinctive type of deductive argumentation called
the hypothetical syllogism. In this chapter we will investigate the nature
and types of hypothetical syllogism. Since our consideration here will
presume all that we saw previously in reference to the hypothetical
proposition, the student is advised to review carefully the chapter on the
hypothetical proposition.
A hypothetical proposition is a compound proposition which involves
a commitment to a nexus or link between its component propositions
without asserting the truth status of any one of them.
The component propositions in a hypothetical proposition are so
related that the truth status of any one of them has a bearing on the
truth status of the others. For this reason, once the intellect sees the truth
or falsity of any one of the components of a hypothetical proposition,
it is moved to affirm or deny the other element or elements. For example,
the truth of John is a Communist, in the hypothetical proposition If
John is a Communist, his children are in spiritual danger, forces assent to
the truth of the proposition John’s children are in spiritual danger. Here
we have a clear-cut case of hypothetical reasoning which can be expressed
in the following argument:

If John is a Communist, his children are in spiritual danger.
But John is a Communist.
Therefore, John’s children are in spiritual danger.
The simple conditional premise in this syllogism asserts that its elements
are so related that if the first is true the second must also be true. The
categorical premise asserts the truth of the first element of the conditional.
222

THE HYPOTHETICAL SYLLOGISM

223

The truth of the second element of the conditional follows as a cate­
gorical conclusion. This is an instance of a hypothetical syllogism. The
hypothetical premise functions as major premise and the categorical
premise functions as minor precisely because the categorical premise is
taken under the hypothetical and contracts it, in a sense, to the con­
clusion. There are, of course, no major, minor, and middle terms.
Hypothetical argument works in virtue of the affirmation or denial of
sequentially connected propositions and not, as is the case with the
categorical syllogism, in virtue of the triple identity of terms.
Every hypothetical syllogism can be resolved into three propositions:
the major premise, the minor premise, and the conclusion. The major
premise is a hypothetical proposition. The minor premise is a proposition
(either categorical or compound) which affirms or denies one or more of
the elements of the major. The conclusion is a proposition (either
categorical or compound) which affirms or denies the other elements of
the major. The simplest and most frequently employed form of hypo­
thetical argumentation has a two-membered hypothetical major, a cate­
gorical minor affirming or denying one of the two elements of the major,
and a categorical conclusion affirming or denying the other element. The
example suggested above is an instance of this relatively simple form
of hypothetical syllogism. Throughout this chapter we shall limit our­
selves for the most part to a consideration of this elementary' type of
hypothetical syllogism, reserving an investigation into syllogisms with
multimembered majors on the one hand and majors with compound
components on the other to a special section of the chapter.
We have seen that there are five types of hypothetical proposition,
namely, the simple conditional, the reciprocal conditional, the inclusive
alternative, the exclusive alternative, and the disjunctive. Each of these
can be used as a major premise in a hypothetical syllogism. Each specifies
a different type of syllogism with a different set of rules. We shall take
up each type of hypothetical syllogism separately, carefully discerning the
special set of rules proper to each.
II. The Figures and Moods of the Hypothetical Syllogism

There are theoretically two different figures for each type of hypothetical
syllogism, although some hypothetical syllogisms admit of a valid con­
clusion in only one of these two figures. The figure of the hypothetical
syllogism is determined by the minor premise. 'Hie minor premise either
affirms or denies an clement of the major. When the minor premise
affirms an element of the major, the syllogism is said to be in the first

224

THE LOGIC OF THE THIRD OPERATION

figure. Wien the minor denies an element of the major, the syllogism
is in the second figure. Thus, it is relatively easy to determine the figure
of any hypothetical syllogism.
However, a word of caution is necessary. Students are easily misled into
thinking that the figure of a hypothetical syllogism depends absolutely
upon the quality of the minor premise. Some think that an affirmative
minor as such constitutes the first figure and that a negative minor as
such constitutes the second figure. This is not the case. The minor
considered absolutely does not determine the figure of the syllogism.
It is the minor in its determinate reference to some element of the
major which determines the figure of the hypothetical syllogism. An
affirmative minor determines the first figure only if it represents an
affirmation of some one of the component parts of the major. An
affirmative minor bearing upon a negative element of the major constitutes
a denial of that element of the major and determines a second figure
syllogism. In similar fashion a negative minor gives the second figure only
if it bears upon an affirmative element of the major premise. A negative
minor gives the first figure whenever it bears upon a negative element
of the major. The affirmation of an affirmative proposition is affirmative.
Its denial is negative. The denial of a negative proposition is affirmative,
and its affirmation is negative.
A minor premise in a hypothetical syllogism sufficiently affirms an
element of the major only if it affirms that element in its entirety. Thus
an A element cannot be considered affirmed on the basis of a minor
premise asserting the truth of the corresponding I or A'. As a matter
of fact, the truth of the I or the A] is compatible with the truth of the
corresponding O, which is the contradiction of the A. The minor premise
in a proposed hypothetical syllogism can legitimately constitute a first
figure syllogism only if it represents a total affirmation of an element of
the major, precisely because anything less than this is compatible with
the contradiction of that element. There is, of course, no reason why
one could not conclude to an I proposition in place of an A so long
as the premises warranted the A, or to an O in place of an E. Here,
as in the case of the categorical syllogism, the inferior conclusion is con­
tained in the superior and is allowed wherever the superior is legitimate.
À minor premise in a second figure hypothetical syllogism functions
adequately so long as it represents at least a simple denial of an element
of the major premise. The minor premise in a second figure hypothetical
syllogism can be either the contradiction or the contrary of the element
it denies. However, when the premises in a hypothetical syllogism demand
a conclusion which denies an element of the major, that conclusion can

THE HYPOTHETICAL SYLLOGISM

225

only be the contradiction of the element it denies or the inferior of that
contradiction. The contrary of a proposition is more than a simple denial
because it denies not only that proposition but its inferior as well, and
the premises of a hypothetical proposition are able to generate nothing
more than the affirmation or simple denial of some element of the
major premise. The contrary can be used as a minor premise in so far
as it involves at least a simple denial. The contrat)' cannot be used as a
conclusion because it involves more than a simple denial.
The moods of the hypothetical syllogism depend upon the quality of
the component propositions in the hypothetical major premise. Two­
membered majors allow for four moods, while multimembered majors
allow for proportionately more. In a two-membered major both elements
can be affirmative, both negative, the first affirmative and the second
negative, or the first negative and the second affirmative. These four
combinations constitute the four moods possible for any hypothetical
syllogism with a two-membered major premise. The moods of the hypo­
thetical syllogism are not as significant as those of the categorical
syllogism, and little reference will be made to them from now on.
As an illustration of differences in figure and mood note the following
four hypothetical syllogisms. Each is valid. The first and third are simple
conditional syllogisms. The second is an exclusive alternative. 'The fourth
is a reciprocal conditional. The first two arc in the first figure. The last
two are in the second figure. The first three arc in different moods.

1. If the weather is bad, the game is off.
But the weather is bad.
Therefore, the game is off.
2. Either the weather is bad or the game is not off, but not both.
But the game is not off.
Therefore, the weather is not bad.
3. If the weather is not bad, the game is on.
But the game is not on.
Therefore, the weather is bad.
4. If and only if the weather is not bad, the game is on.
But the weather is bad.
Therefore, the game is not on.
III. The Simple Conditional Syllogism

The fundamental type of hypothetical proposition is the conditional.
Similarly, the fundamental type of hypothetical syllogism is the condi-

226

THE LOGIC OF THE THIRD OPERATION

tional. The simple conditional syllogism is a hypothetical syllogism with a
simple conditional major premise and a minor premise which affirms or
denies one of the elements of the major. The conclusion of the simple
conditional syllogism affirms or denies the other element of the major
according to the demands of the special rules applying to the simple
conditional syllogism:

An affirmation of the antecedent requires an affirmation of the
consequent, and a denial of the consequent requires a denial of
the antecedent.

j No conclusion follows from a denial of the antecedent or from an
affirmation of the consequent.
Tire reason for these rules is found in the nature of the simple con­
ditional proposition. Tlic simple conditional proposition is defined in
terms of a sequential nexus between its antecedent and its consequent.
It asserts that its antecedent cannot be true without its consequent being
true. It does not assert any reciprocal sequential nexus between its
consequent and its antecedent. Thus the simple conditional proposition
demands that its consequent be true whenever its antecedent is true. But
it allows for the truth of its consequent even if its antecedent is false.
This means that nothing follows either from the affirmation of its
consequent or from the denial of its antecedent. Tire consequent can be
true with a true or false antecedent, and the antecedent can be false with
a true or false consequent. The falsity of the antecedent, however, follows
from the falsity of the consequent. If the consequent could be false and
the antecedent true, then the simple conditional could not assert, as it
does, that the truth of the antecedent necessitates the truth of the
consequent.
'Hie first two of the following four syllogisms are invalid because they
break the rules for the simple conditional syllogism. The last two are the
only valid syllogisms which could be constructed with the major premise
which has been used as common to all four.

1. If Freud’s psychological theory is completely correct, then man
is not a morally responsible agent.
But man is not a morally responsible agent.
Therefore, Freud’s psychological theory is completely correct.
(Invalid; Nothing follows from the affirmation of the consequent
in a simple conditional syllogism.)
2. If Freud’s psychological theory is completely correct, then man is
not a morally responsible agent.

THE HYPOTHETICAL SYLLOGISM

227

But Freud’s psychological theory is not completely correct.
Therefore, a man is a morally responsible agent.
(Invalid: Nothing follows from the denial of the antecedent in
a simple conditional syllogism.)
3. If Freud’s psychological theory is completely correct, then man
is not a morally responsible agent.
But Freud’s psychological theory is completely correct.
Therefore, man is not a morally responsible agent.
(Valid: The affirmation of the consequent does follow from the
affirmation of the antecedent.)
4. If Freud’s psychological theory is completely correct, then man
is not a morally responsible agent.
But man is a morally responsible agent.
Therefore, Freud’s psychological theory is not completely correct.
(Valid: The denial of the antecedent does follow from the denial
of the consequent.)
IV.

The Case of the "Disguised" Categorical Major

In the chapter on the compound proposition we spoke about proposi­
tions that could be understood as categorical although expressed gram­
matically in hypothetical form. They should, we said, be taken as
categorical rather than hypothetical, because they function better as major
premises in syllogisms which follow the rules of categorical rather than
hypothetical argumentation. Take the proposition If anyone is disposed to
act unjustly, he will make an uncertain friend. This can be coupled with
an A, A1, or I minor affirming its “antecedent” and yielding respectively
an A, A’, or I conclusion affirming its “consequent.” Thus each of the
following is a valid argumentation:

çt '

1. If anyone is disposed to act unjustly, he will make an uncertain
friend.
But every’ man is disposed to act unjustly.
Therefore, every man will make an uncertain friend.
2. If anyone is disposed to act unjustly, he will make an uncertain
friend.
But some men are disposed to act unjustly.
Therefore, some men will make uncertain friends.
3. If anyone is disposed to act unjustly, he will make an uncertain
friend.
But John is disposed to act unjustly.
Therefore, John will make an uncertain friend.

228

THE LOGIC OF THE THIRD OPERATION

However, hypothetical argument requires the minor in a first figure
syllogism to affirm an element of the major in its entirety. Thus syl­
logisms which validly admit A, A', and I propositions as legitimate minor
premises (when the antecedent of the major is not clearly an I proposi­
tion) for the same major premise cannot without difficulty be taken as
hypothetical argumentations. Tire three syllogisms above can be better
understood to work if their common major premise is understood as
Everyone who is disposed to act unjustly makes an uncertain friend
rather than the corresponding conditional. With this as their major
premise the three syllogisms can be readily seen to be respectively
legitimate AAA, AH, and AA'A' categorical syllogisms in the first figure.
The rules for the categorical syllogism clearly allow for the passage from
the universal major to the particular or singular conclusion by way of a
particular or singular minor. The rules for the hypothetical syllogism,
however, demand total affirmation for any minor which is taken as the
affirmation of some element of the major. The point is not that “If
anyone is disposed to act unjustly, he will make an uncertain friend”
cannot be taken as a hypothetical proposition. The point is that it can
be understood as a categorical proposition and that it is ordinarily pref­
erable to take it in this way.
In our chapter on the compound proposition we noted that these
“disguised” catcgoricals can take the form of alternatives and disjunctives
as well as conditionals. There are apparent alternative and disjunctive
syllogisms as well as apparent conditional syllogisms which are better
understood as expressions of categorical syllogisms. Each of the following
syllogisms might be more clearly expressed as categorical syllogisms:

J

1. First expression:
Either a man is virtuous or he is not truly happy.
But John is truly happy.
Therefore, John is virtuous.
Second expression:
Every man who is truly happy is virtuous.
But John is a man who is truly happy.
Therefore, John is virtuous.
2. First expression:
One cannot be both a miser and a spendthrift.
But some men are misers.
Therefore, some men are not spendthrifts.
Second expression:

229

THE HYPOTHETICAL SYLLOGISM

No one who is a miser is a spendthrift.
But some men are misers.
Therefore, some men are not spendthrifts.

Before proceeding to the second type of hypothetical syllogism it
should be of value to show how simple conditional syllogisms can be
represented symbolically. Here as elsewhere, when the concern is one of
formal logic, symbols can lx: of great assistance. The student is referred
to the chapter on hypothetical propositions, where the symbolic repre­
sentation of the hypothetical is explained and put into practice. It is
important to recall that the literal symbols in the hypothetical proposition
stand for categorical propositions and not for terms as they do in the
categorical proposition itself. The following simple conditional syllogism
can be accurately and conveniently symbolized in either of the ways
illustrated:
If children are characteristically free of discipline in the home,
juvenile delinquency can be expected to be the rule.
But children arc characteristically free of discipline in the home.
Therefore, juvenile delinquency can be expected to be the rule.

a->b
But a
Therefore, b

or

[(a->b) • a]->b

V. The Reciprocal Conditional Syllogism

The reciprocal conditional syllogism is a hypothetical syllogism with
a reciprocal conditional major premise and a minor premise which affirms
or denies one of the elements of the major. The conclusion of the
reciprocal conditional syllogism affirms or denies the other element of
the major according to the demands of the special rule applying to the
reciprocal conditional syllogism:

An affirmation of either element requires the affirmation of the
other element, and a denial of either element requires the denial
of the other.
The reciprocal conditional proposition by definition involves a reciprocal
conditional connection between its elements. Its antecedent cannot be
tme without its consequent being true, and its consequent cannot be
true without its antecedent being true. Thus, if either one is true, the
other must be true, and if either one is false, the other must be false.

230

THE LOGIC OF THE THIRD OPERATION

Neither element can be true while the other is false because the truth
of either implies the truth of the other. Neither can be false while the
other is true; otherwise the truth of cither would not be, as the reciprocal
conditional asserts, incompatible with the falsity of the other. The differ­
ence between the simple conditional proposition and the reciprocal con­
ditional proposition is that the former can have a valid conclusion only
if its antecedent is affirmed or its consequent denied whereas the latter
gives valid conclusions if either of its elements — antecedent or con­
sequent — is affirmed or denied.
The following four syllogisms illustrate valid reciprocal conditional
syllogisms. Since they share a common major premise they illustrate the
four different conclusions virtually contained in any reciprocal conditional
proposition.

1. If, and only if, the invasion is successful, the reputation of the
commanding general is secured.
But the invasion is successful.
Therefore, the reputation of the commanding general is secured.
(Valid: The affirmation of the consequent does follow from the
affirmation of the antecedent in a reciprocal conditional syl­
logism.)

2. If, and only if, the invasion is successful, the reputation of the
commanding general is secured.
But the reputation of the commanding general is secured.
Therefore, the invasion is successful.
(Valid: Tire affirmation of tire antecedent does follow from the
affirmation of the consequent.)
3. If, and only if, the invasion is successful, the reputation of the
commanding general is secured.
But the invasion is not successful.
Therefore, the reputation of the commanding general is not
secured.
(Valid: The denial of the consequent does follow from the denial
of the antecedent.)
4. If, and only if, the invasion is successful, the reputation of the
commanding general is secured.
But the reputation of the commanding general is not secured.
Therefore, the invasion is not successful.
( Valid: The denial of the antecedent does follow from the denial
of the consequent.)

231

THE HYPOTHETICAL SYLLOGISM

Symbolically these four syllogisms would appear as:

1. a<->b
But a
Therefore,
2. a*-*b
But b
Therefore,
3. a*-»b
But —a
Therefore,
4. a<->b
But — b
Therefore,

or

[(a<->b) *a]->b

or

[(a«-»b) • b]->a

or

[(a*->b) • —a]->—b

or

[(a<-»b) • — b]->—a

b

a

—b

—a

VI. The Inclusive Alternative Syllogism

The inclusive alternative syllogism is a hypothetical syllogism with an
inclusive alternative major premise and a minor premise which denies one
of the elements of the major. Its conclusion affirms the other element of
the major according to the special rules applying to the inclusive alter­
native syllogism:

The affirmation of either clement yields no conclusion.

J The denial of either element necessitates the affirmation of the
other.
The nexus asserted between the elements of an inclusive alternative
proposition is such that if one is false the other must be true. It asserts
no more than this, allowing for the possibility that the elements may
both be true. Thus the affirmation of either element in tlie minor leaves
the conclusion in doubt, since the other element may or may not be true.
But the denial of one of the elements of the major yields a conclusion
which affirms the other, for the elements cannot both be false. Hence,
the inclusive alternative syllogism allows for a conclusion in the second
figure, but it allows for no conclusion in the first figure.
Each of the following four syllogisms is an example of an inclusive
alternative syllogism. The first two are in the first figure and hence are
invalid. The last two are in the second figure, and they are valid.
1. Either Aristotle and Plato were good philosophers or modern
Scholasticism is academically unsound.

THE LOGIC OF THE THIRD OPERATION

232

But Aristotle and Plato were good philosophers.
Therefore, modern Scholasticism is not academically unsound.
(Invalid: No conclusion follows in the first figure of an inclusive
alternative syllogism.)
2. Either Aristotle and Plato were good philosophers or modern
Scholasticism is academically unsound.
But modem Scholasticism is academically unsound.
Therefore, Aristotle and Plato were not good philosophers.
(Invalid: No conclusion follows in the first figure.)
3. Either Aristotle and Plato were good philosophers or modern
Scholasticism is academically unsound.
But Aristotle and Plato were not good philosophers.
Therefore, modem Scholasticism is academically unsound.
(Valid: 'rhe denial of one element necessitates the affirmation of
the other.)
4. Either Aristotle and Plato were good philosophers or modern
Scholasticism is academically unsound.
But modern Scholasticism is not academically unsound.
Therefore, Aristotle and Plato were good philosophers.
(Valid: The denial of either element necessitates the affirmation
of the other.)

These four syllogisms can be symbolically represented as follows
1. a\/b
But a
Therefore,
2. a\/b
But b
Therefore,
3. a\/b
But —a
Therefore,
4. a\/b
But —b
Therefore,

or

[(a\/b) • a]->—b

or

[(a\/b) • b]->—a

or

[(a\/b) • -a]->b

or

[(a\/b) • — b]—>a

—b

—a

b

a

VII. The Exclusive Alternative Syllogism

The exclusive alternative syllogism is a hypothetical syllogism with an
exclusive alternative major premise and a minor premise which affirms or

THE HYPOTHETICAL SYLLOGISM

233

denies one of the elements of the major. The conclusion of the exclusive
alternative syllogism denies or affirms the other element of the major
according to the special rules for the exclusive alternative syllogism.

' The affirmation of cither element necessitates the denial of the
other.

. The denial of either element necessitates the affirmation of the
other.
In an exclusive alternative proposition the connection between its
elements is such that one must be true and any other false. Thus in a
two-membered exclusive alternative the falsity of one element demands
the truth of the other, and the truth of one demands the falsity of the
other. Jf the minor affirms an element of the major, the conclusion
must deny the other. If the minor denies an clement of the major, the
conclusion must affirm the othfiL
Each of the following four syllogisms is an example of a valid exclusive
alternative syllogism. The first two arc in the first figure, and the last
two are in the second figure. Although the inclusive alternative syllogism
works only in the second figure, the exclusive alternative allows for a
conclusion in both figures.

'

1. Either logic is an indispensable tool for philosophy or philosophy
is not rigorously scientific, but not both.
But logic is an indispensable tool for philosophy.
Therefore, philosophy is rigorously scientific.
(Valid: The affirmation of one element necessitates the denial
of the other.)
2. Either logic is an indispensable tool for philosophy or philosophy
is not rigorously scientific, but not both.
But philosophy is not rigorously scientific.
Therefore, logic is not an indispensable tool for philosophy.
(Valid: The affirmation of one element necessitates the denial of
the other.)

3. Either logic is an indispensable tool for philosophy, or philosophy
is not rigorously scientific, but not both.
But logic is not an indispensable tool for philosophy.
Therefore, philosophy is not rigorously scientific.
(Valid: The denial of one element necessitates the affirmation
of the other.)

234

THE LOGIC OF THE THIRD OPERATION

4. Either logic is an indispensable tool for philosophy, or philosophy
is not rigorously scientific, but not both.
But philosophy is rigorously scientific.
Therefore, logic is an indispensable tool for philosophy.
(Valid: The denial of one element necessitates the affirmation
of the other.)

These four syllogisms can be presented symbolically in this fashion:
a/\—b
But a
Therefore,
2. a/\—b
But — b
Therefore,
3. a/\—b
But —a
Therefore,
4. aZ\—b
But b
Therefore,
1.

or

[(a/\—b) ■ a]->b

or

[(a/\—b) ■ — b]a

or

[(a/\—b) • —a]->—b

or

[(a/\—b) • b]->a

h

—a

—b

a

VIII. The Disjunctive Syllogism

The
major
major.
of the

disjunctive syllogism is a hypothetical syllogism with a disjunctive
premise and a minor which affirms one of the elements of the
The conclusion of a disjunctive syllogism denies the other element
major according to the special rules for the disjunctive syllogism:

The affirmation of cither element requires the denial of the other.
The denial of either element yields no conclusion.
The nexus between the component propositions of a disjunction is such
that one at least must be false. This is the extent of the disjunctive com­
mitment. It alloua for the possibility that all of its components might be
false. If any element is affirmed the other must be denied, for one at
least must be false. Nothing can be concluded from the falsity of any
element, since all can be false, but need not be.
Whereas the inclusive alternative works only in the second figure, the
disjunctive syllogism works only in the first figure. The exclusive alterna­
tive, we saw, works in both the first and second figures. It is interesting
to note that a two-membered exclusive alternative proposition can be

THE HYPOTHETICAL SYLLOGISM

235

resolved into a copulative proposition composed of corresponding in­
clusive alternative and disjunctive propositions. Thus, a/\b = (a\/b) •
n — (a - b). Along the same line, recall that the simple conditional syllo­
gism works sometimes in the first figure and sometimes in the second figure,
whereas the reciprocal conditional always works in both figures. It can
do so because it can be resolved into a copulative proposition composed of
simple conditionals with inverted antecedents and consequents. Thus:
a^b = (a—>b) • (b->a).
As an illustration of the disjunctive syllogism consider these four
examples. The first two are in the first figure, and they are valid. The
last two are in the second figure, and they are, accordingly, invalid.

1. It cannot be true both that Communism is a true philosophy and
that Christianity is a true religion.
But Communism is a true philosophy.
Therefore, Christianity is not a true religion.
(Valid.- The affirmation of either element necessitates the denial
of the other.)
2. It cannot be true both that Communism is a true philosophy and
that Christianity is a true religion.
But Christianity is a true religion.
Therefore, Communism is not a true philosophy.
(Valid: The affirmation of either element necessitates the denial
of the other.)
3. It cannot be true both that Communism is a true philosophy and
that Christianity is a true religion.
But Communism is not a true philosophy.
Therefore, Christianity is a true religion.
(Invalid: Nothing follows from the denial of cither element.)
4. It cannot be true both that Communism is a true philosophy and
that Christianity is a true religion.
But Christianity is not a true religion.
Therefore, Communism is a true philosophy.
(Invalid: Nothing follows from the denial of either element)
Symbolically these syllogisms can be rendered as follows:
1. n —(a b)
or
[n — (a • b) • a]-> — b
But a
Therefore, —b
2. n — ( a • b )
or
[n — ( a • b ) • b] -» —a
But b
Therefore, —a

236

THE LOGIC OF THE THIRD OPERATION
3. n — (a•b)

[n — (a • b) • —a]->b

But —a
Therefore, b
4. n — (a • b)
But — b
Tlicrefore, a

[n — (a • b) • — b]->a

IX. The Reduction of Alternative and Disjunctive Syllogisms
to the Conditional Syllogism

The conditional proposition, as we know, is the fundamental type of
hypothetical proposition; to it alternative and disjunctive propositions
can be reduced. Consequently, the conditional syllogism is the funda­
mental type of hypothetical syllogism and alternative and disjunctive
I syllogisms can be reduced to equivalent conditional syllogisms.
The inclusive alternative proposition can be reduced to an equivalent
simple conditional. Thus, a\/b = —a->b. Two valid syllogisms can be
constructed with a\/b as the major premise. These are:
1. a\7b
But —a
Therefore, b

2. a\/b
But — b
Therefore, a

If the conditional equivalent of the major premise is combined with
the same minor two equivalent simple conditional syllogisms can be
formed. These are:

1. — a-»b
But —a
Therefore, b

2. —a->b
But — b
Therefore, a

The disjunctive proposition is also reduced to a corresponding simple
conditional. We saw n — (a • b) e a->—b. The two valid disjunctive
syllogisms which can be constructed with n — (a • b) as the major premise
are:
2. u — (a ■ b)
1. n — (a • b)
But b
But a
Therefore, —a
Therefore, —b
The simple conditional syllogisms to which they can be reduced are:
1. a->— b
But a
Therefore, — b

2. a->—b
But b
Therefore, —a

237

THE HYPOTHETICAL SYLLOGISM

The exclusive alternative proposition is reduced to_a reciprocal con­
ditional. Thus, aAb«ae-b; a/\b admits of four valid conclusions
in syllogisms which can be reduced to valid reciprocal conditional syl­
logisms with a<->—b as the major premise. The four exclusive alternatives
which can be constructed from a/\b and the four reciprocal conditionals
to which they can be reduced are as follows:
2. a/\b
3. aAb
1. a/\b
But —a
But
b
But a
Therefore, —a Therefore, b
Therefore, — b
3. ae-b
2. a<->—b
1. a<-»—b
But
b
But —a
But a
—
a
Therefore,
b
Therefore,
Therefore, — b

4. a/\b
But —-b
Therefore, a
4. a<->—b
But —b
Therefore, a

X. Hypothetical Syllogisms With a Multimembered Major

Thus far we have considered only hypothetical syllogisms with a two
membered major premise. Alternative and disjunctive propositions, how­
ever, can have more than two component propositions. Hypothetical
syllogisms with multimembered majors are evidently more difficult to
evaluate than those with only a two-membered major. Nevertheless, the
rules we have been discussing apply proportionally to them if properly
modified. Here we will see in what these modifications consist.
The Alternative Inclusive Syllogism
The alternative inclusive proposition states that at least one of several
component propositions must be true in the light of the others. Any or
all of the others may be true or false. Tims, if any element of a multi­
membered alternative inclusive syllogism is affirmed, the others are
collectively and singly in doubt. Nothing at all follows in the first figure.
But, if one of the elements is denied, a conclusion follows which asserts,
in another alternative inclusive proposition, the truth of at least one of
the others, e.g.:

a\/b\/c\/d
But —a
Therefore, b\/c\/d
When we deny several alternatives we must conclude to an inclusive
alternative which asserts the truth of at least one of those remaining.

