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LIBRARY

UNIVERSITY OP
CALIFORNIA

SANWKJO



A SURVEY OF
SYMBOLIC LOGIC



BY

C. I. LEWIS



UNIVERSITY OF CALIFORNIA PRESS

BERKELEY

1918



PRESS OF

THE NEW ERA PRINTING COMPANY
LANCASTER, PA.



TABLE OF CONTENTS

PREFACE v

CHAPTER I. THE DEVELOPMENT OF SYMBOLIC LOGIC. 1
SECTION I. The Scope of Symbolic Logic. Symbolic Logic
and Logistic. Summary Account of their

Development 1

SECTION II. Leibniz 5

SECTION III. From Leibniz to De Morgan and Boole 18

SECTION IV. De Morgan 37

SECTION V. Boole 51

SECTION VI. Jevons 72

SECTION VII. Peirce 79

SECTION VIII. Developments since Peirce 107

CHAPTER II. THE CLASSIC, OR BOOLE-SCHRODER AL-
GEBRA OF LOGIC 118

SECTION I. General Character of the Algebra. The Postulates

and their Interpretation 118

SECTION II. Elementary Theorems 122

SECTION III. General Properties of Functions 132

SECTION IV. Fundamental Laws of the Theory of Equations . .' . 144

SECTION V. Fundamental Laws of the Theory of Inequations. 166
SECTION VI. Note on the Inverse Operations, "Subtraction"

and "Division" 173

CHAPTER III. APPLICATIONS OF THE BOOLE-SCHRODER

ALGEBRA 175

SECTION I. Diagrams for the Logical Relations of Classes .... 175

SECTION II. The Application to Classes 184

SECTION III. The Application to Propositions 213

SECTION IV. The Application to Relations 219

CHAPTER IV. SYSTEMS BASED ON MATERIAL IMPLI-
CATION 222

SECTION I. The Two- Valued Algebra 222

Hi



IV



Table of Contents



SECTION II. The Calculus of Prepositional Functions. Func-
tions of One Variable 232

SECTION III. Prepositional Functions of Two or More Variables . 246
SECTION IV. Derivation of the Logic of Classes from the Calcu-
lus of Prepositional Functions 260

SECTION V. The Logic of Relations 269

SECTION VI. The Logic of Principia Mathematica 279

CHAPTER V. THE SYSTEM OF STRICT IMPLICATION... 291
SECTION I. Primitive Ideas, Primitive Propositions, and Im-
mediate Consequences 292

SECTION II. Strict Relations and Material Relations 299

SECTION III. The Transformation {-/-} 306

SECTION IV. Extensions of Strict Implication. The Calculus
of Consistencies and the Calculus of Ordinary

Inference 316

SECTION V. The Meaning of "Implies" 324

CHAPTER VI. SYMBOLIC LOGIC, LOGISTIC, AND MATHE-
MATICAL METHOD 340

SECTION I. General Character of the Logistic Method. The

"Orthodox" View 340

SECTION II. Two Varieties of Logistic Method : Peano's Formu-
laire and Principia Mathematica. The Nature
of Logistic Proof 343

SECTION III. A "Heterodox" View of the Nature of Mathe-
matics and of Logistic 354

SECTION IV. The Logistic Method of Kempe and Royce 362

SECTION V. Summary and Conclusion 367

APPENDIX. TWO FRAGMENTS FROM LEIBNIZ 373

BIBLIOGRAPHY 389

INDEX. . 407



PREFACE

The student who has completed some elementary study of symbolic
logic and wishes to pursue the subject further finds himself in a discouraging
situation. He has, perhaps, mastered the contents of Venn's Symbolic
Logic or Couturat's admirable little book, The Algebra of Logic, or the
chapters concerning this subject in Whitehead's Universal Algebra. If he
read German with sufficient ease, he may have made some excursions into
Schroder's Vorlesungen iiber die Algebra der Logik. These all concern the
classic, or Boole-Schroder algebra, and his knowledge of symbolic logic is
probably confined to that system. His further interest leads him almost
inevitably to Peano's Formulaire de Mathematiques, Principia Mathematica
of Whitehead and Russell, and the increasingly numerous shorter studies
of the same sort. And with only elementary knowledge of a single kind of
development of a small branch of the subject, he must attack these most
difficult and technical of treatises, in a new notation, developed by methods
which are entirely novel to him, and bristling with logico-metaphysical
difficulties. If he is bewildered and searches for some means of further
preparation, he finds nothing to bridge the gap. Schroder's work would
be of most assistance here, but this was written some twenty-five years
ago; the most valuable studies are of later date, and radically new methods
have been introduced.