238

THE LOGIC OF THE THIRD OPERATION

If but one remains, a categorical proposition follows in which this one
is affirmed, e.g.:
1.

2.

a\/b\/c\/d
But —a • — b
Therefore, c\Zd
a\/b\/c\/d
But —a • — b • —c
Therefore, d

The Exclusive Alternative Syllogism
The exclusive alternative proposition asserts hypothetically that one of
a number of component propositions is true while the others cannot be
true. Thus, in a multimembered type of exclusive alternative syllogism,
if any element of the major is affirmed, each of the others must be
denied, e.g.:
But a
Therefore, — b • —c • —d
If one element is denied, one, and only one, of the remaining is true,
and this is asserted in an exclusive alternative proposition composed of
the remaining elements, e.g.:

a/\b/\c/\d
But —a
Therefore, b/\c/\d
If several elements are denied, there are again two possibilities. If but
one remains, this must follow as true. If there are several remaining, an
exclusive alternative, which asserts that one and only one of these must
be true, follows as a conclusion, e.g.:
1. a/\b/\c/\d

But —a • — b • —c
Therefore, d
2. a/\b/\c/\d
But —a • — b
Therefore, c/\d

The Disjunctive Syllogism
The disjunctive proposition is a hypothetical assertion that at least
one of a number of component propositions is false. Thus, if one of the

THE HYPOTHETICAL SYLLOGISM

239

elements of a multimcmbered disjunctive proposition is affirmed, a dis­
junctive conclusion which asserts that one of the others must be false
follows, e.g.:

n — (a • b • c • d)
But a
Therefore, n — ( b • c • d )
If several arc affirmed respectively as true, there arc two possibilities. If
only one remains, it must be false. If several remain, then a disjunctive
proposition which asserts that at least one is false follows as a conclu­
sion, e.g.:

1. n — (a • b • c • d)
But a • b • c
llicrcfore, —d
2. n — (a • b • c • d)
But a • b
Therefore, n — ( c • d )
XI. Major Premises With Compound Components

We noted earlier that the less complex compound propositions involve
categorical elements and that the more complex involve component
propositions which are themselves compound: a-»b is an example of a
relatively incomplex compound proposition; (a • b)-> (c/\d) is an exam­
ple of a more complex compound proposition. The first is a simple con­
ditional with categorical components. The second is a simple conditional
whose antecedent is a copulative proposition and whose consequent is an
exclusive alternative proposition. a->b can stand symbolically for almost
all the simple conditionals we have used as examples so far, for in most
cases wc have limited ourselves to conditionals with categorical com­
ponents. (a-b)-»(c/\d) could be used symbolically to stand for a
proposition such as If the Communists arc militarily powerful and the
free world is militarily unprepared, then either the free world will fall
to the Communists or God will inten'ene on the side of the free world,
but not both.
We have seen that conditional, alternative, and disjunctive propositions
with categorical components can serve as major premises in hypothetical
syllogisms. Similarly, hypothetical propositions with compound compo­
nents can be used as majors in hypothetical argumentation. When this is
the case the minor premise itself will ordinarily be a compound proposi­

240

THE LOGIC OF THE THIRD OPERATION

tion affirming or denying a compound element of the hypothetical
major, e.g.:
(a • b)-»(c/\d)
But —(c/\d)
Therefore, —(a'b)

This is a simple conditional syllogism. The antecedent of its major
premise is a copulative proposition and its consequent is an exclusive
alternative proposition. Since the minor denies the consequent of the
major, the conclusion must deny its antecedent. To illustrate how complex
the situation can become with a hypothetical major premise as complex
as this one, let us note various additional wav's in which the consequent
might be denied. If we refer to the truth table for hypotheticals, which
cannot be used to confirm hypotheticals taken strictly but sometimes can
indicate disconfirmation, we note that c • d and —c • —d both imply
—(c/\d). Hence, there arc at least three other minors which can yield
the same conclusion as — (c/\d). These are c-d, —c- —d, and (c-d)
/\(-C-— d). Nor arc these the only ones. cAds-c^d, and
(—C«->d) -> (—c-»d). Thus, — (c/\d)
— (— c<->d), and — (—c—>d) ->
_(—c<->d). Hence, both — (—c<-»d) and — (—c-»d) can also be used as
minor premises to yield the same conclusion as —(c/\d).
There is no need for us to venture further into the possibilities of
increasing complexity. We have seen enough to know that hypothetica]
propositions, and consequently hypothetical syllogisms, can become ex­
tremely complex. However, so long as the basic nature of the hypotheticals
is understood and the fundamental rules for hypothetical discourse ac­
curately enforced, there is no reason why hypothetical argumentation
should ever become unreasonable.
Exercise XVII

1. Define: hypothetical proposition, hypothetical syllogism, simple condi­
tional syllogism, reciprocal conditional syllogism, inclusive alternative
syllogism, exclusive alternative syllogism, disjunctive syllogism.
2. What is the distinguishing feature of the hypothetical proposition? How
many types of hypothetical proposition are there? How do they differ from
one another?
3. What is the rationale, generally speaking, of hypothetical argumentation?
4. How are the figures and the moods of the hypothetical syllogism
determined?
5. Why is it legitimate to conclude from the affirmation of the consequent
in a reciprocal conditional syllogism, but not in a simple conditional
syllogism?

THE HYPOTHETICAL SYLLOGISM

241

6. Why is it illegitimate to proceed from the contrary of the consequent
in a simple conditional syllogism to the contrary of its antecedent, when
it is legitimate to proceed from the contradiction of the consequent to
the contradiction of the antecedent?
7. What type of hypothetical syllogism can be considered to have a cate­
gorical equivalent? Illustrate.

8. In what sense is the exclusive alternative syllogism a composite of cor­
responding inclusive alternative and disjunctive syllogisms?
9. Classify each of the following hypothetical syllogisms according to type
and figure. Check each one for validity.
9.1. If a woman is beautiful, she will have many admirers; but Susan
has many admirers; therefore, Susan must be beautiful.
9.2. Either a plane figure has three angles equal to two right angles or
it is not a triangle; but no rectangle has three angles equal to two
right angles; therefore, no rectangle is a triangle.
93. Unless some men are students, no men are teachers; but some
men are not teachers; therefore, some men are not students.
9.4. If color is the specific difference of redness, then not every essential
predicate is categorically substance; but color is not the specific
difference of redness; therefore, every essential predicate is cate­
gorically substance.
9.5. Either a liquid boils at 100 degrees centigrade or it is not water;
but this liquid boils at 100 degrees centigrade; therefore, this
liquid is water.
9.6. Only if a man is intelligent is he a good leader; but John is intel­
ligent; therefore, John is a good leader.
9.7. No universal is both the genus and the difference of the same
inferior; but biped is not the genus of the inferior man; therefore,
biped is the difference of man.
9.8. If and only if a being is alive, it has a soul; but even' maple tree is
alive; therefore, every maple tree has a soul.
9.9. Either the principle of contradiction holds or no propositions are
certain; but some propositions are certain; therefore, the principle
of contradiction holds.
9.10. If the citizens of a country are generally of low moral caliber, that
country is in danger of disintegration from within; but the citizens
of Ireland are not generally of low moral caliber; therefore, Ireland
is not in danger of disintegration from within.

10. Once again classify each of the following hypothetical syllogisms according
to type and figure, and then check each one for validity':
10.1. a\/— b
But b
Therefore, —a
10.2. —a<->b
But a
Therefore, b

242

THE LOGIC OF THE THIRD OPERATION

n — (a---- b)
But a
Therefore, —b
10.4. a\/b\/c
But —a —b
Therefore, c
10.5. a/\—b
But b
Therefore, a
10.6. a/\—b/\—c
But —a • b
Therefore, c
10.7. —a<->b
But a
Therefore, b
10.8. n — (—a--- b)
But —b
Therefore, a
10.9. n—(a-b’C)
But a
Therefore, n — (b • c)
10.10. —a/\b
But —a
Therefore, —b
(a ■ b )->c] • —(a • b)} -> —c
10.11.
10.12. —a<->b) —bi -> a
(a/\b) ->c] • —e} -> —(a/\b)
10.13.
10.14. a\/b\/—c\/—d) • (c • d)]->a • b
10.15. n —(—a • —b) • a] -> —b
11. Practice the reduction of alternative and disjunctive syllogisms to the
conditional by reducing the alternatives and disjunctives in the preceding
two questions to equivalent conditionals.

10.3.

CHAPTER

XVIII

Complex Syllogistic Patterns

I. Complex Syllogisms

Thus far our study of argument has been limited to a generally uncom­
plicated form of syllogism. We have, to be sure, studied many different
syllogistic patterns, but each has involved only a major premise and a
minor premise, together yielding a conclusion. However, we would be
seriously misled were we to conclude that every syllogism must be
exclusively of one type, with one figure and one mood, and that each
must be composed of one conclusion following from two and only two
premises. The student will find that many syllogisms are relatively
complex, representing a more or less complicated combination of several
of the syllogisms of the type wc have so far studied. It is true that the
logical force of these complex syllogisms stems from the simpler syllo­
gisms from which they are constructed. Thus, we have already investigated
all that we need fundamentally in order to evaluate complex syllogisms.
Nevertheless, it is helpful to consider at least some of the possible complex
syllogistic patterns. Otherwise they can prove to be stumbling blocks
for a student when he meets them in other courses. Some complex
syllogistic patterns are rarely encountered. Others are almost common­
place. We shall consider the most significant of the latter in this chapter.
II. The Abbreviated Syllogism

Before taking up the most significant types of complex syllogism, let us
consider what is spoken of as the abbreviated syllogism. Whatever an
abbreviated syllogism is, certainly it is not one of the types of complex
syllogism spoken about in the preceding section. There we indicated that
by a complex syllogism we refer to a more or less complicated syllogistic
pattern involving a combination of fundamentally simple or incomplex
categorical and/or hypothetical syllogisms. Thus, the types of complex
syllogism are always expanded — never abbreviated — syllogisms.
243

244

THE LOGIC OF THE THIRD OPERATION

Some logicians speak of the abbreviated syllogism as a special type of
syllogism. They call it an enthymeme and define it as a syllogism in
which one proposition is not expressed but nevertheless understood. This
calls for some comment. First of all, the term "syllogism” can be used for
both the mental deductive argumentation and its oral or written expres­
sion. It is possible to omit the oral or written expression of a proposition
which is nonetheless understood. But it is impossible to understand a
proposition and at the same time to fail mentally to express it, for it is
understood precisely in so far as it is mentally expressed. Thus, we can
legitimately speak of an abbreviated syllogism only in reference to the oral
or written expression of a logical entity. When we talk about the special
types of syllogism in this chapter we are concerned with more than
verbal expression.
The practice of some logicians to speak of the abbreviated syllogism
as a special type of syllogism and to call it an enthymeme calls for a
second comment. The term "enthymeme” is borrowed from Aristotle,
who used it for the syllogism which is characteristic of rhetoric. It is not
our purpose here to discuss the typical characteristics of the rhetorical syl­
logism. It is enough to say that the difference between the syllogism of the
rhetorician and any other syllogism is not a difference in form, but rather
a difference in matter. The rhetorician argues from premises which
express probabilities and signs, but unless he argues from at least two
premises he can come to no conclusion. It is true that in orally expressing
himself the rhetorician frequently uses an abbreviated form. More often
than not he may leave one or other of his premises unexpressed (orally,
that is, but not mentally). It is because of this that the term "enthymeme”
has become associated with the notion of the abbreviated syllogism, and
through this association it has come to be used by some generally to refer
to all abbreviated syllogisms. We should note that to use the term in this
way is not to use it as Aristotle used it. What is more important, we
must note that it is meaningless to speak at all of an abbreviated
syllogism except in the context simply of the grammatical expression of
a logical expression which defies mental abbreviation.
Although there is no such thing as an abbreviated syllogism from the
strictly logical point of view, we must be prepared to meet abbreviated
grammatical expressions of syllogisms, and we should be able to use them
ourselves. Usually a premise is the element not expressed in so-called
abbreviated syllogisms, although at times the element not expressed is the
conclusion. In any event these syllogisms can be evaluated only after the
unexpressed premise or conclusion is supplied. Then, depending on the
type of syllogism involved, the resulting syllogism must be evaluated in

COMPLEX SYLLOGISTIC PATTERNS

245

the light of the appropriate rules for syllogistic procedure. If the syllogism,
with the unexpressed premise or conclusion supplied, measures up to the
rules appropriate to it, it is valid. If it fails to measure up, it is invalid.
To argue that no plant is able to suffer because ever}' plant is nonsentient is to argue validly, so long as the unexpressed premise which is
understood is Nothing which is nonsentient can suffer. Fully expressed,
this is a valid first figure categorical syllogism concluding directly.
Nothing which is nonsentient can suffer.
But every plant is nonsentient.
Therefore, no plant can suffer.
(Valid.)

However, to argue that the number seven is odd because it is a whole
number is invalid, if the unexpressed premise is If a number is odd, it is
a whole number. Fully expressed, this is a first figure simple conditional
syllogism which is invalid because it attempts to go from the affirmation
of the consequent to the affirmation of the antecedent.
If a number is odd, it is a whole number.
But the number seven is a whole number.
Therefore, the number seven is an odd number.
(Invalid: One cannot affirm the consequent in order to affirm
the antecedent.)

‘JJvery creature is dependent on the Creator for existence and some
creatures are men” is a valid categorical syllogism concluding directly in
the third figure, if the unexpressed conclusion is understood to be Some
men are dependent on the Creator for existence. It is an invalid syl­
logism with an illicit minor term if the conclusion which is understood is
Even' man is dependent on the Creator for existence.
1. Every creature is dependent on the Creator for existence.
But some creatures are men.
Therefore, some men arc dependent on tlie Creator for existence.
(Valid.)
2. Every creature is dependent on the Creator for existence.
But some creatures are men.
Therefore, every man is dependent on the Creator for existence.
(Invalid: The syllogism has an illicit minor term.)
III. The Syllogism With a Justification for Its Premises

Perhaps the expanded syllogism most frequently used is one (either
categorical or hypothetical) which adds a justification to one or both of

246

THE LOGIC OF THE THIRD OPERATION

its premises. This occurs whenever a premise is expressed precisely as a
conclusion or as the main component in a causal proposition. To illustrate:
1. No polygon is a circle, because every polygon is rectilinear, and
no circle is rectilinear.
But every square is a polygon.
Therefore, no square is a circle.
2. All concepts identical in comprehension are identical in extension.
But man and rational animal are identical in comprehension since
the latter is the essential definition of the former.
Therefore, man and rational animal are identical in extension.
3. If any A proposition has the definition of its subject for its
predicate, that A proposition can be effectively simply converted,
because the definition is convertible with the defined.
But Every man is a rational animal is an A proposition with a
predicate which is the definition of its subject, since animal is
the proximate genus of man and rational is the specific differ­
ence of man.
Therefore, Every man is a rational animal can be effectively simply
converted.
Logicians have called this type of expanded syllogism an epicheirema,
although once again we should note that this is not the way in which
Aristotle used this term. Aristotle used the term to refer to a dialectical
syllogism proceeding from probable premises. However, since this is a
legitimate type of syllogism, we shall speak of it as an epicheirema, for
want of a better term, even though this is not the way in which Aristotle
used this term. Thus understood, the epicheirema is basically a simple
syllogism involving a major premise and a minor premise, with some
justification expressed for one or both of its premises. The epicheirema
can be checked for validity by separately checking the sequence of its
basic syllogistic pattern on the one hand and the sequence between its
premises and their antecedents on the other. The epicheirema is valid
only if there is no invalid sequence in its total structure.
The following syllogism (1) is a valid epicheirema. It has a basic
syllogistic structure (2) which is a valid first figure categorical syllogism
concluding directly. It has a major premise (3) which is the conclusion
of a valid second figure categorical syllogism concluding indirectly.

1. No polygon is a circle, because every polygon is rectilinear, and
no circle is rectilinear.
But every square is a polygon.

COMPLEX SYLLOGISTIC PATTERNS

247

Therefore, no square is a circle.
(Valid epicheirema.)
2. No polygon is a circle.
But every square is a polygon.
Therefore, no square is a circle.
(Valid first figure categorical syllogism concluding directly.)
J 3. Every polygon is rectilinear.
But no circle is rectilinear.
Therefore, no polygon is a circle.
. (Valid second figure categorical syllogism concluding indirectly.)

'Ilie following is an epicheirema with a valid basic categorically syl­
logistic pattern in the first figure concluding directly. Itjs invalid because
it has a major premise which is the conclusion of an invalid second figure
categorical syllogism with an unextended middle term.
Every man is capable of speech, because everyone capable of
speech is rational, and every man is rational.
But John is a man.
Therefore, John is capable of speech.

Our final example is an invalid epicheirema. It is basically an invalid
categorical syllogism suffering from an unextended middle term despite
a minor premise which is a legitimate causal proposition.
Every animal is sentient
But some organisms arc sentient, since everything sentient is an
organism.
Therefore, some organisms are animal.
IV.

The Polysyllogism

The polysyllogism is a clear-cut case of an expanded syllogism. It is
compounded from several simple syllogisms in such fashion that the
conclusion of one simple syllogism is used as the major premise for the
next until finally the conclusion of the polysyllogism taken as a whole
is reached. The component syllogisms in the polysyllogism can be either
categorical or hypothetical. To illustrate:
1. No bcing-of-the-reason is categorized.
But every- logical relation is a being of the reason.
Therefore, no logical relation is categorized.
But every predicable as such is a logical relation.
Therefore, no predicable as such is categorized.

248

THE LOGIC OF THE THIRD OPERATION

2. If seven is a whole number, seven is either odd or even.
But seven is a whole number.
Therefore, seven is either odd or even.
But seven is not even.
Therefore, seven is odd.
To check the validity of a polysyllogism simply reduce the polysyllogism
to the simple syllogisms out of which it is constructed, and then check
each component simple syllogism against the appropriate rules. If any
of the simple syllogisms is invalid, the polysyllogism as a whole is invalid.
If all of the component syllogisms are valid, the polysyllogism is valid.
The preceding arguments are valid polysyllogisms. The first is resolved
into two valid first figure categorical syllogisms concluding directly. The
second is resolved into a valid conditional syllogism in the first figure and
a valid alternative syllogism in the second figure. The following arguments
arc invalid polysyllogisms:

1. If every animal is mortal, then every animal is a substance.
But every animal is a substance.
Therefore, every animal is mortal.
But every man is an animal.
Therefore, ever}' man is mortal.
2. Every free agent is morally responsible for his actions.
But every man is a free agent.
Therefore, every man is morally responsible for his actions.
But Tom is morally responsible for his actions.
Therefore, Tom is a man.

The first of these invalid polysyllogisms includes two simple syllogisms:
the first is an invalid simple conditional syllogism in the first figure,
and the second is a valid categorical syllogism in the first figure concluding
directly. The second polysyllogism includes, first of all, a valid categorical
syllogism in the first figure concluding directly and, second, an invalid
categorical syllogism in the second figure concluding directly.

V. The Sorites

The sorites is a special type of extended syllogism which is ordinarily
an expanded categorical syllogism with more than three terms and more
than two premises. It is, in fact, nothing but a telescoped form of
polysyllogism. There are two ways of telescoping a polysyllogism so that
it becomes a sorites: one results in what is known as the Aristotelian

COMPLEX SYLLOGISTIC PATTERNS

249

sprites; the other, the Goclenian sorites.1 Tlic Aristotelian sorites is con­
structed of a number of propositions so arranged that the predicate of
one becomes the subject of the next, until finally the conclusion is
reached in which the predicate of the last premise is cither affirmed of
or denied of the subject of the first premise. In the Goclenian sorites
the subject of one proposition becomes the predicate of the next, until
the conclusion is reached in which the predicate of the first premise is
affirmed or denied of the subject of the last premise. The Aristotelian
sorites has a series of middle terms with increasing extension, while the
Goclcnian sorites has a series of middle terms with decreasing extension.
The following argumentations are respectively examples of the Aristotelian
and Goclenian sorites:
1. Every’ man is an animal.
But every animal is an organism.
And every organism is a body.
And every body is a substance.
Therefore, every man is a substance.
(Valid Aristotelian sorites.)

2. Every body is a substance.
But every organism is a body.
And every animal is an organism.
And every man is an animal.
Therefore, every man is a substance.
(Valid Goclenian sorites.)

The validity of a sorites, like that of a polysyllogism, depends upon
the validity of each of the simple syllogisms out of which it is constructed.
If any of them is invalid, the sorites itself is invalid. However, unlike
the polysyllogism, the sorites has a special set of rules against which
it can be measured. These rules are simply restatements of the rules for
categorical syllogism contracted precisely to the special case of the sorites.
Clearly each of them holds Ixxause its violation ultimately violates one
of the general rules for categorical syllogism. There are three rules for
a valid sorites:

Rule 1. No premise may be particular except the one which contains
the subject of the conclusion (i.e., the first premise in the Aristotelian
sorites and the last premise in the Goclenian sorites).
1 Aristotle did not use the term "sorites,” but he did suggest the form of syllogism
known as the Aristotelian sorites. The so-called Goclenian sorites was proposed by the
logician Goclenius, a sixteenth-century commentator on the logical works of Aristotle.

250

THE LOGIC OF THE THIRD OPERATION

The following Aristotelian sorites breaks the rule:
Every A is B.
But every B is C.
And some C is D.
Therefore, some A is D.
(This is invalid because C, which is used as a middle term, is never
universal. If the second premise is made negative in order to make C
universal, then the sorites involves an affirmative conclusion coming
from a negative premise. If the conclusion is made negative to escape
this, then D in the conclusion is more universal than D in the last
premise. 'Hiere is no way to escape the charge of invalidity so long
as any premise except the one which contains the subject of the
conclusion is particular.)

Rule 2. No premise may be negative except the one which contains
the predicate of the conclusion (i.c., the last premise in the Aristotelian
sorites and the first premise in the Goclcnian sorites).
The following Goclenian sorites breaks this rule:

Every A is B.
But no C is A.
And ever}' D is C.
Therefore, no D is B.
(This is invalid because B is universal in the conclusion and only par­
ticular in the first premise. If the first premise is made negative to
escape this, there are two negative premises. If the conclusion is made
affirmative to escape the same thing, there is a negative premise and
an affirmative conclusion. In any event, there is some defect in the
argumentation so long as any premise except the one containing the
predicate of the conclusion is negative.)

Rule 3. A particular premise necessitates a particular conclusion, and
a negative premise necessitates a negative conclusion. This is true, of
course, precisely in virtue of the general rules for the categorical syllogism
which demand precisely the same thing.
The following sorites break this third rule:
1. Some A is B.
But every B is C.
And everj' C is D.
Therefore, every A is D.

COMPLEX SYLLOGISTIC PATTERNS

251

(This is invalid precisely because A is universal in the conclusion and
only particular in the first premise.)
2. Every A is B.
But every C is A.
And no D is C.
Therefore, every D is B.
(This is invalid precisely because no affirmative conclusion is possible
following a negative premise.)
Although a sorites is ordinarily an expanded form of categorical argu­
mentation, it is possible to construct a conditional sorites. In the con­
ditional sorites several premises are so constructed that the consequent of
one becomes the antecedent of the next. In accord with the rules for
conditional argumentation the conclusion which follows from these
premises must combine the antecedent of the first premise with the
consequent of the last. To illustrate:

If a plane figure is a square, it is a rectangle.
But, if it is a rectangle, it is a parallelogram.
And, if it is a parallelogram, it is four-sided.
. Therefore, if a plane figure is a square, it is four-sided.

Because the denial of the consequent in a conditional proposition
implies the denial of its antecedent, the preceding conditional sorites
can be restated in the following fashion. This reveals a second form of
conditional sorites in which the antecedent of one premise becomes the
consequent of the next until a conclusion is reached in which the con­
sequent of the first is joined to the antecedent of the last.
If a figure is not a rectangle, it cannot be a square.
zVnd, if a figure is not a parallelogram, it cannot be a rectangle.
And, if a figure is not a four-sided figure, it cannot be a paral­
lelogram.
Therefore, if a figure is not a four-sided figure, it cannot be a
square.
VI. The Dilemma

The dilemma is a complicated form of syllogistic argument combining
conditional and alternative propositions in an antecedent which yields
either a categorical or an alternative conclusion. The major premise in any
dilemma is a complex copulative proposition composed of simple condi-

252

THE LOGIC OF THE THIRD OPERATION

tionals. The minor premise is an alternative proposition which alternately
affirms the antecedents or denies the consequents of the conditionals
in the major. There are four different forms that a dilemma can take.
These constitute the simple constructive dilemma, the complex construc­
tive dilemma, the simple destructive dilemma, and the complex destruc­
tive dilemma. A simple dilemma differs from a complex dilemma in that
the conditional propositions in its major premise have either a common
antecedent or a common consequent, whereas the conditional propositions
in the major of the complex dilemma have both different antecedents and
different consequents. The constructive dilemma differs from the destruc­
tive in that the minor premise in the former alternately affirms the
antecedents of the conditionals in the major, while the minor premise
in the latter alternately denies the consequents of the conditionals in
the major premise.

The Simple Constructive Dilemma
In the simple constructive dilemma each of the conditionals in the
copulative major premise has the same consequent. Their antecedents
are the component propositions in the alternative minor. The major and
minor premises together yield as a conclusion the consequent common to
the component conditionals of the major premise.
For example:

(a->c) -(b->c)
But a\/b
Therefore, c
The Simple Destructive Dilemma
In the simple destructive dilemma the conditional major premise
expresses two consequents following from a common antecedent. The
minor premise alternately denies each of these consequents. The denial
of the antecedent common to the two consequents follows as a conclusion.

To illustrate:

a->(b • c)
But —b\Z—c
Therefore, —a
The Complex Constructive Dilemma
In the complex constructive dilemma the copulative major is composed
of conditionals with different antecedents, each implying different con­

COMPLEX SYLLOGISTIC PATTERNS

253

sequents. 'l’ire alternative minor premise alternately affirms the ante­
cedents. A conclusion follows which alternately affirms the consequents.
As an example:
(a->c) • (b-*d)
But a\Xb
Therefore, c\/d

The Complex Destructive Dilemma
The major premise of the complex destructive dilemma is composed
of conditionals with respectively different antecedents and consequents.
The minor alternately denies each of the consequents. A conclusion fol­
lows which alternately denies each of the antecedents.

For example:
(a->c) • (b-»d)
But — c\/—d
Therefore, —a\Z—b
The dilemma is a valid form of syllogistic argumentation. It is composed
of conditional and alternative propositions, and it is constructed strictly
in accord with the rules for the conditional and alternative syllogisms. So
long as a dilemma is proposed in one of its four distinctive forms, and
so long as the hypothetical propositions out of which it is formed are
worthwhile, the dilemma is a legitimate complex syllogism. Defective
hypothetical components will, of course, render a dilemma itself defective.
Logicians speak of destroying a dilemma either by slipping between its
homs, or by taking it by its homs. To slip between the horns of a
dilemma is to show that its minor premise is illegitimate by pointing
out an additional component for the alternative premise. To take the
dilemma by the homs is to show that one or other of the consequents in
the conditionals comprising the major premise is not implied by its
supposed antecedent.
VII. Combinations of Complex Syllogisms

The complex syllogisms we have investigated in this chapter are com­
binations of the simple syllogisms previously considered. There is no
reason why these complex syllogisms cannot themselves be combined
into still more complicated complex patterns. So long as each of the
component syllogisms in the complex syllogism which results measures

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THE LOGIC OF THE THIRD OPERATION

up to its own set of rules, the complex syllogism as a whole is valid.
Consider the following example of a complex syllogism which finally
yields the conclusion Khrushchev cannot be trusted:
If a Communist is a good Communist, he cannot be trusted, be­
cause on principle he is committed to the thesis that the end justifies
the means. If a Communist is a bad Communist, he cannot be
trusted, because he is not true even to his own principles. But a
Communist must be cither a good Communist or a bad Communist.
Hence, no Communist can be trusted. Thus, since Khrushchev is a
Communist, Khrushchev cannot be trusted.
This is an example of a valid polysyllogism constructed from a simple
constructive dilemma and a first figure categorical syllogism concluding
directly. Tire conclusion of the dilemma is the major premise of the
categorical syllogism. The dilemma itself is an example of an cpichcircma,
since there is a justification added to each of the conditionals comprising
the major premise of the dilemma.
Exercise XVIII (A)

1. Define: complex syllogism, abbreviated syllogism, epicheirema, polysyl­
logism, sorites, dilemma.
2. What is the logical status of the abbreviated syllogism?
3. What are the rules for the validity of an epicheirema?
4. What are the rules for the validity of a polysyllogism?
5. What is the difference between the Aristotelian and the Goclenian
sorites? Defend the rules for a valid sorites.
6. What is the general plan of a dilemma? What are the different forms
of the dilemma? When is a dilemma a legitimate form of argumentation?
7. Suggest examples of both valid and invalid epicheiremas, polysyllogisms,
sorites, dilemmas, and any combinations of these. Explain the defect in
the invalid examples.
8. Identify the special type of syllogism exemplified in each of the following.
Evaluate each as far as validity is concerned, and explain what is wrong
with each inx^alid argumentation:
8.1. Every' A is B, because some B is A.
But some A is C, because every C is A.
Therefore, some C is B.
8.2. No A is B.
But every B is C.
And every C is D.
And every' D is E.
And some E is G.
Therefore, some A is not G.

COMPLEX SYLLOGISTIC PATTERNS

83.

8.4.

8.5.
8.6.

8.7.
8.8.

8.9.

8.10.

255

Every A is B.
But no C is A.
Therefore, no C is B.
But some C is D.
Therefore, some D is not B.
If a proposition is self-evident, it is immediate, since it can be
seen apart from premises.
But The White Mountains are beautiful is immediate, since it
can be seen apart from premises.
Therefore, The White Mountains arc beautiful is self-evident.
(a-*—c) • (—b—>d)
But —c\/d
Therefore, a\/—b
Whatever is free is intelligent.
But whatever is intelligent is spiritual.
And whatever is spiritual is incorruptible.
Therefore, whatever is incorruptible is free.
—a\/b, since a-»b
But —b, since —(b • c)
Therefore, —a
Ever}' polygon is a rectilinear plane figure.
But every parallelogram is a polygon.
And every rectangle is a parallelogram.
And no circle is a rectangle.
Therefore, no circle is a rectilinear plane figure.
(a->c) • (—b—>—d)
But a\/—b, since a->b
Therefore, c\/—d, since —c • d
But d
Therefore, c
(a->b)-> (c\/d)
But a-*b, because e->b, and e=c
Therefore, c\/d
But —c, because f->c, and —f
Therefore, d

Exercise XVIII (B)

This is a general exercise in formal logic for all types of syllogistic argu­
mentation. You are to identify the type of syllogism which is attempted in
each of the following argumentations. Wherever possible indicate its figure
and its mood. For this you may find it helpful to restate the syllogisms in
strictly logical form wherever this is not already the case. You may find it
helpful also to restate each argumentation symbolically. Check each syllogism
for validity, and explain what is wrong with each invalid syllogism. Supple­
ment this exercise by examining this textbook, as well as your other textbooks
and reference books, for attempts at syllogistic argumentation, classifying and

256

THE LOGIC OF THE THIRD OPERATION

evaluating each in the manner suggested above for the following examples of
argumentation:
1. Unless a predicable is necessary, it is not essential. Thus, the predicable
property is essential, because it is necessary.
2. One must surely accept what one can plainly sec. Hence, these words must
be accepted, since they can be plainly seen.
3. Acts of worship are not contrary to nature, since no act is both contrary to
nature and pleasing to God, and acts of worship are pleasing to God.
4. Since one cannot be both an old-guard Republican and politically liberal,
the politician in question must be an old-guard Republican, because he
is certainly not politically liberal.
5. A coward cannot help but hurt himself; since cowards lie, and liars cheat,
and cheaters hurt themselves.
6. Either a composite expression is a perfect composite expression or it is
not a proposition, for every proposition is a perfect composite expression.
Therefore, no definition is a proposition.
7. Either a composite expression is a proposition or it is not a perfect
composite expression, for every proposition is a perfect composite expres­
sion. Thus, no definition is a proposition, because no definition is a perfect
composite expression.
8. Hell strikes fear in our hearts because it is a frightening word, and
frightening words strike fear in our hearts.
9. The human soul cannot be corrupted, because it is not composed of
entitative parts, and every corruptible substance is composed of entitative
parts.
10. Some polygon is rectangular, and every polygon is rectilinear; hence, even
rectangle is rectilinear.
11. Only animals are sentient. Thus, man is sentient, because man is animal.
12. If 20 is higher than 15, it is higher than 10; but 20 is higher than 10;
therefore, 20 is higher than 15.
13. Either the human will is completely determined, or it is completely un­
determined, or it is partially determined. If it is completely determined,
we cannot be held accountable for our choices. If it is completely un­
determined, we cannot choose at all. Since we do choose and arc held
accountable for our choices, the human will must be partially determined.
14. Some Romans are good singers, since all Romans arc Italian, and many
Italians are good singers.
15. No circles arc rectilinear, because all rectangles arc rectilinear, and no
circles are rectangles.
16. If and only if a figure has interior angles equal to 180 degrees, it is a
triangle. 'Ilius, since every isosceles figure is a triangle, every isosceles
figure has interior angles equal to 180 degrees. Accordingly, since no
scalene triangle is isosceles, no scalene triangle has interior angles equal
to 180 degrees. This, however, implies that no scalene triangle is a
triangle.

COMPLEX SYLLOGISTIC PATTERNS

25"

17. John cannot be a Marine, because his military record is anything but
enviable, and the Marines have an enviable military record.
18. No one can be simultaneously a father and a son. Thus, since Tom is a
son, Tom is not a father.
19. Seven must be an odd number, since it is not even.
20. Since a Fascist government cannot be Communist, and a Communist
government cannot be Fascist, then a non-Communist government must
be Fascist, and a non-Fascist government must be Communist. Thus, ever}'
government must be either Fascist or Communist.
21. Immorality is the greatest of all evils. Thus, cheating on an exam is the
greatest of all evils, since cheating on an exam is an immorality.
22. Our government is no stronger than our chief executive; hence, a strong
president insures a strong government.
23. An American either is loyal to his country, or he is not loyal to his country.
If he is a Communist, he is not loyal to his country. But he must either
be a Communist or not be a Communist. Hence, either an American is a
Communist or he is a loyal American.
24. No immediate proposition is proved, because every self-evident proposition
is immediate, and no self-evident proposition is proved.
25. Every science is a quality, for every habit is a quality, and every intellectual
virtue is a habit, and every science is an intellectual virtue.
26. If a man is psychotic, he is not normal. Thus, John is psychotic, because
he is not normal; while Peter is normal, since he is not psychotic.
27. Running is a transitive operation, because it is categorized as action, and
cvciy transitive operation is categorized as action.
28. To make political broadcasts sympathetic to an enemy is to collaborate,
and to collaborate is to do something to be ashamed of; thus, many of our
prisoners of war from the Korean War have nothing to be ashamed of,
since many of them did not make political broadcasts sympathetic to the
enemy.
29. Since a liberal favors foreign aid, a politician who does not favor foreign
aid is not a liberal, and a politician who does favor foreign aid is a liberal.
50. All animals are sentient, since no plants are sentient, and no animals arc
plants.

CHAPTER

XIX

Demonstrative and Dialectical Discourse

I. The Syllogism Materially Considered

Thus far we have considered the syllogism solely from the point of
view of formal logic. Wc have investigated exclusively what it takes to
make a syllogism valid. But validity docs not guarantee truth. Conse­
quently, it is hardly enough to master merely the rules of valid argument.
Logic looks beyond itself to totally sound discourse in the other dis­
ciplines. The logic student must learn not only what makes a conclusion
follow from its premises, but what makes it follow determinately either
as certainly true or as probably true. Thus, it remains for us to investigate
the logical properties belonging to syllogisms from the point of view of
their matter. These are the properties which determine, in the general
way proper to logic, the truth, certainty, and explanatory force of the
syllogism. Several syllogisms identical in form can have conclusions with
significantly different force as far as truth is concerned. One may involve
logical properties guaranteeing at best a probably true conclusion, which
admits of the possibility of its opposite. A second may involve logical
properties which guarantee a factually true conclusion, which most
certainly is true but could, given another factual situation, not-be true.
A third may involve logical properties which guarantee a necessarily true
conclusion, which is true precisely because it cannot not-be true. The
logical properties which determine the force of a conclusion in reference
to truth and certitude are the subject matter of this chapter.
II. Science and Demonstration

The word “science” is used in many ways. It is used most commonly
today to refer to the experimental or laboratory disciplines, such as
physics, chemistry, and biology, which are perhaps most aptly described
as empiriological. This is an acceptable usage of the term “science,” but
it is not the only one. Aristotle used it in a_diffcrcnt way, and his usage
258

DEMONSTRATIVE AND DIALECTICAL DISCOURSE

259

has remained respectable although it is much less popular than the one
common today. In Aristotle’s sense, to know a thing scientifically is to
know it in its necessary causes. Scientific knowledge, understood in this
sense, is a relatively perfect kind of knowledge, a knowledge in which the
intellect can rest because it is both certain and causally explained. It is a
kind of knowledge achieved, unfortunately, only with great difficult}' and
all too rarely. Yet it can be reached, and when it is, it comes as the con­
clusion to a syllogism which is perfect not only formally but materially
as well. A syllogism which can generate a conclusion scientific in the
Aristotelian sense — that is, a certain proposition proven through causes —
is called a demonstration. .Throughout this chapter we shall speak of
“science” in Aristotle’s sense, and we shall refer to demonstration as the
scientific syllogism.
Demonstration itself is an analogous notion. It is realized proportionally
in different types of syllogisms which yield certain knowledge proven
through causes in respectively different ways. Wc shall distinguish between
the types of demonstration very shortly. However, first we must consider
the most perfect type of demonstration. Since demonstration is an analo­
gous notion, the others can be understood only by reference to its
primary instance. Science, in its strictest sense, is achieved as the con­
clusion to a syllogism which establishes the necessary connection between
a subject and one of its specific properties. It does this in the light of
premises which show the necessary connection between this subject and its
definition and between this definition and the property in question. The
following argument is an example of the strictest form of demonstration:
Every rational animal is capable of speech.
But ever}' man is a rational animal.
Therefore, every man is capable of speech.

This is a categorical syllogism in the first figure concluding directly, with
two A premises and an A conclusion. The minor term is a subject of
scientific inquiry’. The major term is a specific property of this subject.
The middle term is simultaneously the necessary and proximate cause
of the property of the subject and the essential definition of the subject
itself. Since the premises are necessarily true the conclusion follows as
necessarily true. Since the premises express the cause of the inherence of
the property in the subject the conclusion follows as fully explained. The
example may well disappoint the student. It seems all too easy, and
perhaps somewhat less than crucial. However, it is probable that the
demonstration is not yet fully appreciated by the student. Jt is one
thing to admit that the premises are necessary and explanatory’ and that

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THE LOGIC OF THE THIRD OPERATION

the conclusion is therefore necessary and explained. It is another to
see the necessary and explanatory character of the premises. This involves
keen intellectual insight, which comes perhaps only as the result of
lengthy and laborious intellectual effort. However, once the premises arc
fully appreciated for what they are in themselves, it is relatively simple to
view them as premises and to appreciate the scientific status of the con­
clusion generated from them. For this reason the demonstration itself may
seem to be anticlimactic. We shall say more, in this chapter and in the
next, about the premises in the demonstration and the effort involved
in achieving them.

III. Prescientifk Knowledge

We have seen that scientific knowledge, in Aristotle’s sense of the term,
is achieved as a conclusion in a demonstrative syllogism. Because it is a
conclusion, the scientific proposition presupposes prior knowledge. In this
section we shall consider what must be known before it is possible to
demonstrate a conclusion. First of all, the terms of any proposition must
be known before that proposition itself is known. Second, the premises
must be known before the conclusion based on them can be known.
For demonstration, then, there must be foreknowledge of the subject and
predicate of the conclusion of the demonstration, as well as foreknowledge
of its premises. Tliis means, for the strictest form of demonstration, that
we must have a prior knowledge of its scientific subject, of the property
in question, and of the premises of the demonstration.
'Hiere are two things we can know about an object. We can know
what it is, and we can know that it is. To know what it is, is_to be able
to define it. To know that it is, is to know of its existence. The incomplex
object of simple apprehension can be known in both way's. We can
know what it is cither in a nominal or a real definition, and we can know
that it is. But the complex object of judgment is different. Propositions
cannot be defined, and thus we cannot know what they are. But we can
know that they are — at least in the way in which propositions arc, that
is, as being true.
With these distinctions in mind, let us consider precisely what must
be known before we can demonstrate a conclusion in the strictest form
of demonstration:

1. About the subject —we must know that the subject exists and we
must be able to define it in a real and causal definition. In the strictest
form of demonstration we attempt to show that a scientific subject

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261

exists as necessarily possessed of a given property. We show this by
establishing a necessary cause-and-effect relationship between the defini­
tion of the subject and the property. The proof itself does not bear
upon the existence of the subject. This is presupposed. It does bear
upon the inherence of the property in the subject. This is show
through a middle term which is the real definition of the subject and
the cause of its property. Thus, a real and causal definition of the
subject is also presupposed.
2. About the property —we must know at least its nominal definition.
We do not presuppose knowledge of its existence or its essential
definition. A property is categorically an accident; as such, it is essen­
tially defined in terms of its proper subject. To know that an attribute
is defined essentially in terms of a given subject is to know that it
necessarily inheres in that subject, and that is what is expressed in
the scientific conclusion. This need not, and cannot, be presupposed.
Nor need the existence of the property be presupposed. Once the
premises have established that a property, known only through a nom­
inal definition, actually does flow necessarily from the definition of a
subject known to exist, the conclusion must affirm the actual existence
of that property in the subject. Thus all that need be presupposed
about the property is its nominal definition.
3. About the premises —we must know that they are necessarily true.

IV. The Requirements for the Premises of Demonstration

The definition of demonstration as a syllogism productive of science
is a definition in terms of final cause. This definition of demonstration
through final cause leads us to a definition of demonstration through its
matter, namely, a syllogism with premises that are true, priman', im­
mediate, prior to the conclusion, better known than the conclusion, and
cause of the conclusion. Unless a syllogism has such premises, it cannot
yield a conclusion measuring up to the strict notion of science. Let us
sec why this is so by considering these characteristics of the premises
one by one:
1. The premises must be true, for only true premises can cause the
conclusion to be true. False premises allow for, but in no sense ever
guarantee, a true conclusion.
2. The premises must be primary. This means that they must be prin­
ciples of demonstration which are themselves not demonstrated (or
ultimately reduced to such propositions). The alternative to this is an

262

3.

4.
5.

6.

THE LOGIC OF THE THIRD OPERATION

absurd infinite regress in premises. In the strict sense a premise is a
primar}' principle of demonstration only if it is proper to its conclu­
sion, that is, only if it is strictly proportioned to the scientific subject.
This requires, for example, that whenever a conclusion is about a
mathematical subject the middle term in the premises be expressed
precisely in a mathematical fashion. A statement out of logic or out of
physics would not function properly as a primary premise in mathe­
matical discourse.
The premises must be immediate. Tins means that the predicate in a
premise must belong immediately (that is, not by way of any middle
term) to its subject. Again, unless we have this, at least reductively,
there is an impossible infinite regress in premises.
The premises must be prior to the conclusion because they are prin­
ciples from which the conclusion proceeds.
The premises must be better known than the conclusion because they
serve as principles which manifest the conclusion, which in itself
lacks the evidence they supply.
The premises must be the cause of the conclusion. In one sense this
is true of all premises. It is the very nature of argument that the
conclusion be virtually in the premises as an effect in a cause. Premises
and conclusion are always so related that a knowledge of the premises
as premises cause a knowledge of the conclusion. The premises in the
strict form of demonstration are related to the conclusion as cause to
effect in a very special way. The premises not only cause the conclusion
to be known, but they represent the real cause which explains the
reality represented in the conclusion.

The conclusion to a strict demonstration is a universal proposition
expressing the necessary inherence of a property in a scientific subject.
This conclusion can be necessary only in so far as some necessity in the
premises is communicated to the conclusion, for nothing is in the con­
clusion which is not at least virtually in the premises. To say that the
premises must be true, primary, and immediate is to insist that one or
both of the premises be self-evident. It is not necessary that both premises
in a demonstration be self-evident. In the strictest form of demonstration
both will be self-evident, but it is possible to demonstrate less strictly
with one self-evident and one factually evident proposition. However, at
least one premise must be self-evident, at least reductively, or else the
conclusion cannot be necessary. This is true because the only absolutely
necessary propositions are those which are self-evident or those which
follow from strictly self-evident propositions. We have already spoken

DEMONSTRATIVE AND DIALECTICAL DISCOURSE

263

about self-evident propositions, and we shall discuss them more at length
in the following chapter. From the positive point of view a self-evident
proposition is a proposition whose truth is evident in the very terms
involved. From the negative point of view a self-evident proposition is
a universally necessary proposition which involves no middle term between
its predicate and its subject.
V. Universality, Perseity, and Convertibility

Three significant logical relations found in the propositions making
up the strictest form of demonstration deserve special mention. Not every
proposition in every demonstration involves all three. However, a demon­
stration in which any of these properties is missing is less perfect than
one in which they are found. The three relations in question arc uni­
versality, perseity, and convertibility. Let us consider them one by one:

1. Universality
We have spoken of universality from the formal point of view in
previous chapters. In the present reference we add something to this
notion. A proposition involves a kind of universality whenever the
predicate is said (in any way) of each and every' instance of the subject.
A proposition involves perfect universality whenever the predicate is said
necessarily of each and every instance of the subject. It is this perfect
universality which is a property of demonstration.
AV
c^c I.
2. Perseity
Perseity is a logical property belonging to a universal proposition in so
far as the predicate is said necessarily of the subject precisely in virtue
of what is essential to the subject. The opposite of a per sc connection
between predicate and subject is a per accidcns connection, in which the
predicate is connected with the subject only accidentally. The modes of
perseity have been traditionally numbered as four. However, only the first,
second, and fourth concern us. The third is a mode of being, not a mode
of predication.
A proposition is said to be in the first mode of perseity whenever its
predicate is an element in the essential definition of its subject (i.e., either
the genus, difference, or essential definition itself). A proposition is in
the second mode of perseity whenever its subject is an element in the
definition of its predicate. This occurs when the predicate of the proposi­
tion is related to the logical subject as an accident to its proper real
subject, e.g., as risible is related to man. A proposition is in the fourth

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THE LOGIC OF THE THIRD OPERATION

mode of perseity whenever its subject is related to its predicate as the
proper (i.e., proximate and precise) and necessary cause of the predicate.’
Consider the example we have taken of a strict type of demonstration:
Ever}’ rational animal is capable of speech.
But every man is a rational animal.
Therefore, ever}' man is capable of speech.

'Hie major premise is in the fourth mode of perseity. The minor premise
is in the first mode of perseity. And the conclusion is in the second mode
of perseity.
3. ConvertibilityA proposition is convertible in this reference whenever its subject and
predicate are necessarily coextensive. A convertible proposition is said to
lie commensurately universal. The predicate belongs to every instance
of the subject, and only to this subject. The propositions in our example
of the strict type of demonstration are not only universal, but they are
commensurately universal as well.

VI. Types of Demonstration

We have concentrated so far on the strictest form of demonstration.
There are, in fact, three basically different types of demonstration, with
variations possible within each. Each is, in its own way, a demonstration
because each generates certain knowledge proved through causes. Demon­
stration is an analogous notion realized primarily in the kind of demonstra­
tion we have discussed so far. We shall call the primar}’ instance of
demonstration explanatory demonstration. The other forms of demonstra­
tion are less strict instances of demonstration. They are more or less
perfect as the}’ more or less closely approach the perfection of the
explanatory demonstration.
Demonstration can be divided into types from two different points
of view. From one point of view, demonstration is either explanatory
demonstration (propter quid) or demonstration of the fact (quia). Ex­
planator}' demonstration not only manifests with certitude the fact that
something is necessarily true, but it expresses the proper and adequate
reason why it must be true and could not not-be true. Demonstration of the
fact simply manifests with certitude the fact that something is necessarily
1 Traditionally the third mode of perseity has been taken to be the manner in
which a primary substance exists in itself and not in another. This is not significant
for our theory of demonstration.

DEMONSTRATIVE AND DIALECTICAL DISCOURSE

265

true. From another point of view, demonstration is either a priori or a
posteriori. A priori demonstration moves from a necessary cause to its
effect, and a posteriori demonstration proceeds from an effect to its
necessary cause. Combining the dividing members of these two divisions
of demonstration, it would seem perhaps that there are four types of
demonstration, namely, the a priori explanatory’, the a posteriori explana­
tory, the a priori demonstration of fact, and the a posteriori demonstration
of fact. However only three of these four arc real possibilities. An a
posteriori explanatory demonstration is not possible. The existence of an
effect can point to the existence of a cause but it can never explain the
cause, for the cause is, by definition, the explanation of the effect. Let
us consider the three types of demonstration one by one, beginning with
the most perfect type, that is, the a priori explanatory demonstration.
Since there can be no a posteriori explanatory demonstration it is not
necessary in naming the most perfect type to say any more than “explana­
tory demonstration.”

A. Explanatory' Demonstration
The name “explanatory’ demonstration” is reserved for the most perfect
tvpe of demonstration in which a conclusion is proved through a middle
term which adequately expresses that on account of which (propter
quid) the conclusion is what it is and must be as it is. Formally, an
explanatory’ demonstration is a first figure categorical syllogism concluding
directly. Materially, the middle term of the explanatory demonstration is
related to the major term as a necessary’ and proximate cause to a con­
vertible effect. There are three different ways to construct an explanatory’
demonstration:

1. The first is to prove an immediate property’ of a subject through a
middle term which is simultaneously the definition of the subject and
the proximate cause of this property. This type of demonstration can
proceed by way of a middle term which defines in terms of any one of
the four causes (material, formal, efficient, and final) for natural
science, of one cause (formal) for mathematics, and of any one of
three causes (formal, efficient, and final) for metaphysical science, e.g.:
Every rational animal is capable of speech.
But every man is a rational animal.
Therefore, every man is capable of speech.
We have already analyzed this example (pp. 259, 264). Summarizing
our findings briefly, we can make the following observations. Formally
this is a first figure categorical syllogism concluding directly. It is a valid

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THE LOGIC OF THE THIRD OPERATION

syllogism with an AAA mood. On the side of matter: The minor term
is a scientifically knowable subject. The major term is a specific property
of the scientific subject. The middle term is the essential definition
of the subject and the proximate and necessary cause of the property.
Each proposition in the demonstration is universally necessary, and
each is convertible. The major premise is a self-evident proposition
in the fourth mode of perseity. The minor premise is a self-evident
proposition in the first mode of perseity. The conclusion is a mediate
proposition (shown to be necessarily true through the premises) in the
second mode of perseity.
2. The second is to prove a mediate property2
*5 of a subject through a
middle term which is an immediate property of the subject and the
proximate cause of the mediate property, e.g.:
Everything having intellect and imagination can learn geometry.
But ever}' man has intellect and imagination.
Therefore, every man can learn geometry.

3. The third is to prove one cause of its subject through a middle term
expressing another cause of this same subject when the former cause
is related to the latter as effect to cause, e.g.:

Every argumentation productive of science is from premises
which are ultimately necessary and immediately true.
But every demonstration is an argumentation productive of
science.
Therefore, ever}' demonstration is from premises which are
ultimately necessary and immediately true.
In this example, a definition of demonstration itself in terms of
material cause is demonstrated by way of a middle term which defines
demonstration in terms of final cause. The causality of the material
cause depends on the prior three causes. Hence, the material cause can
be demonstrated by the final, efficient, or formal cause. The causality
of the formal cause depends on the efficient and final causes. Hence,
2 Some properties find their proximate cause in the principles intrinsic to the essence
of their subjects. A second type of property finds its proximate cause in a property
of the first type and belongs to the essence through this prior property. The first type
of property is immediate; the second type is mediate. The example of a mediate and
an immediate property used here is borrowed from the translators of The Material
Logic of John of St. Thomas, translated by Yves R. Simon, John J. Glanville, and
G. Donald Hollenhorst (Chicago: University of Chicago Press, 1955), p. 617. The
footnotes supplied by the translators to Chapter 6, pp. 616-625, are recommended as
extremely illuminating for the theory of demonstration.

DEMONSTRATIVE AND DIALECTICAL DISCOURSE

267

the formal cause can be demonstrated by either efficient or final cause.
And the efficient cause depends upon the final cause, so that the
efficient cause is able to be demonstrated by way of the final cause.

B. A Priori Demonstration of the Fact
An a priori demonstration of the fact proves that the predicate of the
conclusion belongs to its subject by relating them through a middle
term which expresses the remote or mediate cause of the predicate. In so
far as the middle term is not the proximate cause of the effect, this falls
short of the perfection demanded for explanatory demonstration. But
in so far as this is a demonstration from cause to effect, this type of
demonstration is a priori. There arc two ways to construct an a priori
demonstration of fact. In the first case the cause through which the effect
is proved is remote in the sense of being once removed in the real order.
In the second case the effect is proved through a cause remote in the
sense of being generic, that is, once removed in the order of abstraction.

1. An effect is demonstrated by way of a remote or mediate cause when­
ever a mediate property is proved of its subject through a middle term
which defines this subject. Since the definition of the subject expresses
the proximate cause of the immediate property in question, this defini­
tion expresses only the remote or mediate cause of the property proved
to belong to the subject in this demonstration, e.g.:
Every rational animal can learn geometry.
But every man is a rational animal.
Therefore, every man can learn geometry.

2. A significantly different way to prove an effect by way of a remote or
mediate cause is in a demonstration which proves an effect from a
cause which is mediate and nonconvertible, but exclusive (exclusive
in the sense that, though it is not universally a cause of the effect, the
effect cannot be achieved save that it function as a generic cause), e.g.:
Everything capable of speech is an animal.
But no plant is an animal.
Therefore, no plant is capable of speech.