What such a student most needs is a comprehensive survey of the sub-
ject one which will familiarize him with more than the single system
which he knows, and will indicate not only the content of other branches
and the alternative methods of procedure, but also the relation of these to
the Boole-Schroder algebra and to one another. The present book is an
attempt to meet this need, by bringing within the compass of a single
volume, and reducing to a common notation (so far as possible), the most
important developments of symbolic logic. If, in addition to this, some
of the requirements of a "handbook" are here fulfilled, so much the better.

But this survey does not pretend to be encyclopedic. A gossipy recital
of results achieved, or a superficial account of methods, is of no more use
in symbolic logic than in any other mathematical discipline. What is
presented must be treated in sufficient detail to afford the possibility of real
insight and grasp. This aim has required careful selection of material.



vi Preface

The historical summary in Chapter I attempts to follow the main thread
of development, and no reference, or only passing mention, is given to
those studies which seem not to have affected materially the methods of
later researches. In the remainder of the book, the selection has been
governed by the same purpose. Those topics comprehension of which
seems most essential, have been treated at some length, while matters less
fundamental have been set forth in outline only, or omitted altogether.
My own contribution to symbolic logic, presented in Chapter V, has not
earned the right to inclusion here; in this, I plead guilty to partiality.
The discussion of controversial topics has been avoided whenever possible
and, for the rest, limited to the simpler issues involved. Consequently,
the reader must not suppose that any sufficient consideration of these
questions is here given, though such statements as are made will be, I hope,
accurate. Particularly in the last chapter, on "Symbolic Logic, Logistic,
and Mathematical Method ", it is not possible to give anything like an
adequate account of the facts. That would require a volume at least the
size of this one. Rather, I have tried to set forth the most important and
critical considerations somewhat arbitrarily and dogmatically, since there
is not space for argument and to provide such a map of this difficult terri-
tory as will aid the student in his further explorations.

Proofs and solutions in Chapters II, III, and IV have been given very
fully. Proof is of the essence of logistic, and it is my observation that stu-
dents even those with a fair knowledge of mathematics seldom command
the technique of rigorous demonstration. In any case, this explicitness can
do no harm, since no one need read a proof which he already understands.

I am indebted to many friends and colleagues for valuable assistance in
preparing this book for publication : to Professor W. A. Merrill for emenda-
tions of my translation of Leibniz, to Professor J. H. McDonald and
Dr. B. A. Bernstein for important suggestions and the correction of certain
errors in Chapter II, to Mr. J. C. Rowell, University Librarian, for assistance
in securing a number of rare volumes, and to the officers of the University
Press for their patient helpfulness in meeting the technical difficulties of
printing such a book. Mr. Shirley Quimby has read the whole book in
manuscript, eliminated many mistakes, and verified most of the proofs.

But most of all, I am indebted to my friend and teacher, Josiah Royce,
who first aroused my interest in this subject, and who never failed to give
me encouragement and wise counsel. Much that is best in this book is
due to him. C. I. LEWIS.

BERKELEY, July 10, 1917.



CHAPTER I

THE DEVELOPMENT OF SYMBOLIC LOGIC

I. THE SCOPE OF SYMBOLIC LOGIC. SYMBOLIC LOGIC AND LOGISTIC.
SUMMARY ACCOUNT OF THEIR DEVELOPMENT

The subject with which we are concerned has been variously referred
to as "symbolic logic", "logistic", "algebra of logic", "calculus of logic",
"mathematical logic", "algorithmic logic", and probably by other names.
And none of these is satisfactory. We have chosen "symbolic logic"
because it is the most commonly used in England and in this country, and
because its signification is pretty well understood. Its inaccuracy is
obvious: logic of whatever sort uses symbols. We are concerned only
with that logic which uses symbols in certain specific ways those ways
which are exhibited generally in mathematical procedures. In particular,
logic to be called "symbolic" must make use of symbols for the logical
relations, and must so connect various relations that they admit of "trans-
formations" and "operations", according to principles which are capable
of exact statement.