The first kind of a priori demonstration of fact is expressed in the first
figure concluding directly. However, the second type can be expressed
only in the second figure, and, accordingly, allows only for a negative
conclusion. The reason for this is that the effect proved through the
remote cause is convertible with that cause in the first type of a priori

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THE LOGIC OF THE THIRD OPERATION

demonstration of fact, while it is less extended than the cause in the
second type. Any attempt to argue affirmatively in the second instance
results in an unextended middle term.
C. A Posteriori Demonstration of Fact
An a posteriori demonstration of fact is achieved by way of a middle
term which is related to the predicate of the conclusion as effect to cause.
One instance of this type of demonstration involves an effect convertible
with its cause, while another involves an effect less extended than its cause.
1. The first instance of an a posteriori demonstration of fact is a syllogism
proving a cause through an effect convertible with this cause. Examples
of this type of demonstration can be achieved by interchanging the
middle and major terms in examples of explanator)' syllogisms, e.g.:
Whatever is capable of speech is a rational animal.
But every man is capable of speech.
Therefore, every man is a rational animals

We have said that demonstration is an analogous notion, which is
perfectly realized only in explanator)' demonstration. lire requirements
for the less strict types of demonstration are not identically what they
arc for explanatory demonstration. In this example of an a posteriori
demonstration of the fact we have a fine illustration of this point.
This example is like the first example offered for the explanatory
demonstration, except that here the middle and major terms of that
example are interchanged. In our first example foreknowledge of the
essential definition of the scientific subject is a necessary prerequisite
for demonstration. However, in this present example only the nominal
definition of the subject is presupposed. To know the essential defini­
tion of the subject here is already to know the conclusion, which simply
predicates this definition of the subject.
2. A second kind of a posteriori demonstration of fact proves a cause
through an effect which is less extended than, but yet exclusive to,
that cause, e.g.:
Whatever is capable of speech has a soul.
But every man is capable of speech.
Therefore, every man has a soul.

We should point out that any attempt will be logically defective which
seeks to prove a cause of a subject through an effect less extended than

DEMONSTRATIVE AND DIALECTICAL DISCOURSE

269

that subject or through an effect which is not exclusive to the cause.
In the first case the syllogism attempted has an illicit minor term,
and in the second case it has an unextended middle term.
Each of the types of demonstration treated admits of a form weaker than
the one described and illustrated above. For each there is a weaker form
in which the middle term remains the same while the minor term be­
comes a subjective part of the original minor. This is a weaker form since
neither the minor premise nor the conclusion is commensurately universal.
In addition, the explanatory' demonstration and the a priori demonstration
of the fact which proceeds from the definition of tlic scientific subject
to a mediate property of this subject admit of a weaker form which
employs the original minor term as middle and a subjective part of the
original minor term as minor. Consider, for example, the following ex­
planatory demonstration (No. 1) and its weaker forms (Nos. 2 and 3):

1. Every argumentation productive of science is from premises which
are ultimately necessary and ultimately true.
But every demonstration is an argumentation productive of
science.
Therefore, every’ demonstration is from premises which are ulti­
mately necessary’ and ultimately true.
(The conclusion is convertible, and the middle is proximate both
to the minor term and the major term.)

2. Every argumentation productive of science is from premises which
are ultimately necessary and ultimately true.
But every explanatory’ demonstration is an argumentation produc­
tive of science.
Therefore, every explanatory demonstration is from premises which
are ultimately necessary and ultimately true.
(The conclusion is not convertible, and the middle is not
proximate to the minor term although it is proximate to the
major.)
3. Every demonstration is from premises which are ultimately neces­
sary and ultimately true.
But every' explanatory demonstration is a demonstration.
Therefore, every explanatory demonstration is from premises which
are ultimately necessary' and ultimately true.
(The conclusion is not convertible, and the middle is not
proximate to the major term although it is proximate to the
minor. )

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THE LOGIC OF THE THIRD OPERATION

VII. The Middle Term in Demonstration

The middle term in the scientific syllogism is the most significant single
principle of demonstrative discourse. The fundamental task of anyone who
attempts to demonstrate is to find a middle term which can manifest the
necessary connection between the predicate and the subject of a scientific
conclusion. This middle term must in every case be the cause of the
knowledge of the connection between the extremes in the conclusion. In
an a posteriori proof the middle term is a cause only of the knowledge
of the connection. In a priori proof it is both a cause of the knowledge
of the connection and the real cause of that connection. In an explana­
tory demonstration the middle term is the adequate and proximate real
cause of the connection between the predicate and subject of the con­
clusion. In an a priori demonstration of fact the middle term is a real
cause, but a cause which is either less than adequate (because only
generic) or less than proximate (because once removed) in reference to
the effect in question.
A more penetrating study of demonstration than that suited to an
introductory logic course would demand a detailed consideration of middle
terms and the modes of defining proper to them. We shall, however, only
sketch a plan of inquiry suited to a more advanced study. The most
significant instance of a middle term is the definition of the scientific
subject employed in the explanatory demonstration. The force of defini­
tions in scientific discourse differs to the degree in which the definitions
differ in the way they abstract from matter. From this point of view,
there are three generically distinct modes of defining, and each determines
a generically distinct scientific discipline. The three genera of science are
respectively natural science, mathematical science, and metaphysical
science. Demonstrations in the natural sciences have instances of physical
or sensible (i.e., changeable) being as a subject of inquiry, and they
involve middle terms which define their subjects in the only way these
subjects can exist outside the mind, namely, in matter. Man, for example,
is defined in terms of “flesh and blood,” and no man exists outside the
mind save as concretized in flesh and blood. The subjects of mathematical
demonstrations are ultimately instances of quantified being, and the
middle terms in mathematical demonstrations define their subjects with­
out matter, even though they can exist outside the mind only in matter.
Thus, triangle is defined in abstraction from any physical characteristics,
although triangles can exist outside the mind only as concretized in
triangularly shaped physical things. The subject of metaphysical demon­
stration is ultimately being itself, and the middle term in metaphysical

DEMONSTRATIVE AND DIALECTICAL DISCOURSE

271

demonstration defines its subject in a way in which this subject can exist
outside the mind, that is, without matter. Being is in some cases material,
but being need not be material, and, in fact, is not material in God.
We can distinguish between the subjects proper to physical science,
mathematics, and metaphysics by noting that the first is concerned with
beings dependent on matter both for their existence and for their
intelligibility'; the second, with beings dependent on matter for their
existence but not for their intelligibility; the third, with beings inde­
pendent of matter both for their existence and their intelligibility.
It is not the business of logic further to pursue the differences between
the various sciences, nor to discuss the peculiar characteristics of demon­
stration in the different sciences. It is the business of logic to investigate
the general notion of demonstration and generally its division into types.
Each particular science has its own proper methodology'. To the extent
to which a particular science demonstrates conclusions, it contracts the
general notion of demonstration to the needs of its own proper subject
matter. The general theory' of demonstration is open to contraction in
the face of the scientific subjects respectively proper to the several sciences.
This is true whether the subject be proper to the natural sciences, mathe­
matics, or metaphysics. The general theory must not be confused with any
one of its contractions.
VIII. Dialectical Argument

The strictest form of argument is the demonstration, which yields a
certainly true conclusion proven in the light of necessarily true premises.
However, a demonstration is comparatively rare. For one thing, demon­
stration bears only upon necessary matter, and many items even of
academic significance are contingent rather than necessary'. This is par­
ticularly true, for example, in the area of political action. For another
thing, the human intellect is frequently limited in its attempts to define
a thing and consequently limited in reference to what it might demon­
strate of that thing. For example, demonstrations from essential definition
to property are impossible for natural species save the species man, since
man is the only natural species we can essentially define. Perhaps the most
favorable area for examples of demonstrative argumentation is that of
classical mathematics. Mathematical entities arc not subject to the con­
tingent attributes which are characteristic of natural things, and they are
relatively easy for the human intellect to understand. Nevertheless, even
in mathematics demonstration does not come without some difficulty.
There is a second form of argumentation which is not demonstrative,

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THE LOGIC OF THE THIRD OPERATION

but which is nevertheless legitimate, and, in fact, significantly so. This
kind of argumentation, called dialectical, falls short of a certainly true
conclusion, but it does give conclusions at least probably true. A proposi­
tion is certainly true when the evidence for it rules out the possibility
of its contradiction. A proposition is probably true when there is solid
evidence for it, but when this evidence leaves open the possibility of its
contradiction. A certainly true conclusion is more excellent than a
probably true conclusion. However, a probably true conclusion is better
than no conclusion. Probable argumentation has a significant place both
as a substitute for, and as a supplement to, demonstration.
Generally speaking, the name “dialectic” can be applied to any
form of logical discourse which is less than demonstrative. This includes
the kind of discourse which is characteristic of rhetoric and poetry. In
a more restricted sense dialectical discourse is divided off against demon­
stration on the one hand and rhetoric and poetry on the other. From this
point of view the following points can be made. Demonstration aims at
science understood as certain knowledge proven through causes. This
involves a commitment to one side of a contradiction without fear of
the opposite. Dialectical argumentation aims at opinion. This involves
a commitment of the mind to one side of a contradiction, but with a fear
of its opposite. The rhetorical argument intends to persuade one to accept
one side of a contradiction. Poetry, in so far as it can be considered an
instrument for teaching, seeks to generate an inclination to one side of
a contradiction through a pleasing representation of it. We shall continue
to speak of dialectics without reference to rhetorical argumentation or
poetry. The scientist who demonstrates his conclusion resolves one of two
contradictor,' propositions into self-evident principles which demand
acceptance of this proposition and the rejection of its contradictory'. The
dialectician, on the other hand, resolves one side of a contradiction into
premises which certainly support it as probable but which allow for other
premises which might support its opposite. On the side of form, the
demonstration and the dialectical syllogism are of equal status —each
must be, for its own purposes, indefectibly valid. It is on the side of
matter that they differ. The demonstration proceeds from premises which
cannot be false, because they are ultimately self-evident. The dialectical
argumentation proceeds from probably true premises, which can, of
course, communicate to their conclusions no more than their original
probability. Some of the more usual dialectical premises include those in
contingent matter which are true for the most part but which admit of
exceptions, those which are probably true either because they are com­
monly held by most men or are proposed by some respected authority on

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275

the point, and those which are proper to one field but which lend them­
selves as argumentative principles in another. Each of the following
argumentations is at best dialectical:
1. Mothers love their children. (Probable in the sense of being true
for the most part.)
But Susan is a mother.
Therefore, Susan loves her children.

2. All contemporary natural species have evolved from less perfect
species. (Probable because it is the opinion of experts in the
field of biology.)
But man is a contemporary natural species.
Therefore, man has evolved from a less perfect species.
3. Contraries are within the same genus. (Certain in logic and
applicable in psychology to yield a probable conclusion.)
But love and hatred are contrary passions.
ITierefore, love and hatred are passions of the same appetite.

Dialectic differs from a demonstration in that the former is untermi­
nated while the latter is terminated. Demonstration is terminated because
its conclusion is resolved into propositions which infallibly represent
reality. Dialectical conclusions arc not certain because they are not
resolved into propositions which infallibly represent real things. Dialectic
remains within the mind while tending to the real. For this reason the
dialectician must be able to employ every logical device calculated to
manifest the truth of a proposition. As a dialectician he cannot see the
necessary correspondence between his conclusions and reality. 'Ilius he
must be prepared to explore every avenue of logic open to him in order
to come as close as possible to a certainly true conclusion, although he
can never reach certitude through dialectic. He must be skilled in the
arts of defining and dividing. He must be able to distinguish between
terms on the basis of the most subtle differences in signification and
supposition. He must be able to classify predicates predicably and cate­
gorically so that the meanings of terms and the force of propositions
may be fully evaluated. He must be able to distinguish between
propositions on the basis of formal differences, and, even more so, on
the basis of material differences. He must know the various forms of
argumentation and be aware of the properties determining their probativeforce. He must know the minimum limits for scientific discourse and be
especially skilled in arguments falling between this limit on the one hand
and illogical discourse on the other. He can in a sense be less rigid and

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more flexible, more general and less pointed, less rigorous perhaps, but
nonetheless never less logical, than the one who demonstrates. He is
satisfied with less, for certainty is more satisfying than probability. But he
is more often satisfied, since many more things can be shown dialectically
to be probable than can be shown demonstratively to be certain. The
course in logic has been ordered primarily to demonstration because the
scientific syllogism is the prime instance of logical discourse. It has also
been ordered to dialectical argumentation, in the realization that the
academic intellect does not live on demonstration alone but looks to
dialectics as a necessary complement in the quest for truth.
Perhaps the most significant use of the dialectical method in our day
is its application as “scientific method” in the empiriological disciplines.
We have talked about the several uses of the name “science.” In the
language of most people the term is applied to those disciplines such as
physics, chemistry, and biology, which can be called empiriological.
Strangely enough, the method which characterizes these disciplines is
nonscientific in the Aristotelian sense. It is, as we have used the term,
dialectical rather than scientific. We should not think that there is one
method univocally applied to the subjects respectively proper to the
several empiriological disciplines. Yet there is generally one method
applied most strictly in physics and proportionally in the other em­
piriological disciplines which can be spoken of as the “scientific
method.” In its most simple outline the method involves first of all
the formation of a hypothesis or theory calculated to explain what has
been observed in a given physical, chemical, or biological area of interest.
The method allows as truly significant only those data which can in some
way be quantitatively measured, and it involves a hypothesis which is
always mathematical in form even though it bears upon the physical.
From the hypothesis — which as originally formulated explains only what
has already been observed — conclusions are deduced which are translated
into predictions of results from future experiments. When the results
which arc predicted are in fact realized, the hypothesis is said to be
verified. It is too much here to claim that the hypothesis has been
established as certainly true. In effect, it has been shown only to be a
possibility. Its verification points to probability at best. If it is sub­
sequently verified in a different experiment, it is shown to be more
probable. However, it can never be shown to be more than highly probable
no matter how many times and in how many ways it is experimentally
verified. This is a good example of the unterminated dialectical procedure
which generates a probable conclusion. Let us look more carefully at the
“scientific method” in order to see that this is the case. Given the

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275

hypothesis as an antecedent, the predicted results are related to it as a
conclusion validly deduced from its premises. In the light of the experiment this conclusion is seen certainly to be tme (after the fashion of a
factually evident proposition). Now, as we have seen, a true conclusion can
come either from true or false premises. Thus, beginning with the con­
clusion as evidence, all we can say of the premises is that they could be
true. In the demonstrative argumentation the premises are better known
than the conclusion. Given as certainly true, they generate a certainly
true conclusion. In the empiriological argumentation the conclusion is
better known than the premises. Given as true, it can do no more than
indicate the possibility of the premises. Thus, the method of empiriologi­
cal science is at best dialectical. It can never establish any of its hypotheses
as necessarily true. Although it lacks the perfection of the demonstrative
method as far as certainty is concerned, it is much more widely applicable
than the demonstrative method. It has yielded and it will continue to
yield significant, though only probable, knowledge about a multitude of
things which are more or less impervious to the method of demonstration.
The logical character of the empiriological method is a clear illustration
of dialectical argumentation. Its widespread and profitable employment
is an example of the intrinsic worth of the dialectical procedure.
We have already noted that the demonstrative method presupposes
prior logical discourse which paves the way for the demonstrative proof.
In the sense that this predemonstrative discourse is nondemonstrative
it can be spoken of as dialectical. In this sense of the term, dialectical
discourse can prepare for demonstration in two significantly different ways.
It docs this in a negative fashion when it is employed to refute false
opposing positions. It does this positively by constructively paving the
way for demonstrative argumentation. It may be that much more effort
must be expended dialectically in preparation for a demonstration than
in the demonstration itself. We have mentioned this point already, to
forestall impatience on the part of the student in the face of the apparent
anticlimactic character of demonstration. Once the essential definition of
the scientific subject in a strict explanatory demonstration is known, once
the nominal definition of its property is known, and once the self-evident
character of the premises of the demonstration is appreciated, the con­
clusion is easily and quickly seen as demonstrably guaranteed. Yet there
may be long and difficult processes of division, definition, and argumenta­
tion necessary before the principles necessary for the demonstration in
question are achieved. It is not that an essential definition cannot logically
stand on its own two feet, nor that a self-evident proposition depends
logically on prior propositions. This would be absurd. Yet a definition is

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THE LOGIC OF THE THIRD OPERATION

not easily understood just because it is essential, nor is a proposition
easily grasped because it is self-evident. There usually is the need for
serious and complicated discourse in order to understand fully a definition
or a self-evident proposition. Once the definition or the self-evident propo­
sition is grasped, the dialectical scaffolding erected so that it might be
grasped can be torn down, and the definition or the self-evident proposi­
tion will stand alone. Yet, without the dialectical scaffolding, perhaps
neither would have ever been grasped.
Thus dialectical discourse senes at least three purposes. It is of pre­
demonstrative value in so far as it helps to pave the way for demonstration
where demonstration is possible. It complements the demonstrative
method by supplying arguments of subsidiary value in areas where some
demonstration is possible. And it serves in place of demonstration in
many areas where demonstrative procedure is out of the question for
one reason or another.
Exercise XIX

1. Define: science (empiriological), science (in Aristotle’s sense), demon­
stration, material universality, perseity, commensurate universality' (or
convertibility), explanatory demonstration, demonstration of the fact, a
priori demonstration, a posteriori demonstration, dialectical argumentation,
rhetorical argumentation, poetical argumentation, “scientific method.”
2. What is the difference between the formal and the material consideration
of the syllogism? Illustrate.
3. What is science in Aristotle’s sense? What is the connection between
this and demonstration?
4. What docs it mean to say demonstration is analogous?
5. What must be known prescientifically before one is able to construct a
proof which is an instance of the strictest type of demonstration?
6. Explain how it is that the definition of demonstration in terms of material
cause is necessitated in the light of its definition in terms of final cause.
7. List, and explain, the requirements for the premises of a strict demon­
stra Hon.
8. What is the role of the self-evident proposition in the theory of demon­
stration?
9. Every man is a rational animal is materially as well as formally universal:
it is in the first mode of perseity; and it is commensurately universal or
convertible. Explain. What about Every man is animal? What about
Every stone is colored?
10. Demonstration is logically divided into types by way of two distinct
dichotomies. Nevertheless, there are only three basically different kinds of
demonstration. Explain.

DEMONSTRATIVE .IND DIALECTICAL DISCOURSE

277

11. What are the various ways in which explanatory demonstration can be
achieved? Explain what it is about each which keeps it on the level of
explanatory demonstration.
12. In what way is an a priori demonstration of fact more like an explanatory
demonstration than an a posteriori demonstration? In what way is it
less than explanatory? How do the two remote causes, respectively
distinguishing the two ways of demonstrating in an a priori demonstration
of fact, differ significantly from one another? What do they have in com­
mon so that each is a principle of basically the same type of demonstration?
15. When is a nonconvertible effect able to be a middle term in demon­
stration? When is a nonconvertible effect an illicit principle in demonstra­
tion? Illustrate the difference between an a posteriori demonstration of
fact by way of convertible effect on the one hand and a legitimate non­
convertible effect on the other.
14. Why is it accurate to say that the middle term in the scientific syllogism
is the most significant single principle of demonstrative discourse?

15. In general how are differences in sciences determined according to differ­
ent modes of defining?
16. What is the relation of the logic of demonstration to the methodologies
respectively of the particular sciences?

17. Distinguish between demonstration and dialectical discourse. Carefully
explain the relationships between them.
18. What are the advantages and disadvantages of demonstrative argumenta­
tion on the one hand and dialectical argumentation on the other?
19. Suggest and explain several examples of dialectical discourse.
20. Discuss the “scientific method” and the manner in which it differs from
the method of demonstration. Explain the reasons why it is a good
example of dialectics.
21. Tire following argumentations are suggested as illustrations in a wellknown treatise on demonstrative discourse entitled On Demonstration.‘
Examine each one carefully to see if you can discover the logical points
each is supposed to illustrate.
21.1. Every plane figure bounded by three lines, having an exterior angle
equal to two opposite interior angles, has three angles equal to
two right angles.
But every triangle is a plane figure bounded by three lines, having
an exterior angle equal to two opposite interior angles.
Therefore, every triangle has three angles equal to two right angles.
8 A translation of this treatise by Vincent R. Larkin is found in Readings in Logic,
edited by Roland Houde (Dubuque: William C. Brown Company, Publishers, 1958),
pp. 148—152. The De Demonstratione is a brief work on the theory of demonstration,
traditionally listed among the works of St. Thomas Aquinas. Although it is uncertain
that St. Thomas is the author, it is agreed that the teaching is authentically Thomistic.

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THE LOGIC OF THE THIRD OPERATION

Every natural body illumined by the sun when deprived of light
by the interference of the earth is eclipsed.
But the moon is a natural body illumined by the sun and some­
times deprived of light by the interference of the earth.
Therefore, the inoon is sometimes eclipsed.
21.3. Ever}' triangle has three angles equal to two right angles.
But ever}' isosceles figure is a triangle.
Therefore, every isosceles figure has three angles equal to two right
angles.
21.4. Every plane figure bounded by three lines, having an exterior angle
equal to two opposite interior angles, has three angles equal to
two right angles.
But every isosceles figure is a plane figure bounded by three lines,
having an exterior angle equal to two opposite interior angles.
Therefore, every isosceles figure has three angles equal to two
right angles.
21.5. Every luminous nontwinkling body is situated near the observer.
But a planet is a luminous nontwinkling body.
Therefore, a planet is situated near the observer.
21.6. Every luminous body which is situated near the observer is non­
twinkling.
But a planet is a luminous body which is situated near the observer.
Therefore, a planet is nontwinkling.
21.2.

22. The following examples arc taken from a scholarly article on demonstra­
tion written by William A. Wallace, O.P.4 Examine each one carefully;
then classify and evaluate it in the light of what we have seen about
demonstration.
22.1. What is done always or for the most part, and for what is best
and most suitable, is for a determined goal or for an end (i.e.,
cannot be merely by chance ).
But what is done by nature is done always or for the most part
and for what is best and most suitable.
Therefore, what is done by nature is for a determined goal or for
an end.
22.2. A thing divisible into parts on which depends the motion of the
whole is moved by another.
But whatever is in motion is a thing divisible into parts on which
depends the motion of the whole.
Therefore, whatever is in motion is moved by another.
22.3. A monng thing that is moved by at best a finite series of moved
movers is moved by a first Mover completely unmoved.
But whatever is moved is a moving thing that is moved by at
best a finite scries of moved movers.
Therefore, whatever is moved is moved by a first Mover completely
unmoved.
* "Some Demonstrations in the Science of Nature,” The Thomist Reader, 1957,
pp. 90-118.

DEMONSTRATIVE AND DIALECTICAL DISCOURSE

22.4.

22.5.

22.6.

22.7.
22.8.

22.9.

22.10.

279

A body that waxes and wanes through crescent phases is a spherical
physical body illumined by an external source (i.e., the sun).
But the moon is a body that waxes and wanes through crescent
phases.
Therefore, the moon is a spherical physical body illumined by an
external source (i.e., the sun).
An organism that (not having a photosynthetic process to make its
own food) must seek food from its surroundings is an organism
endowed with sensation.
But an animal is an organism that must seek food from its
surroundings.
Therefore, an animal is endowed with sensation.
A fluid of limited quantity that flows continuously in one direction
is moved circularly.
But blood is a fluid of limited quantity that flows continuously
in one direction.
Therefore, blood is moved circularly.
A rational animal is capable of natural science.
But man is a rational animal.
Therefore, man is capable of natural science.
A being who performs immaterial cognitive acts is a being with
an immaterial cognitive faculty.
But man is a being who performs immaterial cognitive acts.
Therefore, man is a being with an immaterial cognitive faculty.
A being with an immaterial cognitive faculty is a being with an
immaterial substance that is the subject of that faculty.
But man is a being with an immaterial cognitive faculty.
Therefore, man is a being with an immaterial substance that is
the subject of that faculty.
A spiritual substance or an immaterial form is incorruptible or
immortal.
But the soul of man is a spiritual substance or immaterial form.
Therefore, the soul of man is incorruptible or immortal.

23. Examine argumentations found in your textbooks and reference books.
Evaluate them from both the formal and the material point of view, es­
pecially distinguishing the strict proofs from the dialectical argumentations.

CHAPTER

XX

Induction

I. The Insufficiency of Deduction

The basic instance of deduction is the categorical syllogism which
shows how the extremes in the conclusion are related through a middle
term which appears in each of the premises. The connection between this
middle term and the other term in each premise may be evident apart
from a further middle term or it may itself be manifested by way of
another middle term. However, it is impossible to admit an infinite series
of middle terms. Sooner or later the déductive conclusion of the
categorical syllogism must be resolved into premises which are evident
in the light of something other than deductive evidence. We have seen
in an earlier chapter that at least one premise in a categorical syllogism
must be universal, and truly universal, not merely enumeratively so. What
is the character of this universal proposition which is necessary as a prin­
ciple of syllogistic argumentation but cannot be its term? Where does it
come from, and what is the guarantee of its adequacy?
To be more specific, let us consider the most significant instance of
the categorical syllogism, namely, demonstration. 'Hie conclusion in a
demonstration is proved to be necessarily true in the light of premises
which arc necessarily true and manifest the truth of the conclusion.
These premises may be seen to be necessarily true because they are
proved to be true in the light of previous premises. However, unless
some propositions are necessarily true without proof, then no premises
at all can prove any conclusion. An endless series of proved premises is
absurd. Demonstrated conclusions are resolved into premises which are
ultimately principles of proof without being prosed themselves — or there
arc no demonstrations at all. What is the character of these nonproved
principles of proof? Where do they come from, and what is the guarantee
of their adequacy?
The answer to these questions leads us to a discussion of induction.
280

INDUCTION

281

The universal propositions which sene ultimately as principles of deduc­
tive argumentation are seen in the light of their singular instances. The
human intellect is able, so long as there is sufficient sensory- experience,
to rise from the singular instances of a universal proposition to an under­
standing of that proposition itself, either because of a sufficient enumera­
tion of singular instances or because of an insight into the meanings of
the terms involved. In cither case the ascent of the mind from the
singulars to the universal is spoken of as induction. In general the
universal principles of deduction arc products of an inductive movement
of the mind from singular instances of a truth to a universal expression of
this truth which is guaranteed as adequate because it is grounded in an
undeniable experience of singulars. In the special case of demonstration,
as we shall sec, the nonproved principle of proof is a self-evident proposi­
tion, which is in its own way the product of an induction.
II. The Types of Induction

We have said that the movement of the mind from singular instances
of a universal proposition to that universal proposition is induction. As
a matter of fact, the notion of induction can be extended to mean any
passage of the mind from the particular or singular to the universal. This
would include the abstraction of the universal object of simple appre­
hension from the singulars in which it is concretized as well as the
induction of the universal proposition from its singular instances. Induc­
tion, generally understood, terminates either in the incomplex object of
the first operation or the complex object of the second. We have said,
and we shall soon see why this is so, that induction terminates in the
complex object of judgment either as the result of an insight into the
meanings of its terms or as a result of a sufficient enumeration of singular
instances of the universal. The first of these last two inductions is
immediate, while the second is an example of reasoning.
Thus we can distinguish between the following three types of induction :

1. Abstractive induction — the movement of the mind from the concrete
instances of a nature to an understanding of that nature taken ab­
solutely-achieved on the level of simple apprehension.
2. Immediate induction of self-evident propositions — the movement of
the mind from an understanding of the terms involved in a universal
proposition to the necessary truth of that proposition itself — achieved
on the level of the second operation of the intellect in an immediate
judgment.

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THE LOGIC OF THE THIRD OPERATION

3. Mediate induction — the discursive movement of the mind from a
sufficient enumeration of the singular instances of a universal proposi­
tion to the truth of that proposition — achieved in the conclusion to
an inductive reasoning process.

The subject of abstractive induction has been treated elsewhere in this
book. There is no need to go into it again at this point. However, it may
be worthwhile for the student to review what has already been said on the
subject, especially in Chapter II and in Chapter IV. The second kind of
induction, namely, the immediate induction of the self-evident proposi­
tion, is so proximately involved with abstractive induction that it cannot
be understood unless abstractive induction itself is clearly grasped. For
this reason, serious reconsideration of what has already been said on
abstractive induction is recommended as a preliminary to the following
sections on the induction of the self-evident proposition.

III. The Self-Evident Proposition

We have seen that the premises in a scientific syllogism must be
ultimately indemonstrable principles of demonstration. That is, they
must be seen to be universally and necessarily true in themselves and
apart from any propositions brought to bear upon them as evidence for
their truth. They must be premises in an absolute sense, namely, prin­
ciples of proof which are themselves unproved precisely because they
cannot and need not be proved. Propositions of this type must be selfevident propositions, that is, propositions which are known certainly to be
necessarily true once their terms are understood.
The subject and predicate of a self-evident proposition exist in neces­
sary matter and arc so proximately connected with one another that an
understanding of them compels one to assent to the necessary truth
of the proposition in which they are joined, e.g., the proposition The
whole is greater than any one of its parts. No one can understand the
meaning of the terms in this proposition without immediately seeing the
absolutely necessary truth of the proposition itself.
The self-evident proposition is an immediate proposition. This means
that it does not require prior propositions to make it evident. However,
not all immediate propositions are self-evident. We have noted before
that a proposition which finds its evidence in an undeniable factual
situation is also immediate. We have spoken of this second type of
immediate proposition as factually evident in order to distinguish it from
the self-evident proposition. The factually evident proposition is a report

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283

simply on what happens to be the factual situation. As such, the truth of
a factually evident proposition is guaranteed by a factual situation, which,
as factual, might possibly change, e.g., The weather is pleasant today.
Assent to a self-evident proposition is an irrevocable assent to a necessary
connection between a subject and its predicate, in the face of the absolute
impossibility of an assent to its contradiction. Assent to a factually evident
proposition is demanded by what happens to be the ease, and by what
perhaps need not and possibly later on will not be the case. Neither the
self-evident nor the factually evident proposition depends upon prior propo­
sitions to manifest the connection between its extremes. The ven'
meanings of the terms involved does this for the self-evident proposition,
and the factual situation immediately present to the senses does this for
the factually evident propositions. Both are immediate propositions. How­
ever, only the self-evident proposition can be understood to be an ade­
quate absolute premise in demonstration. The factually evident proposition
may find its way into a less strict instance of demonstration. But one
premise, at least, in every demonstration must be ultimately self-evident.
The necessity called for in a scientific proposition can be communicated
to a conclusion only by a proposition which is immediate and irrevocably
necessary. 'Iliis much is true of the self-evident proposition and only
of the self-evident proposition.
We began our discussion of the self-evident proposition by noting the
need for an indemonstrable proposition as an absolute starting point for
demonstration. The self-evident proposition satisfies this need. It is,
first of all undemonstrablc. However, not every undemonstrable proposi­
tion is self-evident. All propositions in contingent matter are undemon­
strable, of course, since demonstration bears only upon the necessary.
However, propositions in contingent matter arc undemonstrable because
of a deficiency in matter. Self-evident propositions are always in necesary
matter, and their indemonstrability springs rather from their excellence
than from any inherent deficiency. Demonstration makes evident what
is not already evident. If something is demonstrable it suffers privation,
since it lacks evidence and must find such evidence in premises. Selfevident propositions do not suffer this privation, since they are already
evident in themselves.
Self-evident propositions are aptly described as the basic truths of the
scientific syllogism. They are basic truths in so far as they are universally
and necessarily true while admitting of no prior propositions necessary
to make them evident. They are truths of the scientific syllogism in so
far as they are principles from which conclusions which have the char­
acter of scientific propositions arc generated.

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THE LOGIC OF THE THIRD OPERATION

IV. The Types of Self-Evident Propositions

Self-evident propositions fall into different classes. Some are said to be
self-evident only in themselves. Others are said to be self-evident to us.
Of these latter some are said to be self-evident in an unrestricted sense.
These are self-evident to all of us. Others are said to be self-evident only
to those skilled in a given scientific area. These are self-evident to the
learned. Let us consider the principles according to which the self-evident
proposition is divided and subdivided in this fashion.
A self-evident proposition is a proposition known certainly to be true
once its terms arc understood. The most perfect instance of this is found
in the proposition in which the predicate is the essential definition or
an clement in the essential definition of the subject.1 Once the subject
is defined, the identity of subject and predicate is grasped and the intel­
lect is moved to commit itself irrevocably to the truth of the proposition.
Suppose, however, that the predicate of a proposition lies within the
definition of its subject, but suppose that this subject is so difficult to
know that it cannot be defined essentially. Such a proposition would
be self-evident in itself, but it would not be self-evident to us. If the
predicate of a proposition lies within the definition of its subject and
if its subject can be essentially defined, this proposition is not only
self-evident in itself but also self-evident to us. If the subject of this
proposition were a common notion, that is, one which can be under­
stood by all men, the proposition would be self-evident to all of us. If the
subject could be defined only by those specially equipped intellectually
to operate within a limited field, the proposition would be self-evident
only to those skilled in that field, that is, self-evident only to the learned.
Thus, it is not too difficult to see, at least in reference to the priman’
instance of the self-evident proposition, the reasons behind the division
and subdivision of the self-evident proposition with which we began
this section.
In defending his argument for the existence of God St. Thomas appeals
to the fact that the proposition God is is not self-evident to us, even
though it is self-evident in itself. Were we to know the essence of God,
1 This instance of the self-evident proposition can be seen to be identified with a
proposition in the first mode of perseity. However, it is not to be understood that the
notion of self-evident proposition (even though these are traditionally referred to as
per se nota propositions) is to be identified with the notion of a proposition involving
a mode of perseity (traditionally called a modum dicendi per se). The conclusion in
a demonstration proving a property of its proper subject is in the second mode of
perseity without, of course, being self-evident, since it is a conclusion.

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285

we could not —nor would we need to — demonstrate His existence, for
His essence is His existence. Yet, since we do not fully comprehend His
essence, this proposition is not self-evident to us. We must, and can,
make it evident by proving that God exists by means of an a posteriori
demonstration which argues from His effects, which are known to us,
to their cause — God Himself. Aristotle and St. Thomas both supply us
with examples of propositions which are self-evident to all of us. These
propositions arc known to all because their terms arc common notions
easily and surely grasped by all men. lire examples include: The same
thing cannot be and not be. The same proposition does not admit
simultaneously of affirmation and denial. The whole is greater than any
one of its parts. Things equal to one and the same thing are equal to one
another. Propositions of this type can be spoken of as axioms. Axioms
must be known and assented to by anyone who engages in rational
discourse. They are the absolutely ultimate and common principles which
guarantee the integrity of all discourse and into which all discourse is
in one way or another resolved. Propositions which arc self-evident only
to the learned are related to the axioms as the proper to the common.
They are known only to a limited number of men precisely because the
terms involved arc less common and more determinate. St. 'Ihomas
illustrates the more restricted type of self-evident proposition by sug­
gesting the proposition All right angles are equal. 'Ibis is a proposition
immediately evident only to one who knows that equality enters into
the definition of right angle, and the definition of right angle is a defini­
tion which escapes the knowledge of many. Another example traditionally
offered is the proposition Incorporeal substances are not situated in place.
We can add to these any proposition in which the essential definition or
part of the essential definition is predicated of a specific subject, such as
Every man is a rational animal. A proposition of this type is sometimes
spoken of as a thesis rather than an axiom. We saw' that the axioms arc
necessary if we are to demonstrate in any scientific area. A thesis proper to
a given area is necessary for demonstration properly within that area.
Axioms may or may not be used explicitly as premises in a demonstration.
Nevertheless, the principle of contradiction, namely, The same proposition
docs not admit simultaneously of affirmation and denial, is presupposed
necessarily as a methodological principle of any demonstration whatsoever
(i.e., as a principle guaranteeing the logical integrity of the demonstra­
tion) even though it is not used explicitly as a premise. For no thesis
can function properly as a principle of demonstration unless it is
firmly understood that its denial and the affirmation of its opposite are
excluded in the face of its own affirmation.

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286

Before leaving this section let us recapitulate by considering the fol­
lowing schema which represents the division and subdivision of self-evident
proposition into its types.
Self-evident
only in itself (e.g., God is.)

Self-evident
proposition
/

To all of
us — axiom

(E.g., The same thing
cannot be and not be.)

To the
(E.g., Man is a rational
learned — thesis animal.)

V. The Induction of Self-Evident Propositions

Self-evident propositions arc, in a sense, natural to the human intellect.
This docs not mean that they arc innately possessed and fully formed
conceptions which arc known prior to any experience of the cxtramental
world. Nor does it mean that they arc mental constructs fabricated by
the intellect out of its own “stuff.” Self-evident propositions are im­
mediate in the sense that they do not depend on prior propositions
for evidence, but they do not arise in us as from a vacuum. They proceed
from prior sensory experience by an immediate induction very intimately
connected with the abstractive induction of the universal concept from its
sensible manifestations.
The truth of a self-evident proposition is seen once the meanings of
its terms are known. These meanings arc grasped in an abstractive
induction of the intelligible object of apprehension from the particular
phenomena of the sensibly existing thing. Thus a self-evident proposition
is grasped without reference to prior propositions, but only after (and,
in a sense, from) sensory experience sufficient to ground an abstractive
induction of terms which are in themselves suited to be the extremes of
a self-evident proposition.
The self-evident proposition is not simply a report on a factual situa­
tion. Yet it is not totally nonempirical. It is, as we have seen, known by
way of an immediate induction from sensible data. Nevertheless, it would
be misleading to say that it depends directly upon empirical data for its
verification. Let us grant that the whole would not be greater than its
parts if all wholes were not in fact greater than their parts. Nevertheless,

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given the meanings of whole and part, wholes must be greater than their
parts, as long as they are at all. Thus, if one understands the meanings
of the terms in the proposition The whole is greater than any of its parts,
one immediately assents to this quite apart from any and all actual
instances of sensibly existing wholes and parts. The motive for assent is,
as it were, built into the content of the proposition itself. We might say
that a self-evident proposition has an intrinsic intelligibility which neces­
sarily constrains the mind to assent. I do not have to have a concrete
whole and its parts in front of me to know the whole is greater than its
parts. It is enough to know the meanings of whole and part. Yet I would
never know those meanings if I never knew any concrete whole and its
parts. And, of course, there is no reality at all to whole or part except
in so far as there are (at leastpossibly) concretely existing wholes and
parts. For this reason, self-evident propositions are inductively attained
only from an experience of singulars, and they make sense only in so far
as they bear upon singulars. Yet they arc only materially dependent upon
experience for their verification. Formally they are verified in their own
necessary intrinsic intelligibility, which precludes the possibility even of
conceiving the opposite.
VI. Mediate Induction

The real world is a world of established regularity. It makes no sense
to conceive of it as though it were fundamentally chaotic, with no
explanation for its phenomena save chance alone. The universe of things
is, as a whole, an ordered universe. Moreover, each of the beings in it is
fundamentally intelligible with a determinate nature, which is what it is
because of its ordination to a determinate end. Chance events occur,
but when they do occur they arc the results of intercepting lines of final
causality. Chance itself is inexplicable except in terms of an underlying
order in nature.
The cxistents in the real world are singular things. They are incommunicably distinct, as things, from one another. However, several indi­
viduals can fall under the same species, and, along with members of
other species, under successively more universal genera and finally under
an ultimate genus, which is a determinate mode of being itself. Each
determinate type of being, generic or specific, is constituted intrinsically
by intelligible principles proper to the essence in question. These essen­
tial principles are multiplied from individual to individual, while remain­
ing, of course, intelligibly identical. These principles are the cause of
properties in things. These properties are, like the principles which cause

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THE LOGIC OF THE THIRD OPERATION

them, numerically distinct from individual to individual while remaining
intelligibly identical. The connection between the principles proper to
one (one in the sense of being intelligibly one) essence and a property
necessarily consequent upon them can be expressed in a universal proposi­
tion. This universal proposition can be spoken of as a law, since it
enunciates an infallible connection between a nature and one of its
attributes which cannot not-be realized in any singular instance of
that nature.
We have seen that some universal propositions arc known immedi­
ately upon the understanding of the terms in the proposition itself.
These arc the self-evident propositions, which are acquired by way of
immediate induction. In some cases, however, very' often because the
subject in question cannot be adequately defined to begin with, it is
not self-evident (at least to us) that a universal law is true. This does
not mean, necessarily, that it is beyond our reach. Precisely because of
the underlying order in reality, whenever a given attribute is found
regularly with a given subject, despite varying circumstances, there is
sound reason to presume that there is a significant connection between
the attribute and that subject. Thus, given a sufficiently discriminative
enumeration of singular instances of a given subject with a given attribute,
it is valid to conclude that the attribute belongs necessarily to the sub­
ject. Here we have induction, namely, the passage from the singular to the
universal. Moreover, we have a special kind of induction, which differs sig­
nificantly from the immediate induction of self-evident propositions. This
is a case of inductive argumentation. It is a mediate induction proceeding
from a sufficient enumeration of singulars to a universal truth which is
particularized in each of the singulars.
Wc must be careful to note very clearly the manner in which mediate
induction differs from the immediate induction of self-evident proposi­
tions. In immediate induction there is no question of any movement
from premises to a conclusion. The self-evident proposition is in no way
a conclusion. It does not look for its evidence to any prior propositions,
but is known to be true in the light of the meanings of its terms. It is a
universal law which is immediately seen to be true. This is not the case
with the proposition achieved by way of mediate induction. This proposi­
tion has the characteristics of a conclusion. Since it is a universal proposi­
tion, it cannot be factually evident (though each of the particulars which
manifest it may be a factually evident proposition). Nor is it a self-evident
proposition, since its truth is not immediately evident in the light of
its terms. It is a proposition which looks beyond this or that fact, as
well as beyond itself, for its evidence. It is seen to be true only in the

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light of a sufficient enumeration of singular instances of itself. The mind
moves legitimately from the knowledge that an attribute belongs to a
given subject in this case, and in that case, and in a sufficient number
of cases, to the knowledge that the attribute belongs universally to every
instance of this subject. The principle of manifestation in mediate induc­
tion is not an insight into the intrinsically necessary intelligibility of a
universal law. It is the sufficiency of an enumeration of singular instances
of this universal law. The evidence for the inductive conclusion is the
enumeration itself. If it happens that an enumeration occasions an
insight into the intrinsically necessary intelligibility of a universal proposi­
tion so that it is seen to be necessarily tme in the light of its terms, the
proposition ceases to be an inductive conclusion and takes on the charac­
ter of a proposition which is self-evident to us. At this point the enu­
meration itself ceases to be of logical significance as a principle of mani­
festation, though it has served materially to ground the intuition of mean­
ing which is now of prime logical significance. When argumentative
induction does not give way to an immediate induction of a self-evident
proposition, the sufficient enumeration of singulars remains the principle
of manifestation which supplies the evidence for the universal law which
is the inductive conclusion.
We must be as careful to distinguish the inductive conclusion from
the deductive conclusion as we have been to distinguish the inductive
conclusion from the self-evident proposition. The inductive and deduc­
tive conclusions are alike only in so far as each is a mediate proposition,
that is, a proposition reached only as the term of rational discourse.
Beyond this, however, they differ radically. The deductive conclusion
depends upon a middle term simultaneously identified with each of
its extremes and thereby manifesting their identity (or if the middle
functions as a principle of separation, their nonidentity). The induc­
tive conclusion looks to an enumeration of singulars, each of which
is a particular instance of it, to manifest the connection between
its extremes. Later we shall note that the inductive enumeration of
singulars lacks the rigor of the deductive middle term, so that, in
practice at least, the inductive conclusion is ordinarily dialectical at
best. For the moment it is important simply to stress the radical
difference between the inductive argument and the syllogism. The two
are irrcducibly distinct forms of argument precisely because they rest upon
distinct logical principles. The basic principle of syllogistic procedure is
the principle of triple identity: two terms each of which is identified
with the same third term are identified with one another. On the basis
of this self-evident logical principle, the categorical syllogism proceeds to

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THE LOCIC OF THE THIRD OPERATION

show the identity between the extremes of its conclusion through a
middle term with which these extremes are both identified, e.g.:

Every animal is mortal.
But every man is animal.
Tliercfore, every man is mortal.
Tlic basic principle of mediate induction, however, is not the principle
of triple identity but rather what we can call the principle of sufficient
enumeration: what can be said of many sufficiently enumerated instances
of a universal subject can be said of that universal subject itself. On
the basis of this self-evident logical principle, the inductive argument
manifests the identity between the extremes of a universal proposition by
enumerating singular instances of this universal proposition sufficient to
warrant the intellectual “leap” from the singular to the universal, e.g.:
Iliis man is mortal; and that man
is mortal; and this other man
is mortal; etc.
Therefore, every man is mortal.

In so far as both deductive and inductive argument “work” in virtue
of distinct logical principles, each involves a distinct mode of inference,
'rhe two may be interrelated, as, for example, whenever the conclusion
of an inductive argumentation is used as a deductive major premise.
Nevertheless, they remain irreducibly distinct forms of argumentation,
each with its own distinctive role to play in the life of learning.
VII. The Problem of the Inductive Enumeration

We have seen that in inductive argumentation an enumeration of
singulars assumes the role played by the middle term in the syllogism.
However, the middle term in the syllogism, at least from the point of
view of formal logic, poses much less of a problem than the inductive
enumeration of singulars. Given a term which is fully extended at least
once, and which is identified with both the major and minor terms (or,
in the negative syllogism, identified with one and separated from the
other), the syllogistic conclusion follows necessarily from the premises.
Furthermore, if those premises are true, the conclusion follows neces­
sarily as true. There is, from the formal point of view, little difficulty
with the middle term in the syllogism. Either the term proposed as the
middle term is a middle, or it is not. It is not difficult to decide this
issue. However, in the case of argumentative induction there are special

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291

problems with the enumeration of singulars which manifests the connec­
tion between the extremes of the inductive conclusion. We have said
that a sufficient enumeration of singulars is required for an inductive
conclusion. This seems to allow for the possibility of an enumeration
which is not sufficient. This is precisely the problem. What constitutes
sufficient enumeration? No a priori rule can be given which will cover
all cases. The best we can say is that an inductive enumeration is suffi­
cient whenever it rules out the possibility of any singular which would
contradict the inductive conclusion. The trouble with this, of course, is
that we arc saying no more than that an enumeration is inductively
sufficient whenever that enumeration is sufficient for induction. Ad­
mittedly, this is redundant, and yet it is the best that we can do. Given
the fundamental order and regularity of the universe, there is a point
at which the human intellect can rise from the singular instances of a
universal proposition to the universal proposition itself precisely because
there is no reason sufficient to explain the recurrence of these singulars
save the truth of that universal law. This is the point of sufficient enu­
meration. It cannot be numerically located. It is not enough simply
to multiply singular cases in order to acquire evidence sufficient for
induction. Tlic observations upon which induction rests must be dis­
criminative rather than numerous. For example, it is not enough simply
to observe that a given attribute is found a great many times with a
given subject to conclude that it belongs universally to that subject. Sup­
pose it happens in each of the many singular cases observed that one
and the same accidental characteristic happens to belong to the subject
along with the attribute in question. Under the circumstances it is
conceivable that the attribute follows upon the accident and not upon
the subject in question. No matter how the observations of this subject
arc multiplied, no conclusion is seriously warranted until the subject
is observed without the accidental characteristic. If the subject, observed
apart from the accidental characteristic, is seen to have the attribute in
question, it is clear that the attribute does not follow from the accident
and may well belong necessarily to the subject. One observation of the
subject without the variable condition of the accident is worth more
than any number of observations with the accidental variable, so long
as there has been some observation of the subject with the accident. Only
when the singulars observed have been discriminatively evaluated so as
to take note of all the variables can they' be deemed sufficient to yield an
inductive conclusion. In most cases there must be a good number of
singulars observed before a conclusion follows. Yet, in no case can we
state a priori just how many this will be. In every case, however, we

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THE LOGIC OF THE THIRD OPERATIO?’

must insist that more than a numerical multiplication of singular in­
stances is necessary in order to ensure sufficiency.
We have spoken of sufficient enumeration, and we have indicated that
it is difficult to explain the point at which enumeration is sufficient for
induction. There is, however, one instance of enumeration which must
be sufficient. This is the case of complete enumeration. Inductive enu­
meration is complete whenever all the subjective parts of a universal arc
included in the enumeration. Clearly if B can be said of all the instances
of A, then every A is B. However, there are several points which must
be brought out here in order to explain the possibility of complete enu­
meration on the one hand and the dangers involved in misunderstanding
it on the other.
First of all, we have consistently spoken only of the enumeration of
singulars. Lest we be misled, let us hasten to note that the inductive
movement can be correctly described as proceeding from the less uni­
versal to the more universal. It is not the case that in every instance of
induction the movement is precisely from singulars to a universal. ITiis
is the primary instance of induction, and the one in the light of which
the others are best understood. However, it is possible to move to a
conclusion about a generic subject from a sufficient experience of the
various species of this genus. It is impossible to observe all the singulars
falling under a universal, since there are an infinite number of singular
inferiors possible for any concept. But it is possible to know all the
species falling under a genus. It is possible to know that B Mongs to
this species of A and to that species of A, and to each and even' species
of A, and from this complete enumeration it is legitimate to conclude
that everj' A is B. There can be no question here of insufficient enu­
meration so long as the enumeration is exhaustive. Yet, when it is
exhaustive, do we have an advance in knowledge from the enumeration
to the conclusion? If X, Y, and Z are the only species of the genus A.
and if it is known that X is B and Y is B and Z is B, it seems that there
is no advance in knowledge expressed in the proposition Every A is B
Since X and Y and Z taken together are extensively equivalent to A.
it seems that to know that X is B and Y is B and Z is B is already to
know that even' A is B. If there is no advance in knowledge here —i.e.,
if Every A is B is not different from X and Y and Z arc B — then there
is no mediate induction. However, it is not necessary to deny an advance
in knowledge in Every A is B over X and Y and Z are B. Only in a
nominalistic scheme of things, in which the so-called universal is no
more than a collection of its so-called subjective parts, must Every A is B
be an alternate expression of X and Y and Z are B (when X, Y, and Z

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293

exhaust the genus A). In a realist scheme A can be understood as a legiti­
mate generic aspect of things realized in each of its inferiors, but known
apart from its specific differences and individuating characteristics. In the
proposition Every A is B the predicate B is affirmed determinately of the
subject A, and this subject represents a legitimate abstraction from the
species X, Y, and Z. A is not simply the collection of X and Y and Z.
A is a nature which can be considered absolutely or, in other words, for
what it is in itself. To know that every A is B is to know that B belongs
to A taken absolutely, and through A to each of the inferiors participat­
ing in A. So long as a universal subject is not understood simply as the
collection of its extensive parts (that is, as a strictly enumerative uni­
versal) but rather as one intelligible object (related to its extensive parts
as genus to species), there is no reason why a complete inductive enu­
meration cannot be sufficient for a truly inductive conclusion.
Complete enumeration is always sufficient enumeration. But, as we have
seen, complete enumeration is not always possible. Whenever the induc­
tion proceeds from singulars in the strict sense, complete enumeration
is never possible. Then we are brought face to face with the problem we
have already considered: when is the enumeration of singulars (admittedly
incomplete) sufficient to warrant the jump from the singulars to the uni­
versal conclusion? The best we have been able to do is to suggest that
this occurs when the enumeration is quantitatively and qualitatively com­
plete enough so that the possibility of an exception to the universal
is ruled out. This occurs when the regularity in the presence of the
predicate in a given type of subject is seen as inexplicable unless the
predicate belongs necessarily to each and every instance of the subject
in question. A conclusion such as this is legitimate only when every
conceivably significant factor is checked out. So long as something acci­
dental to the essence of the subject might possibly be suggested as the
reason for the predicate in question, it is illegitimate to affirm that the
predicate belongs universally to the subject. Under the circumstances it
is clear that sufficient enumeration — save in the case of complete enu­
meration — is extremely difficult. This is at least true for inductive con­
clusions generated as certainly true. However, this is not the case for
probable conclusions to inductive argumentation. Any significant enu­
meration of singulars generates some degree of probability for the uni­
versal that these singulars seem to represent. The more complete the
enumeration, the more probable the conclusion. However, so long as there
is any possibility of an accidental explanation for the seemingly neces­
sary identity of an attribute with a given universal subject, the proposi­
tion expressing this identity is only probable. Because of the extreme

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THE LOGIC OF THE THIRD OPERATION

difficulty in achieving an enumeration sufficient for certitude, most induc­
tive argumentations yield nothing better than a probable conclusion,
though perhaps a conclusion with a very high degree of probability.
Exercise XX

1. Define: induction (generally taken), abstractive induction, immediate
induction of principles, mediate induction of conclusions, self-evident
proposition, axiom, thesis, inductive enumeration, sufficient enumeration,
complete enumeration.

2. What is there about deduction that rules out the possibility that the evi­
dence for all propositions might be deductive?
3. What is the general notion of induction? How is this realized in each
of the three types of induction treated in this chapter?

4. What is there about demonstration which necessitates the existence of
self-evident propositions?

5. Explain each of these statements:
5.1. The self-evident proposition is a premise, but not every premise is
self-evident.
5.2. The self-evident proposition is immediate, but not every immediate
proposition is self-evident.
5.3. The self-evident proposition is necessary, but not every necessary
proposition is self-evident.
5.4. The self-evident proposition is indemonstrable, but not every in­
demonstrable proposition is self-evident.
6. What is the difference between a proposition self-evident only in itself
and a proposition self-evident to us? Illustrate.
7. What is the difference between an axiom and a thesis? What arc the
different roles each has to play in scientific argumentation? Illustrate.
8. Is there any distinction to be made between self-evident propositions and
propositions involving a mode of perseity?
9. In what way do self-evident propositions depend upon the data of sense
experience? How are self-evident propositions verified?
10. What is wrong with a description of the self-evident proposition which
identifies it as a proposition clearly and easily understood?
11. What is there about reality which makes mediate induction possible?
12. What precisely is the difference between the evidence for a self-evident
proposition and an inductive conclusion?
13. What precisely is the difference between the evidence for an inductive
conclusion and the evidence for a syllogistic conclusion?
14. Why are induction and deduction irreducibly different modes of inference?

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295

15. Explain: in the inductive argumentation complete enumeration is always
sufficient, but sufficient enumeration is not always complete.
16. Under what circumstances is complete enumeration a possibility? Illustrate.
17. What docs an inductive conclusion add to a complete enumeration so
that it can be considered truly an advance in knowledge and thus a
conclusion?