If we must give some definition, we shall hazard the following: Symbolic
Logic is the development of the most general principles of rational pro-
cedure, in ideographic symbols, and in a form which exhibits the connection
of these principles one with another. Principles which belong exclusively
to some one type of rational procedure e. g. to dealing with number and
quantity are hereby excluded, and generality is designated as one of the
marks of symbolic logic.

Such general principles are likewise the subject matter of logic in any
form. To be sure, traditional logic has never taken possession of more
than a small portion of the field which belongs to it. The modes of Aristotle
are unnecessarily restricted. As we shall have occasion to point out, the
reasons for the syllogistic form are psychological, not logical : the syllogism,
made up of the smallest number of propositions (three), each with the small-
est number of terms (two), by which any generality of reasoning can be
attained, represents the limitations of human attention, not logical necessity.
To regard the syllogism as indispensable, or as reasoning par excellence, is
2 1



2 A Survey of Symbolic Logic

the apotheosis of stupidity. And the procedures of symbolic logic, not
being thus arbitrarily restricted, may seem to mark a difference of subject
matter between it and the traditional logic. But any such difference is
accidental, not essential, and the really distinguishing mark of symbolic
logic is the approximation to a certain form, regarded as ideal. There are
all degrees of such approximation ; hence the difficulty of drawing any hard
and fast line between symbolic and other logic.

But more important than the making of any such sharp distinction is
the comprehension of that ideal of form upon which it is supposed to
depend. The most convenient method which the human mind has so far
devised for exhibiting principles of exact procedure is the one which we
call, in general terms, mathematical. The important characteristics of
this form are: (1) the use of ideograms instead of the phonograms of
ordinary language; (2) the deductive method which may here be taken
to mean simply that the greater portion of the subject matter is derived
from a relatively few principles by operations which are "exact": and
(3) the use of variables having a definite range of significance.

Ideograms have two important advantages over phonograms. In the
first place, they are more compact, + than "plus", 3 than "three", etc.
This is no inconsiderable gain, since it makes possible the presentation of a
formula in small enough compass so that the eye may apprehend it at a
glance and the image of it (in visual or other terms) may be retained for
reference with a minimum of effort. None but a very thoughtless person,
or one without experience of the sciences, can fail to understand the enor-
mous advantage of such brevity. In the second place, an ideographic
notation is superior to any other in precision. Many ideas which are
quite simply expressible in mathematical symbols can only with the greatest
difficulty be rendered in ordinary language. Without ideograms, even
arithmetic would be difficult, and higher branches impossible.

The deductive method, by which a considerable array of facts is sum-
marized in a few principles from which they can be derived, is much more
than the mere application of deductive logic to the subject matter in
question. It both requires and facilitates such an analysis of the whole
body of facts as will most precisely exhibit their relations to one another.
In fact, any other value of the deductive form is largely or wholly fictitious.

The presentation of the subject matter of logic in this mathematical
form constitutes what we mean by symbolic logic. Hence the essential
characteristics of our subject are the following: (1) Its subject matter is



The Development of Symbolic Logic 3

the subject matter of logic in any form that is, the principles of rational
or reflective procedure in general, as contrasted with principles which
belong exclusively to some particular branch of such procedure. (2) Its
medium is an ideographic symbolism, in which each separate character
represents a relatively simple and entirely explicit concept. And, ideally,
all non-ideographic symbolism or language is excluded. (3) Amongst the
ideograms, some will represent variables (the "terms" of the system)
having a definite range of significance. Although it is non-essential, in
any system so far developed the variables will represent "individuals",
or classes, or relations, or propositions, or " propositional functions", or
they will represent ambiguously some two or more of these. (4) Any
system of symbolic logic will be developed deductively that is, the whole
body of its theorems will be derived from a relatively few principles, stated
in symbols, by operations which are, or at least can be, precisely formulated.

We have been at some pains to make as clear as possible the nature of
symbolic logic, because its distinction from "ordinary" logic, on the one
hand, and, on the other, from any mathematical discipline in a sufficiently
abstract form, is none too definite. It will be further valuable to comment
briefly on some of the alternative designations for the subject which have
been mentioned.