18. Sufficient enumeration is ordinarily sufficient at best for a probable con­
clusion. Explain.
19. The following rules2 are the famous canons for induction proposed by
John Stuart Mill, a nineteenth-century logician who is best known for
his discussion of the inductive method. Discuss the value of each one,
especially with an eye to the discriminative character of the inductive
enumeration.
19.1. If two or more instances of the phenomenon under investigation
have only one circumstance in common, the circumstance in
which alone all the instances agree is the cause (or effect) of the
given phenomenon. (Method of Agreement)
19.2. If an instance in which the phenomenon under investigation
occurs, and an instance in which it docs not occur, have even'
circumstance in common save one, that one occurring only in the
former; the circumstance in which alone the two instances differ
is the effect, or the cause, or an indispensable part of the cause,
of the phenomenon. (Method of Difference)
19.3. If two or more instances in which the phenomenon occurs have
only one circumstance in common, while two or more instances in
which it docs 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. (Joint Method of Agree­
ment and Difference)
19.4. Subduct from any phenomenon such part as is known by pre­
vious inductions to be the effect of certain antecedents, and the
residue of the phenomenon is the effect of the remaining ante­
cedents. (Method of Residues)
19.5. 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. (Method of Concomitant Variations)
20. Evaluate the following attempts at induction. Add to these other exam­
ples of induction taken from your textbooks and reference works in other
classes.
2 John Stuart Mill, System of Logic, Bk. Ill, Ch. VIII (New York: Longmans.
Green and Company, 1930), pp. 255, 256, 259, 260, 263.

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THE LOGIC OF THE THIRD OPERATION

20.1.

20.2.
203.

20.4.
20.5.

20.6.
20.7.
20.8.

20.9.
20.10.

Since the scalene triangle has three interior angles equal to two
right angles, and since the same is true of the isosceles triangle and
the equilateral triangle, it follows that every triangle has three
interior angles equal to two right angles.
Since scotch and soda, rye and soda, brandy and soda, bourbon
and soda, and gin and soda have all been observed to make a man
drunk, it is clear that soda makes a man drunk.
As long as a large number of observations have been made of
liquid boiling at 100 degrees centigrade, it follows that all liquid
boils at 100 degrees centigrade.
Since the senses of sight, hearing, taste, smell, and touch each
demand a corporeal organ, it follows that the senses as such
demand a corporeal organ.
Since all the snow that we have ever seen, or that anyone has
ever seen, has been white, we can conclude that all snow is
naturally white.
Since samples of all known metals have been observed to conduct
electricity, it follows that every metal conducts electricity.
Several metals, for example, lithium, potassium, and sodium, are
lighter than water. Therefore, it is safe to conclude that all metals
are lighter than water.
Since the immediate inferior of species, specific difference, specific
property, and accident is in each case a singular, it follows that the
immediate inferior for every' predicable is a singular.
Since every healthy adult human being speaks, it follows that the
ability to speak belongs by nature to man.
Men die, brutes die, and plants die: hence, all living things die.

CHAPTER

XXI

Fallacious Argumentation

I. The Notion of the Fallacies

Any logically erroneous procedure in rational discourse is spoken of as
a fallacy. Thus, any logically defective argumentation is said to be fal­
lacious. This is the case whether the defect is on the side of form or on
the side of matter. Strictly speaking, an argumentation which is de­
signedly fallacious and intended to deceive is said to be sophistical.
Sophistry can be defined as the art of deceptive argumentation. Sophistry
aims at the appearance of truth, but has no commitment to the truth
itself.
In a sense we have already considered the fallacies. We have investi­
gated the rules for proper logical procedure and have faced the possi­
bility that each might be violated. Tire violation of any one of the rules
for sound discourse is a fallacy. Thus, we may speak of the fallacy of four
terms, the fallacy of the unextended middle tenn, the fallacy of the illicit
major or minor term, and the fallacy of demonstrating from merely
probable premises. We could add to our list of fallacies by naming
the violation of each of the rules wc have already discussed.
There is no limit to the ways in which the rules for sound discourse
can be violated. There is only one legitimate construction for any given
logical form, but there is an indefinite number of ways in which a defect
can be introduced into this construction. We have not been able to sug
gest them all, nor do wc intend to do so in this chapter. However, there
are some rather common fallacies which are traditionally given special
consideration. We shall take them up rather briefly in this chapter. A
direct treatment of some fallacies helps, by contrast, to heighten our
appreciation for licit discourse. A special study of these common fallacies
puts up our guard against them. The fallacies which are traditionally
considered fall into two classes. The first is generally grounded upon
some defect or difficulty on the side of verbal expression. These are the
297

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THE LOGIC OF THE THIRD OPERATION

fallacies of language. 'Fhe second is rooted in something other than a
semantic confusion. These are the fallacies beyond language.
II. The Fallacies of Language

A. The fallacy of equivocation
The fallacy of equivocation is the use of identical oral or written terms
to stand for diverse meanings in the same discourse. This includes the use
of strictly equivocal terms and the illegitimate use of analogous terms.
It also includes the use of terms so vague as to be practically meaningless.
We might even take it to include the use of terms which, though stand­
ing for identical meanings, differ in supposition from proposition to
proposition within an argumentation. Terms are conventional signs, and
it is inevitable that they will be used in different ways at different times.
Logical discourse requires a rigorous determination of the meanings of
the terms involved and a uniformity in the use of these terms in the
propositions of the discourse. Otherwise, the argument will lack integrity,
for it leaves in doubt the status of the terms, which are the fundamental
elements into which any discourse can be finally resolved.
Examples of the fallacy of equivocation are so common that it is not
necessary to supply them here. However, the student should refer to
Chapter X to review the discussion on the unity of signification and
supposition needed for valid discourse, and to reconsider the examples
offered there of the fallacy of equivocation.
B. The fallacy of ambiguity in expression
'Fhe fallacy of ambiguity in expression is the use in discourse of
badly expressed phrases which readily admit of misinterpretation. This
differs from the fallacy of equivocation in that here the confusion is not
in the terms themselves but in their syntactical arrangement. The fallacy
of ambiguity in expression is sometimes called amphiboly.
The following examples of amphiboly should make it very clear that
this fallacy is inconsistent with clear and rigorous expression in logical
discourse:

1. “Logic is necessary for metaphysics because it is scientific.”
(What precisely is scientific — logic or metaphysics?)
2. “The free world the Communist world will survive.”
(Who precisely will survive whom?)
3. “Kathleen loves Susan because she is her mother.”
(Who is mother of whom?)

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299

C. The fallacies of composition and division
The fallacy of composition consists in combining words which should
be understood separately, and the fallacy of division consists in separating
words which must be understood together.
Consider the proposition: “The only noncssential predicables are
properties and accidents.” This makes sense only if “properties” and
“accidents” arc understood separately. It is enough for a predicable to
be a property (or, if not a property, an accident) for it to be a nonessential predicable. To misconstrue the proposition and understand it as
demanding that a nonessential predicable be simultaneously a property
and an accident is to commit the fallacy of composition.
Consider this other proposition: “The major premise and minor
premise are the antecedent in the syllogism.” This proposition is true
only if “major premise” and “minor premise” are taken together as one
logical subject, for they represent the antecedent of the syllogism only
in so far as they are understood as one pair. Neither the major premise
nor the minor premise alone is the antecedent of the syllogism. To take
them separately as though the predicate belongs divisively to each is
to commit the fallacy of division.

D. The fallacy of accent
The fallacy of accent is the misuse of an accent in order to place undue
emphasis upon some clement of a proposition and thereby deceive. The
addition of a special accent to the ordinary proposition can alter its
meaning or add something to it. Consequently, one must be careful in
accentuating propositions and in interpretating propositions involving any
unusual accentuation.
To illustrate how significantly an accent can modify the meaning of
a proposition, consider these two examples:
1. “Logic is not a science of real things.”
2. “Logic is not a science of real things.”

The first example, in which “of real things” is accented, suggests that
logic is a science, but that it is not a science of real things. This is true.
The second, in which “science” is accented, suggests that logic may be
of the real, concerned with real things, but that it is not a science. This
is false. Other examples of this fallacy include:
1. “Some loyal Americans hate Communism.”
(Though an I proposition does not, on its own merits as an I
proposition, imply its subcontrary, this proposition, because of
its accent, implies “Some loyal Americans do not hate Com­
munism.” )

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THE LOGIC OF THE THIRD OPERATION

2. “Aristotle was a good logician."
(The accent added here makes this proposition imply that Aris­
totle was not good at things other than logic. Without the
accent, this proposition leaves this an open question.)

E. The fallacy of similar word construction
This is the fallacy involved in thinking that any similarity in verbal
expression is a sign of a similarity in meaning. Grammatical expression
frequently reflects the logical situation rather accurately, but this is not
always the case. Language is to an extent arbitrary, but logic is not.
To infer that “inflammable” and “invaluable” mean “not flammable"
and “not valuable” just because “insincere” and “invalid” mean (at
least) “not sincere” and “not valid” is to commit the fallacy of similar
word construction. Actually, “inflammable” and “flammable" mean the
same thing, while the “in” in “invaluable” intensifies the adjective instead
of negating it.
These five fallacies do not represent all the fallacies of language,
although they represent the most significant. They arc important in
themselves, since they make themselves felt frequently and subtly. They
are also important as a serious reminder that all fallacies of language
stand in the way of profitable scientific (as well as nonscientific) dis­
cussion. Although there arc many serious problems that cannot be
solved simply by knowing what the terms involved mean, we can never
begin to work on their solution unless we first eliminate completely
whatever semantic confusions originally exist Hath respect to them.
III. Fallacies Beyond Language

A. The fallacy of accident
The fallacy of accident consists in mistaking the accidental (or non
essential) for the essential. Some things are affirmed or denied of a
subject in virtue of something essential to it. Other things are affirmed
or denied of a subject in virtue of something noncsscntial to it. To
confuse these two is to commit the fallacy of accident.
The following examples of this fallacy should be sufficient to indi­
cate the frequency and serious consequences of its use:

1. An argument against a man’s right to life, liberty, and the pursuit
of happiness on the grounds of the color of his skin.
(The right to life, liberty, and the pursuit of happiness is a right
consequent upon human nature as such and not upon an acci­
dent of human nature such as the color of a man’s skin.)

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301

2. An argument for the condemnation of all books because some
books are obscene.
(An obscene book deserves condemnation not in virtue of its
essential character as a book, but rather in virtue of its acci­
dental state of obscenity.)
3. An argument designed to show that a man is not a good artist
because he is not a good man.
(The question of whether or not any man is a good artist is quite
accidental to the question of whether or not he is a good
man. A man is good as a man if his will is rectified. The recti­
fication of the will is compatible with the absence of artistic
talent, while artistic talent is compatible with an unrectified
win.)
B. The fallacy of mistaking the qualified for the absolute
The fallacy of mistaking the qualified for the absolute consists in
monng from something true only relatively or in a special situation to
the unjustified assumption that this is true absolutely and universally.
This fallacy includes the unwarranted jump from the truth of an I or O
proposition to the truth of the corresponding A or E.
The following examples illustrate this fallacy:
1. An argument designed to show that any man can do and say
whatever he chooses to do and say on the grounds that a man
is by nature free.
(Man is free in only a qualified sense. There are limits to human
freedom which are ignored in this argument.)
2. An argument against capital punishment based on the command­
ment of God: Thou shalt not kill.
(This commandment is not to be taken absolutely. There is the
possibility that one man may in justice take the life of another,
e.g., in self-defense. This argument falsely presumes that the
commandment in question applies without qualification.)
3. An argument that a given pan of water will in fact boil at 100
degrees centigrade because all water boils at 100 degrees
centigrade.
(It is true that all water boils at 100 degrees centigrade, but only
at sea level and only without impurities. This pan of water may
not be at sea level and may contain many impurities.)
C. The fallacy of arguing beside the point.
'Ihe fallacy of arguing beside the point is the practice of ignoring the

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THE LOGIC OF THE THIRD OPERATION

proper issue in an attempt to make a point by way of something extrin­
sic to it. Next to the fallacy of equivocation, this is the most common
and persistent fallacy. As a matter of fact, equivocation and arguing be­
side the point can go hand in hand. One reason for ignoring an issue
is to miss it completely through a misunderstanding of the terms used
in it. There are many examples of the fallacy of arguing beside the
point. We shall consider several of them. The species of this fallacy arc
best known by their Latin names, and we shall use these names to
identify them.
a) Argumentum ad hominem (argument aimed at the man):
The argumentum ad hominem confuses the person of one’s opponent
in debate with the real issue of the debate. Whenever one departs from
the true issue to attack one’s opponent, one uses the argumentum ad
hominem. This is the method used, for example, by a man who ques­
tions the political theory of another on the grounds that the other has
been an unsuccessful businessman or has had a criminal past. Again,
it is not relevant to argue that a man’s philosophical stand on a moral
issue is false simply because the man himself is immoral, or because he
was previously shown to be wrong on a question in nuclear physics, or
because he once had a friend who was a Communist. It is never legitimate
to attempt to make a point by discrediting the person of an opponent,
unless the opponent’s argument is in some way based upon his per­
sonal integrity.
b) Argumentum ad populum (argument aimed at the people):
The argumentum ad populum is an appeal to popular passion, school
spirit, racial enthusiasm — that is, to some kind of partisan prejudice —
in order to make a point which has nothing to do with passion or
prejudice. It may be quite easy, regardless of the facts, to convince a
Frenchman that a war against Germany is justified, or a Southerner that
racial integration should be indefinitely put off, or a Marquetc University
student that Marquette’s basketball team is the country’s finest. How­
ever, so long as these arguments rely on some prejudice in their hearers
and lack solid premises which are to the point, they arc simply instances
of an argumentum ad populum.
c) Argumentum ad baculum (argument aimed at the stick):
The argumentum ad baculum is an attempt to win acceptance of a point
by the threat of force rather than by backing it up with solid premises
establishing a conclusion. A logic teacher who argues that students should
study logic in order not to fail the course, or a practical politician who
solicits votes by threatening to do violence to those who refuse to vote
for him, or an executive who wins approval for his policies by threaten­

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303

ing subordinates with dismissal —all these employ the argumentum ad
baculum.
d) Argumentum ad verecundiam (argument aimed at respect):
The argumentum ad verecundiam is an appeal to misplaced authority in
an attempt to establish a point. It is a legitimate dialectical method to
argue from the opinion of an expert in a given field to a conclusion in
that field. However, it is not legitimate to appeal to an authority in one
field to establish a point in another. To do this is to employ the argu­
mentum ad verecundiam. Examples of this fallacy are thrust at us in
our magazines and newspapers, and over our radios and television sets.
Movie actresses give advice on auto insurance, baseball players on shav­
ing equipment, and jazz musicians on wall-to-wall carpeting. There are,
more unfortunately, examples of this fallacy found even in academic
circles, where experts in one field frequently presume upon their
authority in this field in order to make a point in some distinctly different
academic area. Thus an expert psychologist may expect a hearing on
philosophy, and an expert physicist on theology, or an expert mathe­
matician on law.
e) Argumentum ad misericordiam (argument aimed at pity):
The argumenturn ad misericordiam is an appeal to a man’s sympathy
rather than to his reason in order to make a point. This fallacy is very
similar to the argumentum ad populum. Both depend upon the fact
that very frequently the “heart leads the head.” The timeworn picture
of the eloquent lawyer pleading for the life and liberty of a widow
(perhaps, in fact, a widow by her own hand) accused of a crime by
pointing to her poverty, her grief, her defenselessncss, her femininity, etc.,
is the picture of a man employing an argumentum ad misericordiam. The
widow’s life and liberty should depend on guilt or innocence, and neither
guilt nor innocence is evidenced by grief or poverty or the like. A student
who argues for a higher grade lest he incur the displeasure of his parents,
as well as the youngster who answers a parental accusation with a sob,
also employ the argumentum ad misericordiam.
D. The fallacy of begging the question
The fallacy of begging the question consists in assuming in the
premises of an argument the conclusion which ought to be proved. It is
true that the premises in a valid argumentation cannot be true without
the conclusion being true: this is the very essence of validity. Yet the
truth of a premise cannot rest on the truth of the conclusion. The
reverse of this must be the case. So long as a premise is simply a restate­
ment of the conclusion, or is understood only in so far as it follows from
the conclusion, or is in any way dependent upon the conclusion for its

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truth, that premise cannot function validly as a principle for establishing
that conclusion. This fallacy is especially tempting to one who is avidly
concerned to defend a proposition. A man sometimes so intently wants
his position accepted that he assumes it in practice and uses it in its
own defense.

'ITie following examples are illustrations of this fallacy:
1. An argument for the immortality of the soul from the fact that the
soul cannot die.

2. An argument calculated to show that a dog can think because
dogs are intelligent.
3. An argument to prove tliat at least one premise in a categorical
syllogism must be affirmative since they cannot both be negative.
Perhaps the most evident instance of begging the question is found
in an argument which involves the so-called vicious circle. One argues
in a vicious circle whenever he attempts to prove a first proposition by
a second proposition, and then that second proposition by the first. The
following examples illustrate that particular type of begging the ques­
tion represented by the strictly circular argument:

1. An attempt to establish credence in a man’s political position on
the basis of his reputation as a political theoretician — coupled
with an attempt to establish his reputation as a theoretician pre­
cisely on the basis of this political position.
2. An attempt to prove that subcontraries can’t be false together
precisely because contraries can’t be true together — coupled
with an attempt to prove that contraries can’t be true together
because subcontrarics can’t be false together.
We have spoken about the problem of begging the question earlier
in the book. We saw that a categorical syllogism with an enumeratively
universal major premise begs the question, for its conclusion expresses
something which is already expressed in its major premise.

E. The fallacy of the false cause
There is frequently the temptation to argue that one thing happens
because of another because it happens after that other, the post hoc,
ergo propter hoc type of argument. Tins is to argue fallaciously by
attributing a false cause to an event or fact. It may be true that some­
thing which happens after something else is caused by it. Yet temporal

FALLACIOUS ARGUMENTATION

305

sequence is no guarantee of causal sequence. With nothing but temporal
sequence as evidence, no conclusion (save that of possibility) is warranted.

The following examples illustrate this fallacy:
1. An argument that Hoover caused the great depression because it
came upon the country shortly after Hoover took office as
president.
2. An argument designed to show that a given medicine has cured
a certain illness because a person suffering from the Alness was
cured after taking this medicine.
A slight variation of this fallacy is found in an argument which con­
fuses an accompanying attribute with a cause and in an argument which
mistakes an occasion for a cause. I cannot argue that my breathing causes
me to think just because I must breathe at the same time that I think.
Nor can I say that a party causes me to gossip, although it does offer
the occasion for my doing so.
F. The fallacy of the consequent
With the exception of reciprocal conditional propositions, the truth of
a consequent docs not imply the truth of its antecedent, nor does the
falsity of an antecedent imply the falsity of its consequent. To proceed
as though they do this is to commit the fallacy of the consequent. We
have said enough on this in our chapter on hypothetical argumentation.
G. The fallacy of the complex question
To pose a question calling for a “yes” or “no” answer in such a way
that any answer given to it implies more than a simple affirmation or
denial is to commit the fallacy of the complex question. It is unfair to
demand a simple “yes” or “no” answer to questions such as this: “Was
his effort an honest and successful one?” The simple negative answer
seems to imply that the effort was neither honest nor successful. A simple
affirmation seems to imply it was both. If the effort was honest but not
successful, neither the “no” nor the “yes” answer suffices. Each seems to
imply more than the truth. A distinction should be made, to allow for
the possibility that his effort was truly honest but not successful. It is
also unfair to ask a man whether or not he is still as dishonest as ever,
or still stupidly engaged in some enterprise, or not yet over a bad habit.
Both a “yes” and a “no” answer will imply at least something unpleasant
for the man who replies. Perhaps he has never been dishonest. Perhaps
his enterprise was never stupid. Perhaps the habit spoken about is a
good habit. In these cases neither a “yes” nor a “no” answer can do
justice to the facts. Each implies something which is not true.

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Exercise XXI

1. Define: fallacy, sophistry, fallacy of equivocation, fallacy of ambiguity
in expression (amphiboly), fallacy of composition, fallacy of division,
fallacy of accent, fallacy of similar word construction, fallacy of accident,
fallacy of mistaking the qualified for the absolute, fallacy’ of arguing beside
the point, argumentum ad hominem, argumentum ad populum, argu­
mentum ad baculum, argumentum ad verecundiam, argumentum ad
misericordiam, fallacy of begging the question, argument by way of a
vicious circle, fallacy of the false cause, fallacy of the consequent, fallacy
of the complex question.

2. Distinguish strictly between fallacious and sophistical argumentation.
3. Relate the chapter on fallacies to the previous chapters in the book.

4. Distinguish between fallacies of language and fallacies beyond language.
5. Examine each of the following examples in order to uncover the fallacy
(or fallacies) involved. In many cases the examples represent argumenta­
tions which have been actually and seriously attempted.
5.1. A Russian scientist has argued that successful flights of earth satel­
lites and rockets cast doubt on the existence of God and angels.
The reason he has given is that rockets and satellites in flight have
encountered no angels nor discovered a supreme being.
5.2. A well-known educator suggests as an argument for coeducational
schools the fact that the family is naturally coeducational.
53. A segregationist has argued for “all-white” schools in the South on
the principle that “mixed” schools violate the individual’s right to
freedom of association, that is, his right to associate with whom he
pleases.
5.4. A New England “Yankee,” who has always suspected as much,
bakes pleasure in the admission on the part of a Hibernian that
the Irish arc thick around Boston.
5.5. A nationally known economist has argued that the progressive in­
come tax is Marxian because it takes a disproportionately high
amount from high incomes.
5.6. A poor philosopher argues that since we cannot prove our ability
to know anything with certitude we must either blindly assume that
we can or admit to skepticism.
5.7. A well-known Midwestern brewery suggests in its advertising that
its beer is of a superior quality because it is brewed and bottled
only in Wisconsin.
5.8. A man long on an interest in cats and short on logic argues that
one cat has ten lives, because no cat has nine lives, and one cat has
one life more than no cat.
5.9. A phone company has advertised that one phone is not enough in
a modern home because a modem home has plenty of phones.

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307

5.10. A mathematician disputes the right of a philosopher, as philosopher,
to discuss the nature of mathematics on the grounds that it is the
business of the mathematician and not the philosopher to know
mathematics.
5.11. An objection is posed against the Thom istic position that there is
an interior sense, which can be spoken of as the common sense, in
addition to the five exterior senses of sight, hearing, taste, smell,
and touch. The objection rests upon the fact that the common is
not divided off against the proper. The species of a genus arc
divided off against one another, but not against the genus itself.
5.12. An American Communist argues for the right to teach Marxian
economic theory in our American schools on the grounds that our
constitution guarantees freedom of thought and speech to all its
citizens.
5.13. Tliis same Communist argues for the integrity of Marxian eco­
nomic theory on the grounds that it opposes capitalistic oppression
and exploitation of the working class.
5.14. A skeptic expresses amazement that a Thomist philosopher claims
to be, sometimes at least, absolutely certain in his knowledge of
things. The Thomist, he argues, should know that only God has an
absolutely certain knowledge of things.
5.15. The ontological argument for the existence of God finds the
guarantee of God’s existence in the concept of God. Since the
concept of God is the concept of a being which cannot not-exist,
then, since the concept of God exists, God must exist.
5.16. A pharmaceutical house has advertised a salve by noting that some
arthritic patients who have used the salve along with physical
therapy have experienced relief shortly afterward.
5.17. Aristotle’s claim that logic should be a tool for the other sciences
seems to suffer in the face of the fact that for the most part his
only logical works are those included in the Organon.
5.18. One might be tempted to approve the devastation wrought by the
divisions of Hitler, since it is certainly true that the destruction of
Hitler’s war machine was justified.
5.19. A pagan wonders about the slogan of Christians: “Put Christ back
into Christmas.” How else can you spell Christmas except that the
first syllable be Christ?
5.20. Some modem psychologists question St. Thomas’ argument for the
spirituality of the human intellect, which is based upon the fact
that the human intellect knows an object abstracted from its indi­
viduating characteristics. They say that the modern appreciation of
the effect of damage to the cerebral cortex upon the thinking process
is more sophisticated than that possessed by St. Thomas.
5.21. It is absurd to distinguish between philosophy and the history of
philosophy, since philosophy never exists except in the mind of a
philosopher, who, as a philosopher, becomes a figure in the histon'
of philosophy.

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5.22. Wherever there are a number of things, these things can be
counted. Thus, a man with countless blessings must be a man who
is completely unblessed.
5.23. If one is a criminal, he is responsible for his actions; and if one is
responsible for his actions, he is a man. Therefore, all men are
criminals.
5.24. Geometry is rigorously logical and metaphysics is in no way geo­
metrical; thus, metaphysics is in no way logical.
5.25. It is a good thing to relieve the suffering of an innocent person
whenever this is possible. 'Phus, mercy killing is a good thing
whenever it is possible.
5.26. Men and women have a right to an equal status in all things since
men and women are essentially on a par with one another.
5.27. The most significant philosophical thought in Greece must have
found its cause in the teaching of the Sophists, since the}' immedi­
ately preceded the greatest Greek philosophers, namely, Socrates,
Plato, and Aristotle.
5.28. It has been argued that the position which the Catholic Church
takes on birth control is erroneous precisely because the Catholic
clergy is committed to a life of celibacy.
5.29. 'rhe president of any university must be an extremely respected
man, since the position of university president is an extremely
respectable position.
530. A skeptic argues that no one can deny that no proposition can be
accepted without fear of its opposite, since every proposition admits
of contradiction.

6. Carefully examine issues of the daily newspapers, copies of magazines,
even your textbooks, for examples of fallacious argumentation. Make a
list of these examples, naming the fallacy wherever possible, and indicat­
ing precisely what rule of correct logical procedure is violated in each case.

CHAPTER

XXI I

The Nature of Logic

I. The Purpose of This Chapter

In the strict sense it is not precisely the business of logic to concern
itself with the nature of logic. This is the task of that branch of phi­
losophy called epistemology which investigates the nature of knowledge,
the various types of knowledge, and the relationships existing between
those types. It is not the business of the mathematician, as mathematician,
to know the nature of mathematics — but to know the mathematical
subject matter. Similarly, it is not the business of the logician, as logician,
to know the nature of logic —but rather to know the subject matter
of logic, which we shall speak of as the second intention. Nevertheless,
it is of value for any scientist, including the mathematician and the
logician, to know something about the nature of his science, to be able
to evaluate its significance, and to see precisely its limits. Though this
is not properly within the sphere of the science in question, it is im­
portant for the well-being of that science in the scientist in question. The
limits of mathematics are not mathematical, but the mathematics of a
mathematician who ignores them may be of questionable integrity. So
too with logic and the logician. Knowledge about logic is not to be con­
fused with knowledge in logic. Yet the well-being of knowledge in logic
(which is the science of logic) suffers in the absence of sound knowledge
about logic (which is epistemological).
Some effort was made in Chapter I to introduce the student to the
nature of logic. However, the student’s inexperience in logic prior to
his course in logic makes any extended treatment of its nature imprac­
ticable at the beginning of the course. Tire nature and limits of logic are
not easily understood, and they can easily be misunderstood by one who
attempts to investigate them without any experience in logic. It is suffi­
cient at the beginning of the course to discuss only generally the nature
of logic and then briefly to indicate the divisions of logic which deter­
mine the manner in which the course is ordered. At the end of the
309

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course, however, the student should have sufficient experience to under­
stand more determinate!/ the fundamental notions concerning the nature
of logic and its division into parts.
II. Art and Science

Logic is both an art and a science. In order to sec that this is so, let
us consider briefly what it is to be an art and what it is to be a science.
We can then measure logic up to these notions.
Art and science are alike in that each is an intellectual perfection which
is ordered to some properly human activity. They differ in that each is
ordered to a different activity. Man is by nature potentially a knower
and potentially a maker. His intellect is naturally adapted to confront a
reality which exists independently of him and to know that reality as
it is, even to know it scientifically in its causes. His intellect is further
adapted by nature to direct him in the production of works which arc
his own “creation,”1 but which are in one way or another an imitation
of the works of nature. These works “created” by man are spoken of as
works of art. The basic ordination of the intellect to knowledge on the
one hand and to productive activity on the other is sound. However,
the intellect can deviate from the straight path and fall into error in
both, since the intellect is not completely determined by nature for either
scientific knowledge or artistic effort. Fortunately, it can acquire deter­
mination in both respects. In fact, it can be habituated badly or well
in either case. A bad habit in cither respect is an intellectual vice. A
good habit, on the other hand, is an intellectual virtue. The intellectual
virtue which perfects the intellect in its quest for scientific conclusions
is called science. The intellectual virtue which perfects the intellect so
that it can direct a man in his artistic effort without error is called art.
Art differs from science, and from every other intellectual habit, in
being essentially ordered to the production, the making, of a work. Since
the work is the final cause of the art, the truth of any artistic judgment
which bears upon the production of the work depends upon a conformity
with the character of the work to be produced. The raw materials for any
artistic work are given in nature, and the artist imposes an artificial (rather
than natural) form upon these materials in order to produce his work of
art. Ordinarily the matter for the artistic work is found in the world of
physical things which exist outside the intellect. Here the painter finds
1 In the strict sense, creation is a production of something out of nothing. Only
God can create in the strict sense. The production we speak of here — the artistic
production of men — presupposes something given in nature to which an artificial form
can be added.

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311

his canvas and his paints; the sculptor finds his clay or bronze; and the
builder finds wood or bricks or steel. In these eases the work of art
depends upon an intellectual effort of the mind, but it is produced in
a medium which is exterior to the mind. In other cases the matter of
the artistic work is found within the intellect itself. In these instances,
of course, the matter for the work of art is not material in precisely the
way in which paints, clay, and wood are material. However, it is possible
for something to exist in the mind, and in the mind to lack an artificial
form which it could take on. When we spoke on the four causes in our
chapter on definition we described the material cause as an indeterminate
but determinable substratum of form. In a general sense this description
fits both the matter for a work of art which is exterior to the intellect
and that which is interior to the intellect. 'Fhose arts which are ordered
to a work existing outside the intellect arc traditionally spoken of as
servile arts. They include fine arts such as painting and sculpture, as well
as useful arts like medicine and carpentry. Those arts which are ordered
to an interior work produced within the intellect are called liberal arts.
We shall see that logic is a liberal art. Other examples of the liberal
arts are grammar and music.
Science (in the Aristotelian sense) is a habit of the intellect perfecting
it in its quest for knowledge demonstrated through causes. Scientific
knowledge, like all knowledge, admits of a distinction between the
speculative and the practical. Speculative knowledge is knowledge which
is of value for the sake of knowledge. Practical knowledge is knowledge
which is of value for the sake of an activity beyond knowing. In the
strict sense, speculative knowledge is knowledge of an object which is
given in nature and which cannot be produced by man. Being is an
example of such an object. Thus, metaphysics, which is the science of
being, is an example of a speculative science. Being is an object well
worth knowing. But this knowledge is in no way intended to be used
in order to make or modify being. It is purely speculative. Strictly speak­
ing, practical knowledge is knowledge of an object which can be produced
in some way by man. It is knowledge of an object which is of value pre­
cisely in so far as it can be ordered to an activity beyond knowing. The
object of medicine is the curable body, and the object of ethics is the moral
act. In each case the object is somehow ablc-to-be-brought-about, and
in each case the end is an activity which is different from the activity
of knowing. Metaphysics is a speculative science. It represents, in an
unequivocal fashion, knowledge for the sake of knowledge. Medicine and
ethics arc practical sciences. Each is an example of knowledge for the
sake of a noncognitive activity.

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III. Logic as an Art

It is not difficult to see that logic is an art, and that it is a liberal art.
The human intellect is by nature an ability to reason. We spoke about
the native capacity of man for logical discourse, and we called this natural
logic. Natural logic, however, lacks the determination which would
guarantee an orderly, easy, and infallible movement from the known to
the unknown. For this the human intellect stands in need of logic. Logic
is an intellectual habit which enables the intellect to impose that order
upon its objects which will ensure the orderly, easy, and correct movement
from previous knowledge to new knowledge. Logic “firms up” the intellect
for the construction (within the intellect, of course) of integral discourse.
It is a perfection directing the intellect in the production of interior
works such as definitions, divisions, syllogisms, and demonstrations. As a
perfection of mind directive of productive activity, logic is an art. Because
the work produced by logic is an interior work, logic is a liberal art.
We saw that the truth of an artistic judgment is measured by the good
of the work to be produced. A work of art is not wholly arbitrary. Just
as the final cause of a work in nature determines its matter and form, so
the end of an artistic work determines its matter and form. Given the
final cause of a house, for example, it is clear that only some materials
are fitted for the production of a house and that the ways in which these
materials can be put together are determinately limited. On the side of
the matter art demands a certain indetermination so that the matter can
take on an artificial form. On the side of the artist art demands possession
of certain and determinate rules according to which some determination
or form can be produced in the matter. This is precisely the reason why
logic must be both an art and a science. We shall see that it cannot be
the art of sound rational discourse unless it is also the science of logical
relations or second intentions. The reason for this, as we shall see, is that
an adequate understanding of the rules of rational discourse is identical
with a scientific grasp of logical relations or second intentions.
IV. Logic as a Science

Logic as an art is ordered to rational discourse. It is a perfection of
mind which enables us to reason well. It is not enough for adequate
reasoning or argumentation that a conclusion be correct, not even that it
be correct both formally and materially. It must be correct, and, at the
same time, reflexively assured. By this we mean that it must be able to be

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defended. We have seen that no work of art is completely arbitrary'.
This is especially true of rational discourse. Rational discourse is sound
only if it conforms to strict canons which determine its construction. No
one can adequately reason unless he can defend himself in terms of
these canons. And no one can defend himself in terms of these canons
unless he sees the necessity which imposes them upon his discourse.
The rules of argumentation are for the most part particular canons of
logical procedure, which are not immediately evident but which find
their evidence ultimately in self-evident principles in the logical order.
Thus, the six rules of valid procedure for any figure of the categorical
syllogism are particular rules whose necessity is seen only if it is shown
that they follow necessarily from the self-evident principles of triple
identity or the separating third. Similarly, the rule which states that
subcontraries cannot be false together is adequately appreciated as neces­
sary only in so far as it is resolved into the self-evident logical principle
which asserts that propositions which deny one another cannot be true
together. Thus, the art of integral discourse is achieved in a knowledge of
the rules of argumentation which have been resolved into the first prin­
ciples of logical procedure. We have seen that the resolution of particular
conclusions into self-evident propositions is demonstration or scientific
argumentation. Thus, if logic is to be the art of integral discourse, it must
be at the same time a scientific discipline. The rules without which logic­
is not an art can be adequately possessed only if they are known
scientifically.
Every science has its proper object and its proper end. We have seen
that for logic to be the art that it is, logic must be simultaneously a
science. What is the object of logic as a science? What is its end? These
questions might be rephrased so as to be more pointed for us. What does
the student in logic class study? Why does he study this, and what docs
he intend with the knowledge he acquires?
The rules of logical procedure are simply canonical statements of those
logical relationships which exist between objects in the mind and which
govern the ordering of these objects in rational discourse. These logical
relationships are technically spoken of as second intentions. Since logic­
as a science demonstrates the mies of logical procedure, we can say that
the object of logic is the second intention. What precisely is a second
intention?
We have seen that natures enjoy a twofold existence. They exist first as
concretely individualized in real things. They exist secondly in a state
of abstraction, i.e., as known in the mind. The comprehensive notes of
a nature belong to that nature taken absolutely, i.e.. in itself and open to

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THE LOCIC OF THE THIRD OPERATION

existence both in things and in the mind. Moreover, these notes can be
said of the nature as it exists both in the individual and in the mind.
Some attributes belong to the nature precisely in virtue of its existence in
the real individual. We call these attributes individuating characteristics.
Other attributes belong to the nature precisely in virtue of its existence in
the mind as known. As an object in the intellect, and only as such an
object, a nature can be seen to be related in different ways to different
things. These relationships, which an objective concept possesses precisely
as known, arc the second intentions studied by the logician. They are
called second intentions 1)ccause they belong to natures only in so far as
they exist in mind as known, whereas first intentions arc the natures
themselves as the direct object of intellectual knowledge. Moreover,
second intentions are properties which natures acquire only in that
second existence which they enjoy in the intellect as objects of thought,
in opposition to the first existence which those natures have in indi­
vidual real things. .Man, as an object of direct intellection, is a first
intention. The logician is never interested precisely in what it is to be a
man. Man, absolutely taken, is the concern of the psychologist or
anthropologist. However, in the mind man takes on relationships which
arc the concern of the logician. In the mind man takes on the relation
of species to John, of subject in the proposition Every' man is rational,
of middle term in the syllogism, Every man is rational; but John is a
man; therefore, John is rational. The property of being a species, the
property of being a subject, and the property of being a middle term —
these arc all second intentions. Each is a relation belonging to a nature
precisely in the second existence the nature has as object in the intellect.
These are only three of the many second intentions we have studied in
logic. Others include the properties of predicate, essential definition,
logical division, hypothetical proposition, obversion, syllogism, primary
proposition, explanatory’ demonstration, and induction. There is no need
to add to these examples. A complete list would sen e as an outline review­
ing the whole of our course. Our investigation has been essentially a study
of second intentions.
We have said that the art of correct reasoning is simultaneously the
science of second intentions. This can be clear to us only if wc see an
identity between second intentions and the rules of logical procedure, for,
as wc have seen, the art of logic is one with the scientific grasp of the
rules of logical procedure. We have asserted the identity between second
intentions and the rules of logical procedure by saying that the rules of
logic are canonical statements of second intentions. What precisely does
this mean? The point is simply this: to know what a second intention is

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315

is to know how to order an object possessed of this second intention
in logical or rational discourse. The rules governing such discourse are
simply statements in the form of rules or canons of the natures of the
different second intentions. For example, to know what a definition is is
to know how to define a term. To know what a syllogism is is to know
how to proceed validly by way of deduction. To know what a demonstra­
tion is is to know how to prove a conclusion. In the act of discourse
the intellect orders its objects in accord with the relationships which exist
between them. These relationships must be understood or the ordering
may well be erroneous. Without an understanding of these relationships
their use in rational discourse will certainly not be defensible.
Logic can be characterized as a real science but not as a science of real
things. It is a real science in the sense that it is genuinely a science. It is
genuinely a science because it resolves its conclusions into self-evident
premises in its own order. The particular rules of logic are not arbitrary.
They arc not presumed without proof. They are not self-evident. Rather,
as we have seen, they are proved from self-evident propositions. From a
practical point of view, this means that the student is not a logician
simply because he knows what the mies of logic are. 'fhese, presumably,
could be accepted on the authority of a professor or a textbook and
merely memorized. The student becomes a logician only when he comes
to know what the rules of logic are, and why. He is a logician only
in so far as he sees that they cannot not-be what they are by showing
that they follow from absolutely incontrovertible first principles of logical
procedure.
Wc say that logic is not a science of real things even though it is a real
science. It falls short of being a science of real things because its object
is a bcing-of-the-reason and not a real being. The second intention, which
is the object of logic, is a logical relationship belonging to an object pre­
cisely as known. The property of being tired, or old, or hungry can belong
to human nature in the real order. 'Hie property of being subject, or
middle term, or species can belong to man only in so far as man exists
as an object in the mind. Such properties exist only as conferred by the
mind upon objects precisely as objects existing in the mind. Thus, these
properties are not, in the technical sense, real. They are bcings-of-thereason. It is impossible to think of them existing anywhere except in
the mind. Yet they do really exist in the mind. They are not simply
nonentities. Further, they do not come into being in any arbitrary fashion.
Second intentions are not real, but they are founded upon the real. For
example, the logical relation spoken of as predicate may be founded upon
the real relationship between accident and substance. In the proposition,

316

THE LOGIC OF THE THIRD OPERATION

John is white, white is related to John as predicate because in the real
order whiteness is an accident inhering in the substance of John.
We have seen that sciences are either speculative or practical. Logic
is a speculative science. Science is practical only in so far as it is of value
for something other than knowledge. This is not the case with logic.
The logician studies second intentions precisely so that he may be able
to reason well. The end of logic is sound discourse. This is an end
strictly within the sphere of knowledge. Thus logic is a speculative science.
Yet logic differs from most speculative sciences. Ordinarily, speculative
science is of value because a knowledge of the object of the science in
question is worth something in itself. Thus, the metaphysician studies
being because it is good to know about being. However, the logician seeks
knowledge of an object not for the sake of knowledge of that object, but
rather for the sake of knowledge of other objects. The logician does
not study second intentions because it is good to know about second
intentions. He studies second intentions because a knowledge of second
intentions will enable him to proceed scientifically in the disciplines which
seek knowledge of real things. Logic is not an end in itself. It is nothing
if it is not useful. But its use is within knowledge itself, and it remains,
then, knowledge for the sake of knowledge. It is at the same time useful
and speculative. Perhaps the easiest way to sec what we mean by this
is to compare logic with metaphysics on the one hand and medicine
on the other. The object of metaphysics is being, and its end is knowledge
of being. It is speculative without being useful. The object of medicine
is the curable, and its end is the curing of the curable. Medicine is
practical and useful. The object of logic is the second intention. Its end
is knowledge — not of the second intention — but of man in psychology,
being in metaphysics, the continuum in geometry', etc. Thus, logic is
speculative, but useful.
V. The Logic of the First Operation, the Logic of the Second
Operation, and the Logic of the Third Operation

The object of the science of logic is the second intention. Tire second
intention is a relation of the reason which belongs to an object precisely
as known. Some second intentions accrue to objects on the level of simple
apprehension; others on the level of judgment; and still others on the
level of reasoning itself. Tire end of logic is totally sound discourse, with
the emphasis upon the reasoning process. The intellect cannot proceed
in an orderly fashion on tire level of reasoning so long as it is not well
ordered on the level of judgment. Nor can the intellect escape disorder

THE NATURE OF LOGIC

317

on the level of judgment without proceeding in an orderly fashion on the
level of simple apprehension. Thus, there is a natural order to the
logician’s concern for second intentions. First, he must study those which
belong to objects on the level of the first operation, with an eye to the
second and third. Then he must study those second intentions which
objects acquire on the level of the second operation, with an eye to the
third. Only then is he able adequately to study those second intentions
which arise on the level of reasoning itself. The science of logic, then, is
divided into the logic of the first operation, the logic of the second
operation, and the logic of the third operation.
VI. Formal and Material Logic

We have seen that logic is simultaneously the art of sound discourse
and the science of second intentions. Certain second intentions are re
quired for the integrity of discourse from the point of view of its form.
These are second intentions accruing to natures known precisely in virtue
of their logical “position” (i.e., their mode of signification) in discourse.
Scientific knowledge of these second intentions is adequate to the art
of valid or consistent reasoning — but only to the art of valid reasoning.
Other second intentions are required for the integrity of discourse from
the point of view of its matter. These second intentions are acquired by
natures known precisely in virtue of their intelligible content. The logician
who knows these second intentions scientifically is able to determine not
only the validity of his discourse but the force of his discourse in reference
to the truth and necessity of its conclusions. The first type of second
intention specifies that branch of logic spoken of as formal logic. The
second type specifies that branch of logic called material logic.
In order to see the difference between a second intention in formal
logic and a second intention in material logic let us consider the two
propositions Every man is rational and Every man is able to speak.
Rational and able to speak are both related to man as predicate, because
each enjoys the same “position” within the formal pattern of thought.
But rational is an essential predicate related to man as specific difference,
and able to speak is a nonessential predicate related to man as a specific
property. This difference stems from the intelligible content of the con­
cepts rational and able to speak. The property of being a predicate is a
second intention in formal logic. 'Flic property of being a specific differ­
ence and the property of being a predicable property are second intentions
in material logic. If we refer to the theory of argument we can recall that
it is necessary to know what a predicate is simply to be able to reason

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THE LOGIC OF THE THIRD OPERATION

validly. Yet one could recognize a valid argument without knowing the
difference between a specific difference and a property. However, knowl­
edge of the difference between a difference and a property is indispensable
for anyone who would try to distinguish between an explanatory proof
and an a posteriori proof. The one second intention is of formal import,
while the other two are of material import.
The distinction between formal and material logic is most evident (and
of prime importance) on the level of the third operation. However, we
can distinguish between second intentions of both formal and material
import in each of the three operations of the mind. Throughout this
book wc have treated both types of second intentions as we have progressed
successively through the logic of the first operation, the logic of the second
operation, and the logic of the third operation. The following schema
helps to illustrate the point by recalling one or two examples of both
types of second intentions on each level of intellection:
FORMAL IMPORT

MATERIAL IMPORT

First

term

genus

operation

universal

definition

Second

proposition

immediate

operation

converse

commensurately
universal

Third

conclusion

explanatory
demonstration

syllogism

dialectical
argumentation

operation

The names “material logic” and “formal logic” inevitably cause some
difficulty. In a sense both branches of logic arc formal and neither
material. Both are specified by a type of logical form or second intention.
Material logic, no less then formal logic, studies logical relationships which
accrue to concepts. Neither studies the intelligible content of the concept
itself. This is left, of course, to the other sciences. 'Ihe relationship of
predicate which the concept man acquires in the mind when used in
propositions is studied by the formal logician. The relationship of species
which the concept man acquires in the mind with regard to other objects
is studied by the material logician. But the nature of man is studied by the
psychologist. The second intentions in material logic depend proximately

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319

upon the intelligible content of the objects to which they accrue. But
material logic, no less than formal logic, presumes to make no judgment
precisely of the intelligible content of any logical discourse. The material
logician knows what the logical conditions which ensure a certainly true
conclusion must be. But the material logician, as material logician, never
says that this or that proposition is certainly true or not. Logic is related
to each of the other sciences as a tool. It is used by each of them. But it
does not do the work precisely of any one of them. This is as true of
materia] logic as it is of formal logic.
The division of logic into formal and material logic admits of a sub­
division. Some discourse is so disposed materially as to generate a certainly
true conclusion. Other discourse generates only a probably true conclusion.
Still other discourse generates merely suspicion. A final type is propor­
tioned simply to the generation of an intellectual inclination. These four
are respectively: demonstration, dialectical argumentation, rhetorical dis­
course, and poetry. Tire distinction between these four types of discourse
is the reason for a division of material logic on the level of the third
operation into demonstrative logic, dialectics, rhetoric, and poetics.
VII. Logic as a Common Mode

Logic is a science at the same time both speculative and useful. It is
speculative because its end is knowledge (not action), and it is useful
because the knowledge it seeks ultimately as an end is not a knowledge of
its own object but rather of the objects of the other sciences. Because
of this, logic is sometimes called the tool of the other sciences. However,
it is more properly thought of as the tool of the intellect or reason in its
work in the other sciences. Each science has its own object, but the same
reason or intellect is operative in every scientific area. Consequently, there
is one general mode of procedure demanded by the nature of the reason
in its confrontation of real things. Tliis general method of procedure is
logic. Each different science has its own special methodology, a meth­
odology properly suited to the scientific character of its formal object.
Each of these special methodologies is a particular contraction of the
common method. Logic is analogically common to each different type
of scientific methodology. The method of mathematics is different from
the method of psychology; the method of psychology is different from the
method of metaphysics; and so on. Yet each of these methods is logical.
The mathematical methodology is logic in the mathematical mode; the
psychological methodology is logic in the psychological mode; and the
metaphysical methodology is logic in the metaphysical mode. Each

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THE LOGIC OF THE THIRD OPERATION

particular methodology is logical. But each is differently logical, because
each is logical in its own way.
This is an extremely important distinction. It is a distinction fre­
quently missed. One difficulty' for the student arises from the fact that
students must rely heavily on examples to help them to understand
logical theory. It must be kept in mind that each example is a particular
illustration of logic-cmploycd-in-a-spccial-field. Thus each example sug­
gested for the purposes of logic might also be suggested as an example of
some special scientific methodology. For the purposes precisely of logic
the student must slough off on the special character of the example which
comes from the special character of the scientific subject matter of the
discipline from which the example is taken. He must concern himself
only with what is analogically common to this example and similar ex­
amples from other scientific areas. It is impossible to suggest an example
simply of being logical. An example of being logical must always be an
example of being logical in mathematics, or in psychology, or in meta­
physics, or the like. The student must be on his guard lest any example
of being logical in this or that field be taken as an example of being
logical simply.
This last point highlights the useful character of logic. On its own, logic
is worth nothing. It is of value only as it looks beyond itself to the
other disciplines. There is no value in simply being logical — in fact this
is an absurd impossibility. But it is vitally important that in mathematics
one be logical mathematically, and in psychology that one be logical
psychologically, and so on in each of the other disciplines.
Vill. Doctrinal Logic and Logic in Use

'Fhe fact that logic is a useful science is the basis for the distinction
between doctrinal logic (logica docens) and logic in use (logica utens).
Logic, considered simply as the intellectual discipline which is the scien­
tific grasp of second intentions, can be spoken of as doctrinal logic.
Doctrinal logic, as wc have seen, looks beyond itself to the other sciences
for its end. Thus, wc can say that doctrinal logic is virtually useful. It
becomes actually useful when it is put to the use of the reason. Logic
is put to use in one sense whenever it directs the intellect in rational
discourse. This can happen in either one of two ways. First of all, logic
can direct the intellect in demonstrative discourse, which generates cer­
tainly true conclusions by a rigorous resolution to self-evident propositions.
Second, logic can direct the intellect in dialectical discourse, the tentative
discourse productive of a probable conclusion. Logic is put to use in

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321

another sense whenever, as a common science, it supplies a particular
science with propositions that can be used as premises leading to a con­
clusion concerning the subject proper to that particular science. An
example of this last use of logic which we have seen before occurs when
the psychologist argues from the logical proposition that contraries are
in the same genus to the conclusion that the contrary passions of love
and hatred flow from the same appetite.
Logic in use, then, has three meanings: namely, the direction of the
reason in demonstrations proper to the different sciences, the direction
of the reason in different areas of dialectical discourse, and the presenta­
tion of common principles which can serve as premises to manifest
conclusions in the particular sciences. We can speak of the first two uses
of logic as methodological. The third, despite the inevitable terminological
confusion, might be spoken of as a doctrinal use (logica utens ut docens).
In speaking of logic throughout this chapter as a useful science we have
been primarily concerned with the first two uses of logic rather than the
third.
IX. The Question of Symbolic Logic

It is difficult to find a name which adequately describes logic as we have
presented it throughout this book. Inasmuch as the logical works of
Aristotle have been the most significant of the primary texts which have
contributed to it, and inasmuch as it owes a great deal to the works of
men who rank as commentators on Aristotle, it can be spoken of as
Aristotelian logic. Custom would allow, too, that it be spoken of as
traditional logic. Some would speak of it as Scholastic logic.
There is another discipline, besides that of Aristotelian, Scholastic, or
traditional logic, which speaks of itself as a logic. This is, of course, the
mathematical or symbolic logic of many modem mathematicians and
logicians. Some have claimed that this is an extension and perfection of
Aristotelian logic. Some say that it is a universal logic, which either
replaces Aristotelian logic or is related to Aristotelian logic as whole to
part. Some claim that it is a particular logic which plays the role of
instrument in areas foreign to Aristotelian logic. In any event it consists
in a complicated system of symbols which are intended to represent
strictly formal relations between signs. Extremely complicated manipula­
tions of symbols and combinations of symbols are effected with relative
ease and admitted lack of concern for any intelligible content or meaning
behind the symbols. The results are sometimes rather startling in the face
of some of the notions enunciated in our study of what can be called

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THE LOGIC OF THE THIRD OPERATION

traditional logic. We have already noted something of this in our chapter
on hypothetical propositions. For the symbolic logician all hypothetical
propositions are truth-functional, even the conditionals. For him a true
proposition is implied by any proposition, and a false proposition implies
any proposition. This, of course, contradicts the doctrine of the condi­
tional proposition as presented in the framework of Aristotelian logic.
In the light of this difficulty, and a host of others we shall not go into,
what are we to say of the relation between Aristotelian and symbolic logic?
As a matter of fact Aristotelian and symbolic logic are not univocally
logics. Neither is an extension of the other, nor is either related to the
other as whole to part. Aristotelian logic, as we have seen, is an instrument
of the intellect considered precisely as an ability to know the intelligible;
i.e., to grasp the meaningful. Logic, thus considered, cannot be purely
formal in the sense in which the symbolic logician conceives of his logic. It
is true that logical relations can sometimes l>c more easily considered when
represented by symbols. As a matter of fact we have made liberal use
of symbols throughout this book. But the relationships we have symbolized
— even in formal logic —have a real foundation. We can never mistake
the symbol for the relationship and never prescind from the real founda­
tion that specifies the relationship — not even in formal logic. The
symbolic logician docs this. The relationships he is concerned with are
not second intentions. They are not founded upon the real. At best they
are strictly formal relationships between symbols which need not stand
for anything. The formal object of the science of Aristotelian logic is the
second intention. Whatever the formal object of symbolic logic is, it
is not the second intention. Since sciences are specified by their formal
objects, these two sciences cannot be identified in any way. If one is
logic, the other is called logic by analog}' at best. The use of symbols is
highly desirable in logic, even Aristotelian logic, but so-called symbolic
logic is not logic in the sense in which Aristotelian logic is logic. What­
ever it is, and whatever its use, symbolic logic is not an extension of
Aristotelian logic, nor is Aristotelian logic an extension of symbolic logic.
Since symbolic logic is not an instrument of the intellect as it confronts
the real, it should not be considered to be a rival of Aristotelian logic,
whose avowed end is finally scientific knowledge of real things in the
sciences which are concerned with real things. Tin's is not to deny
mathematical or symbolic logic a place in the academic sun. In its own
right, and in reference to its proper end, it may be a very significant
discipline. The point here is simply that Aristotelian logic and symbolic
logic are radically distinct disciplines.

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323

Exercise XXII

1. Define: art, science, liberal art, servile art, speculative science, practical
science, useful science, second intention, first intention, formal logic, ma­
terial logic, demonstrative logic, dialectics, rhetoric, poetics, doctrinal logic,
logic in use, symbolic logic.
2. What does it mean to say that it is not the business of the logician to
investigate the nature of logic?
3. Compare the nature of art with the nature of science.
4. What is the difference between a servile art and a liberal art?
5. What is the difference between a speculative science and a practical science?
6. Why is logic said to be an art? Why is it a liberal art?
7. What does art demand on the part of the matter of the artistic work?
What does art demand on the part of the artist? Apply this to the special
case of logic.
8. Why is logic said to be a science?
9. Why must logic be a science in order to be the art it is?
10. What does it mean to say that logic is a real science but not a science
of the real?
11. What does it mean to say that the rules of logic are canonical statements
of the natures of second intentions?
12. How can the science of logic be speculative and at the same time useful?
Compare logic with medicine on the one hand and metaphysics on the
other in terms of object and end.
13. How can logic be simultaneously a science and an instrument for science?
14. Explain the division of logic into the logic of the first operation, the logic
of the second operation, and the logic of the third operation.
15. Explain the division of logic into formal and material logic. Distinguish
between these in terms of object and end.
16. In what sense are both formal logic and material logic formal? In
what sense is neither purely formal?
17. What is the rationale of the disision of material logic into demonstrative
logic, dialectics, rhetoric, and poetics?
18. Why is logic more properly said to be an instrument of the reason rather
than an instrument of the sciences?
19. How is the proper methodology' of a particular science related to logic?
20. What is the difference between doctrinal logic and logic in use?
21. What are the three uses of logic? Which have we been most concerned
with?
22. Aristotelian logic makes use of symbols, without being symbolic logic.
Explain.