"Logistic" would not have served our purpose, because "logistic" is
commonly used to denote symbolic logic together with the application of
its methods to other symbolic procedures. Logistic may be defined as
the science which deals with types of order as such. It is not so much a
subject as a method. Although most logistic is either founded upon or
makes large use of the principles of symbolic logic, still a science of order
in general does not necessarily presuppose, or begin with, symbolic logic.
Since the relations of symbolic logic, logistic, and mathematics are to be
the topic of the last chapter, we may postpone any further discussion of
that matter here. We have mentioned it only to make clear the meaning
which "logistic" is to have in the pages which follow. It comprehends
symbolic logic and the application of such methods as symbolic logic exempli-
fies to other exact procedures. Its subject matter is not confined to logic.

"Algebra of logic" is hardly appropriate as the general name for our
subject, because there are several quite distinct algebras of logic, and
because symbolic logic includes systems which are not true algebras at all.
"The algebra of logic" usually means that system the foundations of
which were laid by Leibniz, and after him independently by Boole, and



4 A Survey of Symbolic Logic

which was completed by Schroder. We shall refer to this system as the
"Boole-Schroder Algebra ".

"Calculus" is a more general term than "algebra". By a "calculus"
will be meant, not the whole subject, but any single system of assumptions
and their consequences.

The program both for symbolic logic and for logistic, in anything like a
clear form, was first sketched by Leibniz, though the ideal of logistic seems
to have been present as far back as Plato's Republic. 1 Leibniz left frag-
mentary developments of symbolic logic, and some attempts at logistic
which are prophetic but otherwise without value. After Leibniz, the two
interests somewhat diverge. Contributions to symbolic logic were made by
Ploucquet, Lambert, Castillon and others on the continent. This type of
research interested Sir William Hamilton and, though his own contribution
was slight and not essentially novel, his papers were, to some extent at
least, responsible for the renewal of investigations in this field which took
place in England about 1845 and produced the work of De Morgan and
Boole. Boole seems to have been ignorant of the work of his continental
predecessors, which is probably fortunate, since his own beginning has
proved so much more fruitful. Boole is, in fact, the second founder of the
subject, and all later work goes back to his. The main line of this develop-
ment runs through Jevons, C. S. Peirce, and MacColl to Schroder whose
Vorlesungen uber die Algebra der Logik (Vol. I, 1890) marks the perfection
of Boole's algebra and the logical completion of that mode of procedure.

In the meantime, interest in logistic persisted on the continent and
was fostered by the growing tendency to abstractness and rigor in mathe-
matics and by the hope for more general methods. Hamilton's quaternions
and the Ausdehnungslehre of Grassmann, which was recognized as a con-
tinuation of the work begun by Leibniz, contributed to this end, as did also
the precise logical analyses of the nature of number by Cantor and Dedekind.
Also, the elimination from "modern geometry" of all methods of proof
dependent upon "intuitions of space" or "construction" brought that
subject within the scope of logistic treatment, and in 1889 Peano provided
such a treatment in I Principii di Geometria. Frege's works, from the
Begri/sschrift of 1879 to the Grundgesetze der Arithmetik (Vol. I, 1893;
Vol. II, 1903) provide a comprehensive development of arithmetic by the
logistic method.

1 See the criticisms of contemporary mathematics and the program for the dialectic
or philosophic development of mathematics in Bk. vi, Step. 510-11 and Philebus, Step. 56-57.



The Development of Symbolic Logic 5

In 1894, Peano and his collaborators began the publication of the
Formulaire de Mathematiques, in which all branches of mathematics were to
be presented in the universal language of logistic. In this work, symbolic
logic and logistic are once more brought together, since the logic presented
in the early sections provides, in a way, the method by which the other
branches of mathematics are developed. The Formulaire is a monumental
production. But its mathematical interests are as much encyclopedic as
logistic, and not all the possibilities of the method are utilized or made
clear. It remained for Whitehead and Russell, in Principia Mathematica,
to exhibit the perfect union of symbolic logic and the logistic method in
mathematics. The publication of this work undoubtedly marks an epoch
in the history of the subject. The tendencies marked in the development
of the algebra of logic from Boole to Schroder, in the development of the
algebra of relatives from De Morgan to Schroder, and in the foundations
for number theory of Cantor and Dedekind and Frege, are all brought
together here. 2 Further researches will most likely be based upon the
formulations of Principia Mathematica.

We must now turn back and trace in more detail the development of
symbolic logic. 3 A history of the subject will not be attempted, if by
history is meant the report of facts for their own sake. Rather, we are
interested in the cumulative process by which those results which most
interest us today have come to be. Many researches of intrinsic value,
but lying outside the main line of that development, will of necessity be
neglected. Reference to these, so far as we are acquainted with them, will
be found in the bibliography. 4

II. LEIBNIZ

The history of symbolic logic and logistic properly begins with Leibniz. 5
In the New Essays on the Human Understanding, Philalethes is made to
say : 6 "I begin to form for myself a wholly different idea of logic from
that which I formerly had. I regarded it as a scholar's diversion, but I
now see that, in the way you understand it, it is like a universal mathe-

2 Perhaps we should add "and the modern development of abstract geometry, as by
Hilbert, Fieri, and others", but the volume of Principia which is to treat of geometry has
not yet appeared.

3 The remainder of this chapter is not essential to an understanding of the rest of the
book. But after Chapter i, historical notes and references are generally omitted.

4 Pp. 389-406.

6 Leibniz regards Raymond Lully, Athanasius Kircher, John Wilkins, and George
Dalgarno (see Bibliography) as his predecessors in this field. But their writings contain
little which is directly to the point.

8 Bk. iv, Chap, xvn, 9.



6 A Survey of Symbolic Logic

matics." As this passage suggests, Leibniz correctly foresaw the general
character which logistic was to have and the problems it would set itself
to solve. But though he caught the large outlines of the subject and
actually delimited the field of work, he failed of any clear understanding
of the difficulties to be met, and he contributed comparatively little to
the successful working out of details. Perhaps this is characteristic of the
man. But another explanation, or partial explanation, is possible. Leibniz
expected that the whole of science would shortly be reformed by the appli-
cation of this method. This was a task clearly beyond the powers of any
one man, who could, at most, offer only the initial stimulus and general
plan. And so, throughout his life, he besought the assistance of learned
societies and titled patrons, to the end that this epoch-making reform might
be instituted, and never addressed himself very seriously to the more
limited tasks which he might have accomplished unaided. 7 Hence his
studies in this field are scattered through the manuscripts, many of them
still unedited, and out of five hundred or more pages, the systematic results
attained might be presented in one-tenth the space. 8

Leibniz's conception of the task to be accomplished altered somewhat
during his life, but two features characterize all the projects which he
entertained: (1) a universal medium ("universal language" or "rational
language" or "universal characteristic") for the expression of science;
and (2) a calculus of reasoning (or "universal calculus") designed to display
the most universal relations of scientific concepts and to afford some sys-
tematic abridgment of the labor of rational investigation in all fields, much
as mathematical formulae abridge the labor of dealing with quantity and
number. "The true method should furnish us with an Ariadne's thread,
that is to say, with a certain sensible and palpable medium, which will
guide the mind as do the lines drawn in geometry and the formulae for
operations which are laid down for the learner in arithmetic." 9

This universal medium is to be an ideographic language, each single
character of which will represent a simple concept. It will differ from
existing ideographic languages, such as Chinese, through using a combina-

7 The editor's introduction to "Scientia Generalis. Characteristica" in Gerhardt's
Philosophischen Schriften von Leibniz (Berlin, 1890), vn, gives an excellent account of
Leibniz's correspondence upon this topic, together with other material of historic interest.
(Work hereafter cited as G. Phil.)

8 See Gerhardt, op. dt. especially iv and vn. But Couturat, La logique de Leibniz
(1901), gives a survey which will prove more profitable to the general reader than any
study of the sources.

9 Letter to Galois, 1677, G. Phil, vn, 21.



The Development of Symbolic Logic 7

tion of symbols, or some similar device, for a compound idea, instead of
having a multiplicity of characters corresponding to the variety of things.
So that while Chinese can hardly be learned in a lifetime, the universal
characteristic may be mastered in a few weeks. 10 The fundamental char-
acters of the universal language will be few in number, and will represent
the "alphabet of human thought": "The fruit of many analyses will be the
catalogue of ideas which are simple or not far from simple." n With this
catalogue of primitive ideas this alphabet of human thought the whole
of science is to be reconstructed in such wise that its real logical organiza-



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