Index
Absolute supposition, 109, 116
Abstract concept, 35 n
Abstraction, from both individuating char­
acteristics and the act of existing, in
simple apprehension, 19 f, 29; of the
intelligible from the sensible, 4f, 19;
twofold, by way of apprehension and
by way of negative Judgment, 19, 37 f
Abstractive induction, 281 f
Accent, fallacy of, 299 f
Accident, 56; categorical, see Accident
(categorical); fallacy of, 300 f; predi­
cable, see Accident (predicable)
Accident (categorical), 59 ff, 62 ff
Accident (predicable), 49 f; nature of,
42 f; wholly inseparable, 50; wholly
separable, 50
Action, category of, 59 ff, 63
Adversative proposition, 132 f
Albert, St., primary source in logic, ix
Alternative exclusive syllogism, see Ex­
clusive alternative syllogism
Alternative inclusive syllogism, see In­
clusive alternative syllogism
Alternative proposition, 135 ff; exclusive,
136 f; inclusive, 135 f; reduced to con­
ditional, 138
Ambiguity, fallacy of, 298
Analogous terms, presenting special diffi­
culty in discourse, 115; unsuited to the
categories, 58
Analogy, 106; of names, 55
Antecedent, of conditional proposition,
133; and consequent of the argumenta­
tion, 175
Xntepredicamcnts, purpose of, 55 ff
A posteriori demonstration, 265 ff; can­
not be explanatory', 265 ff; of the fact,
two types, 268 f
A priori demonstration, 265 ff; explana­
tory. 265 ff; of the fact, two types,
267 f
Argument, sec Argumentation
Argumentation, antecedent and conse­
quent of, 175; categorical, 179; deduc­
tive, 178 f; demonstrative, 179: demon­
strative, see also Demonstration;
dialectical, 179; dialectical, sec also
Dialectic; and its elements, 174f; ex-

325

pression of object of reasoning, 22, 24;
fallacious, 179, 297 ff; hypothetical, 179;
inductive, 178 f; matter of, 175; sophis­
tical, 179; types of, 178 ff
Argumentum ad baculum, 302 f
Argumentum ad hominem, 302
Argumentum ad misericordiam, 303
Argumentum ad populum, 302
Argumentum ad verecundiam, 303
Aristotelian logic, 142 n
Aristotle, preliminaries to categories, 55;
primary source in logic, ix; on self-evi­
dent propositions, 285; sorites, 248 ff;
use of term "cnthymcmc,” 244
Art, liberal or servile, 311; nature of,
310 f
Axiom, necessary for any science, 285
Begging the question, 303 f
Being, analogically divided into cate­
gories, 61 f; finite, can be categorized.
58; infinite, cannot be categorized, 58;
as quast-genus, 62; real, 56 f; real, actual
or possible, 99; real, can be categorized,
57; of the reason, 56 f, 99; of the
reason, cannot be categorized, 57
Bourke, V., 221
Cajetan, Thomas de Vio, primary source
in logic, ix
Categorical proposition, 120 ff, 129; ex­
pressed as hypothetical, 139
Categorical syllogism, Ixasic principles for,
189 f; defined, 182; elements of, 183 f;
expressed as hypothetical, 227 f; figures
of, 185 f; general rules for, 196 ff; gen­
eral rules for, in reference to basic prin­
ciples for, 196; moods of, 188 f; reduc­
tion of moods to first figure, 190 f;
reduction of moods to first figure, direct
and indirect reduction, 213 ff; reduction
of moods to first figure, directions for
each, 216; special rules for each figure.
205 ff; valid moods, 208 ff
Categories (predicaments) 25; division in­
to ten, 59 ff; division of the universal
essentially taken, 39, 41; nature and
divisions of, 54 ff; as related to the
predicables. 65 f; as ultimate genera. 62;

326
univocally divided into species, 61; what
kind of being can be put in, 57 ff
Causal definition, 77 ff
Causal proposition, 132 f
Cause, efficient, 76; extrinsic, 76; final,
76; formal, 76; four, 75 ff; intrinsic, 76;
material, 76; role of, in explanatory
demonstration, 266 f
Certitude, 11, 179, 258 ff, 319
Codivision, 89
Collective supposition, 112, 117
Commensurate universality (convertibil­
ity), required for strict demonstration,
264
Complex question, fallacy of, 305
Composite expression, 101 f
Composition and division, fallacy of, 299
Compound proposition, 120, 129 ff; more
or less complex, 129
Comprehension, of a concept, 25 ff; meas­
ured in terms of infinite multitude,
29 f; priority over extension, 30; of
terms, in propositions, 125
Comprehensive notes, intelligibly but not
physically distinct, 26; known explicitly
only with difficulty, 27; in less strict
sense, 28; not parts on a par with one
another, 27
Concept, abstract, 35 n; comprehension
of, 25 ff; concrete, 35 n; extension of,
25; formal, 22; objective, 23
Conclusion, in categorical syllogism. 183 f;
direct, 184 f; of hypothetical syllogism,
223; indirect, 184 f; a new truth po­
tentially in its premises, 174
Concrete concept, 35 n
Conditional proposition, 133 ff; reciprocal,
134 f; simple, 133 f
Conjunctive proposition, 129 ff
Consequent, and antecedent of the argu­
mentation, 175; of conditional proposi­
tion. 133; fallacy of, 305
Contradiction, between concepts, 66 f; be­
tween propositions, 148 f; of singular
propositions, 163
Contradictory terms, as used in obversion,
160
Contraposition, between propositions, im­
portance of, 162; of propositions, 161 ff;
questionable for singular propositions,
166
Contrariety, between concepts, 68; be­
tween propositions, 149 f; impossible for
singular propositions, 163
Contrary terms, as used in obversion, 160

INDEX

Converse, 155 ff
Conversion, accidental, of singular propo­
sitions, 165; of propositions, 155 ff; of
propositions, accidental, 156 ff; of prop­
ositions, importance of, 156 f; of propo­
sitions, simple, 155 ff; of propositions,
when legitimate, 157 f; simple, of sin­
gular propositions, 165
Convertend, 155 ff
Convertibility (commensurate universality), required for strict demonstration,
264
Copula, of the categorical proposition,
120; of the compound proposition, 120,
129; establishing a suppositional de­
mand for the subject, 114 f; of propo­
sition, form of verb “to be,” 102
Copulative proposition, 130 ff; occultly,
130 f; openly, 130; plus, 133
Deduction, 178 f; insufficiency of, 280 f
Definition, 67 ff; of an accident, 81; of
an analogical concept, 81; causal, 77 ff;
of characteristically collective terms,
113; of characteristically divisive terms,
113; discloses the comprehension of a
concept, 28; essential, 77 ff; essential,
two modes, 78 f; by extrinsic cause,
77 ff; frequently difficult of attainment,
91; by intrinsic causes, 77 f; modes of,
in reference to abstraction from matter,
270 f; nominal, 73 ff; nonessential, 77 ff;
by property, 77 ff; real. 73 ff; real, types
of, 77 ff; rules for, 80 ff; of an ultimate
genus, 81; value of. along with division,
90 f; by way of combination of acci­
dents, 80; by way of etymology, 74 f
Demonstration, 179, 258 ff; an analogous
notion, 259; a posteriori, 265 ff- a priori,
265 ff; difficult to achieve, 271; ex­
planatory (propter quid), 264 ff; ex­
planatory, three types of, 265 f; of the
fact (quia), 264 ff; from necessary
premises to a ncccssarv conclusion,
262 f; premises of, 261 f; prcscientific
knowledge required for, 260 f; role of
causes in, 286 f; as the scientific svllorism, 259; in strictest form. 259 f,
265 f; terminated in the real. 273; types
of, 264 ff; weaker forms of each type,
269; yields certainly true conclusions,
179
Demonstrative logic, 319
Deprecative statement, 102
Dialectic, 179, 285 ff, 319; leads to prob­

INDEX

able conclusion, 179, 271 ff; as prepara­
tion for demonstration, 275 f; three
purposes of, 276; unterminated, i.e., re­
mains in the reason, 273 f
Dialectical argumentation, see Dialectic
Dichotomous division, 86
Dictum de nullo, 189 f
Dictum de omni, 189 f
Difference, defined, 46 f; generic, 46;
predicable, nature of, 42 f; specific, 46
Dilemma, 251 ff; complex constructive,
252 f; complex destructive, 253; simple
constructive, 252; simple destructive,
252
Discourse, logical, in more and less strict
sense, 84; most strictly realized in rea­
soning, 176; rational, 7; in strict and
less strict senses, 73
Disjunctive proposition, 137; reduced to
conditional, 138
Disjunctive syllogism, with multimem­
bered major, 238 f; special rules, 234 ff
Division, logical, 72, 84 ff; logical, of ana­
logical wholes, 85 f; logical, dichoto­
mous and nondichotomous, 86; logical
essential, 85; logical nonessential, 85;
physical, 87 ff; physical, variety of, 87 f;
rules for, 88 f; for the sake of defini­
tion, 90 f; value of, along with defini­
tion, 90 f
Divisive supposition, 112

Efficient cause, 76
Enthymcmc, a rhetorical syllogism, 244
Enumeration, inductive, complete, always
sufficient, 293; inductive, complete. Only
of species and not of singulars, 292 f;
inductive, must be discriminative to be
sufficient, 291 f; inductive, ordinarily
generates probability at best, 293 f; in­
ductive, sufficiency of, 290; rather than
division, 88; of singulars, principle for
mediate induction, 288 ff
Epicheirema, 246 ff
Equivocal terms, invalidating discourse,
115; unsuited to the categories, 58
Equivocation, fallacy of, 298
Equivocity, 106; of names, 55
Essence, both substantial and accidental,
66; object of simple apprehension, 6,
17, 19, 97
Essential definition, 77 f; two modes, 78 f
Essential division, 85
Evidence, intrinsic or extrinsic, 100 f
Exceptive copulative proposition, 131

327
Exclusive alternative syllogism, special
rules of, 233; with multimembered
major, 238
Exclusive copulative proposition, 131
Existence, actual or possible intended in
propositions, 117; admitting of differ­
ent modes and grounding different judg­
ments, 98 f; known along with essence
in the judgment, 97; object of judg­
ment, not simple apprehension, 19;
possible or actual intended in propo­
sitions, 113 f
Explanatory demonstration, 264 ff; three
types of, 265 f
Exponible proposition, 130
Extension, of a concept, 25, 29 ff; in­
versely proportional to comprehension,
30; of terms in propositions, 110, 125

Faith, assent on extrinsic evidence, lOOf
Fallacy, of accent, 299 f; of accident,
300 f; of ambiguity, 298; of arguing be­
side the point, 301 ff; argumentum ad
baculum, 302; argumentum ad homi­
nem, 302; argumentum ad misericor­
diam, 303; argumentum ad populum,
302; argumentum ad verecundiam, 303;
of begging the question, 303 f; beyond
language, 300 ff; of complex question,
305; of composition and division, 299;
of the consequent, 305; of equivoca­
tion, 298; of false cause, 304 f; of
language, 298 ff; of mistaking qualified
for absolute, 301; of similar word con­
struction, 300; of vicious circle, 304
False cause, fallacy of, 304 f
Falsity, and truth of propositions, 101
Figures, of categorical syllogism, 185 f; of
categorical syllogism, first and most per­
fect, 187; of categorical syllogism, no
fourth, 187 f; of categorical syllogism,
three and only three, 185 f; of hypo­
thetical syllogism, 223 f
Final cause, 76
Finite being, can be categorized, 58
Formal cause, 76
Formal concept, mental expression of ob­
ject known in simple apprehension, 22;
various names for, 24
Formal logic, 317 ff; concerned with valid­
ity, 9 ff
Formal object, 4
Formal supposition, 107 ff; types of, 108 f

Genus, defined, 44 ff; predicable, nature

328
of, 42 f; proximate, 45; remote, 45;
ultimate, 45
Glanville, John J., 266 n
Grammatical expression, not always iden­
tical to logical expression, 123
Grammatical usage, of terms, not to be
confused with logical usage, 111

1 laving possession, category of, 59 ff, 64
Hollenhorst, G. Donald, 266 n
Houde, Roland, 277 n
Hypothetical proposition, 129 f; funda­
mentally conditional, 137 f
Hypothetical syllogism, 222 ff; with com­
pound components, 239 f; conclusion
of, 223; figures of, 223 f; major prem­
ise, 223; minor premise, 223; moods
of, 223 f; with multimembered major
premise, 237 ff; types of, 223; sec also
Disjunctive syllogism, Exclusive alterna­
tive syllogism, Inclusive alternative syl­
logism, Reciprocal conditional syllogism.
Simple conditional syllogism
idea, sec Formal concept
Illicit major term, 198 f
Illicit minor term, 198 f
Immediate induction, of self-evident prop­
ositions, 281 f
Imperative statement, 102
Implication, formal, 142 n; material, 142 n
Inclusive alternative syllogism, special
rules, 231 f; with multimembered major,
237 f
Induction, 178 f; abstractive, 281 f; im­
mediate, of self-evident propositions,
281 f; mediate, 282, 287 ff; mediate,
compared with deduction, 289 f; medi­
ate, compared with induction of selfevident proposition, 288 f; Mill’s canons
for, 295; necessary for deduction, 281;
relation to deduction, 290; three types
of, 281 f
Inference, immediate impossible in strict
sense, 180; strictly taken, limited to
reasoning, 147
Inferior, subjective part, 31 ff
Infinite being, cannot be categorized, 58
Interrogative statement, 102
John of St. Thomas, primary source in
logic, ix
Judgment, 97 ff; attributive, 97; certain
or probable, 100; complexity of its ob­
ject, 17; merely existential, 97; and

INDEX

the motive for assent, 99 ff; the opera­
tion of composition or division, 98;
pre eminence as an intellectual opera­
tion, 99; the second operation of the
intellect, 5 f
Knowledge, effected through formal signs,
21 f; intellectual, 2 ff; sensory, 2ff

Language, fallacy of, 298 f
Larkin, Vincent R., 277 n
Logic, Aristotelian, 142 n; art of, 2; as
“art of arts,” 2; art of rational dis­
course, 7; as common mode to be
contracted to special methodologies,
319 f; demonstrative science of logical
rules, 313; divided into branches by
three operations of intellect, 316 f;
divisions of, according to operations of
intellect, also into formal and material.
9 ff; doctrinal and in use, 320 f; formal
and material, 317 ff (see also Formal
logic. Material logic); as liberal art,
312; mathematical, 142 n, 317 ff, 321 f;
natural, 1 f, 5; nature of, 309 ff; per­
fection of the intellect not the senses,
5; as "rational science,” 2; real science,
but not a science of the real, 315 f;
science of second intentions, 7; as
speculative but useful science, 312 ff.
316; studies logical forms and not in­
telligible content of objects, 318 f; sym­
bolic, 142 n, 317 ff, 321 f; in use,
three uses for. 320 f
Logical relations. the subject matter of
logic, 7 ff
Logical supposition, 108, 116
Logico-real supposition, 109 f. 116
Major premise, in categorical syllogism,
183 f; of hypothetical syllogism, 223
Major term, illicit, 198 f
Material cause, 76
Material logic, 317 ff; branches of, 319;
concerned with probative force, 11 f
Material object, 4
Material supposition, 107 f, 115 f
Mathematical logic, 142 n, 317 ff, 321 f;
no rival of Aristotelian, 322
Matter, proximate and remote, of the
argumentation, 175
Mediate induction. 282
Mental term, see Formal concept
Mental word. see Formal concept
Methodology, scientific, 319 f

INDEX

Middle term, in categorical syllogism,
183 f; in demonstration, 270; example
of a logical relation, 8; unextended,
199 f
Mill, John Stuart, canons of induction,
295; criticism of syllogism, 191 f
Minor premise, of categorical syllogism,
183; of hypothetical syllogism, 223
Minor term, in categorical syllogism,
183 f; illicit, 198 f
Modal proposition, 14
Moods, of categorical syllogism, 188 f;
of categorical syllogism, indirect in sec­
ond and third figures, 211 ff; of hypo­
thetical syllogism, 223 f; valid, of cate­
gorical syllogism, 208 ff
Natural logic, 1 f, 5
Nature, object of simple apprehension, 17,
23; taken absolutely, 36 f; see also
Essence
Natures, complete, can be categorized, 58;
complex, cannot be categorized, 57;
incomplete, categorized only reductivcly, 58; incomplex, can be cate­
gorized, 57
Nominal definition, 73 ff
Nominalists, notion of extension, 30
Nondichotomous division, 86
Nonessential definition, 77 ff
Nonessential division, 85
Noun, element of proposition, 102

Object, formal, 4; material, 4
Objective concept, the object known in
simple apprehension, 23
Obverse, 159 f
Obversion, of propositions, 159 ff; of
propositions, importance of, 160; of
singular propositions, 165 f
Obvertend, 159 f
Operation, immanent, 63; transitive, 63
Opposition, between concepts, 66 ff; be­
tween concepts, modes of, in reference
to division, 85; between propositions.
148 ff; square of, 154
Particular supposition, 110 ff, 117
Passion, category of, 59 ff, 64
Perseity, four modes of, 263 f; not to be
confused with self-evidence, 284 n; re­
quired for strict demonstration. 263 f
Personal supposition, 109f, 112
Physical division. 87 ff

529
Plural, of grammatical concern, not logi­
cal, 123
Poetic argumentation, see Poetics
Poetics, 272, 319
Polysyllogism, 247 f
Porphyry, Neoplatonic commentator on
Aristotle, 51; primary source in logic,
ix; Tree of, 51
Position, category of, 59 ff, 64
Predicables, 41 ff; distinguished from cate­
gories, 54; a division of only some
predicates, 50 f; division of the uni­
versal, 39; division of the universal denominativcly taken, 41 ff; essential, 47;
essential and noncssential, 42 f; as re­
lated to the categories, 65 f
Predicaments, see Categories
Predicate, 97 ff; establishing a supposi­
tional demand for the subject, 114 f;
extension of, 124 f; what it is, 102 f
Predication, essential and nonessential or
denominative, 33
Premises, for demonstration. 272; for dia­
lectical argumentation, 272 f; major, in
categorical syllogism, 183 f; major of
hypothetical syllogism, 223; major, see
also Major premise; minor, in cate­
gorical syllogism, 183 f; minor, of hypo­
thetical syllogism, 223; minor, see also
Minor premise
Principle, of contradiction, a methodologi­
cal axiom. 285; of the separating third.
189 f; of sufficient enumeration, 290;
of triple identity, 189 f, 289
Privation, a mode of opposition between
concepts, 67
Privative terms, as used in obversion, 160
Probability, 11
Probative force, the concern of material
logic, 11 f
Property, defined. 47 ff; generic, 48; predi­
cable, nature of. 42 f: relation to differ­
ence, 48; specific, 48
Proposition, adversative, 132 f; alternative.
135 ff; alternative exclusive, 136 f; al­
ternative inclusive, 135 f; based on in­
trinsic or extrinsic evidence, 100 f; cate­
gorical, 129; causal, 132 f; compound,
120, 129 ff; compound, more or less
complex, 129; compound, symbols for,
139 f; conditional, 133 ff; conditional,
reciprocal, 134 f; conditional, simple,
133 f; conjunctive, 129ff; contraposi­
tion of, 161 ff; conversion of, 155 ff;
copulative. 130 ff; copulative, occultly.

330
130f; copulative, occultly, exceptive,
131; copulative, occultly, exclusive, 131;
copulative, occultly, reduplicative, 131;
copulative, openly, 130; copulative plus,
133; disjunctive, 137; disjunctive, not
simply contradiction of copulative, 140;
divided according to both quantity and
quality, 126 f; division of, 143; elements
of, 120; exponible, 130; expression of
object of judgment, 22, 24; factually
evident, 100 f; having force of singular
with nonpersonal subjects, 124; hypo*
thetical, 129 f; hypothetical, fundamen­
tally conditional, 137 f; hypothetical,
truth functionally taken, 142 f n; modal,
121; modal, with mode of contingency,
121; modal, with mode of impossibility,
121; modal, with mode of necessity',
121; modal, with mode of possibility-,
121; obversion, 159 ff; quality of, af­
firmative or negative, 122 f; quantity of,
universal, particular, or singular, 122 f;
related, mutually indices for truth­
status of one another, 147; self-evident,
100 f; self-evident, the absolute premise
for demonstration, 262 f; self-evident,
basic truth of the scientific syllogism,
283; self-evident, immediate, 282 f; selfevident, indemonstrable, 283; self-evi­
dent, induction of, 281 f, 286 f; selfevident, principle of demonstration,
282 ff; self-evident, types of, 284 ff;
simple or categorical, 120 ff; simply
attributive, 121; strict logical form of,
102 f; truth-functional, 141 ff; tvpcs of,
120 ff

Quality, category of, 59 ff, 62 ff; species
of, 63
Quantity, category of, 59 ff, 62 f
Quiddity, sec Essence

Real being, actual or possible, 99
Real definition, 73 ff; types of, 77 ff
Real supposition. 108, 116; types of, 109 f
Reasoning, nature of, 173 ff; the need for,
173 f; for the sake of judgment, 173 f;
the third operation of the intellect, 5 ff
Reciprocal conditional syllogism, special
rule, 229 f
Reduplicative copulative proposition, 131
Relation, belonging to singular proposi­
tions, 163 ff; between propositions,
147 ff; between propositions, complex,

INDEX

161 f; a category, 59 ff, 63; logical,
second intention, 313 f; mode of opposition Ix-tween concepts, 68
Rhetoric, 319
Rhetorical argumentation, 272
Science, 258 f; empiriological, 274 f; naturc of, 310 f; speculative, divided by
different modes of defining, 270 f;
speculative or practical, 311
Scientific method, of empiriological
science, an instance of dialectic, 274 f
Second intention, 312; canonically ex­
pressed in logical rules, 314; logical re­
lations, 7; nature of, 313 ff; object of
science of logic, 314 f
Sequential nexus, in hypothetical, 130
Sign, formal or instrumental, 21 ff; im­
port in logic, 20 ff; natural or conven­
tional, 20 f; nature and divisions of,
20 ff
Signification, and supposition, 106 f; of
terms, 106, 115
Simon, Yves R., 266 n
Simple apprehension, the first operation
of the intellect, 5 f; incomplexity of
its object as compared to object of
judgment, 17, 55; its object simple or
complex, 17 f; its object, simple or
complex, in reference to categories,
55 f; nature of, 17 ff; a truncated act
of knowing, 97
Simple conditional syllogism, 225 ff; rules
for, 226
Simply attributive propositions, 121
Singular, as characterizing the sense ob­
ject, 4; as opposed to universal, 56
Singularity, of things in the real, 36 f
Singular supposition, 110 ff, 117
Sophistical argumentation, 129
Sophistry, 297
Sorites, 248 f; Aristotelian, 248 ff; Goclenian, 249 ff
Species, from point of view of subjectibility, 46; one per subject, 44; predicable,
nature of, 42 ff
Subalternand (superior), 152
Subaltérnate (inferior), 152
Subalternation, of propositions, 152 f; of
singular propositions, 164 f
Subcontrariety,
between
propositions,
150 f; of singular propositions, 163 f
Subdivision, 89
Subject, logical, 97 ff, 102 f
Substance, 56; category of, 59 ff, 62

INDEX

Superior, logical whole or potential whole,
31 ff
Supposition, absolute, 109, 116; absolute,
virtually universal, 111, 116; collective,
112, 117; divisive, 112; formal, 107 ff;
formal, types of, 108 f; logical, 108,
116; logical, functions as though singu­
lar, 112; logicorcal, 108 f, 116; logicorca), functions as though singular, 111;
material, 107 f, 115 f; particular, 117;
particular or undistributed, 110 ff; per­
sonal, 109 f, 116; of predicates, usually
real and personal, 107; real, 108, 116;
real, types of, 109 f; singular, llOff,
117; of terms, 106ff, 115 ff; universal,
117; universal or distributed, 110 ff
Syllogism, 182; abbreviated, 243 ff; ab­
breviated, grammatically only, 244;
categorical, 182 ff, sec also Categorical
syllogism; categorical, elements of. 183 f;
complex in combination, 253 f; com­
plex, 243 ff; disjunctive, 234 ff, see also
Disjunctive syllogism; exclusive alterna­
tive, 232 ff, see also Exclusive alterna­
tive syllogism; “expository,” 193 f; hy­
pothetical, 222 ff, see also Hypothetical
syllogism; hypothetical, types of, 223;
inclusive alternative, 231 f, see also In­
clusive alternative syllogism; reciprocal
conditional, 229 ff, see also Reciprocal
conditional syllogism; reduction of al­
ternative and disjunctive to conditional,
236 f; simple conditional, 225 ff, see
also Simple conditional syllogism; with
a justification for its premises (epicheirema), 245 ff
Symbolic logic, see Mathematical logic
Terms, expressed indefinitely, 123; ex­
pression of object of simple apprehen­
sion, 24; major, in categorical syllogism,
183 f, see also Major term; middle, in
categorical syllogisms, 183 f; middle,
role in demonstration, 270 f, see also
Middle term; minor, in categorical syl­
logism, 183 f, see also Minor term;
nonsupposing subject, 114 f; and their
signification, 106 ff; and their supposi­

331
tion in propositions, 106 ff; unity- of,
in logical discourse, 115; unity of, in
propositions, 153 f
Thesis, necessary for a given science, 285
Thomas Aquinas, St., 93, 221, 277 n;
definition of logic, 2; on kinds of selfevident propositions, 284 f; primary
source in logic, Lx
Transcendental object, 58
Truth, and falsity of propositions, 101;
found properly in judgment, 101; and
validity. 176 ff
Truth-functional proposition, 141 ff
Truth table, 141 ff; for hypothetical
strictly taken, 142; for hypothetical
truth functionally taken, 142; limited
use for hypothetical strictly taken,
141 f; for square of opposition, 155
Unextended middle term, 199 f
Unity of terms, required in logically re­
lated propositions, 153 f
Universal, as characterizing the intelligible
object for man, 4 f; divided into predi­
cables and categories, 39; does not
falsify the real, 38; nature of, 56;
problem of, 35 ff
Universality, enumerativo, 193; of object
in the intellect, 36 f; required for strict
demonstration, 263; true, rather than
enumcrative, 193
Universal supposition, 110 ff, 117
Univocal terms, readily suited for dis­
course, 115; suited to the categories, 58
Univocity, 106; of names, 55

Validity, the concern of formal logic,
9 ff; and truth, 176 ff
Verb, element of proposition, 102; as
verb-copula, 102; as verb-predicate, 102
Vocative statement, 102
Wallace, William A., O.P., 278
When, category of, 59 ff, 64
Where, category of, 59 ff, 64
Words, as instrumental conventional signs
of meanings, 22 f