SEMICENTENNIAL PUBLICATIONS
OF THE
UNIVERSITY OF CALIFORNIA
1868-1918
A SURVEY OF
SYMBOLIC LOGIC
BY
C. I. LEWIS
UNIVERSITY OF CALIFORNIA PRESS
BERKELEY
1918
/ 3
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 I
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
iii
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 Propositional 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
SECTION IV. The Logistic Method of Kempe and Royce .
SECTION V. Summarv and Conclusion .
APPENDIX. TWO FRAGMENTS FROM LEIBNIZ
BIBLIOGRAPHY
INDEX.
354
362
367
373
389
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 uber 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> L 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 " prepositional 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 iiber 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.
5 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.
6 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. Characteristic a" in Gephardt'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. tit. 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, vii, 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
tion will be reflected in its symbolism.
In spite of fantastic expression and some hyperbole, we recognize here
the program of logistic. If the reconstruction of all science is a project too
ambitious, still we should maintain the ideal possibility and the desirability
of such a reconstruction of exact science in general. And the ideographic
language finds its realization in Peano's Formulaire, in Principia Mathe-
matica, and in all successful applications of the logistic method.
Leibniz stresses the importance of such a language for the more rapid
and orderly progress of science and of human thought in general. The
least effect of it ". . . will be the universality and communication of
different nations. Its true use will be to paint not the word . . . but the
thought, and to speak to the understanding rather than to the eyes. . . .
Lacking such guides, the mind can make no long journey without losing
its way . . . : with such a medium, we could reason in metaphysics
and in ethics very much as we do in geometry and in analytics, because the
characters would fix our ideas, which are otherwise too vague and fleeting
in such matters in which the imagination cannot help us unless it be by
the aid of characters."12 The lack of such a universal medium prevents
cooperation. "The human race, considered in its relation to the sciences
which serve our welfare, seems to me comparable to a troop which marches
in confusion in the darkness, without a leader, without order, without any
word or other signs for the regulation of their march and the recognition of
one another. Instead of joining hands to guide ourselves and make sure
of the road, we run hither and yon and interfere with one another."13
The "alphabet of human thought" is more visionary. The possibility
of constructing the whole of a complex science from a few primitive con-
10 Letter to the Duke of Hanover, 1679 (?), G. Phil., vn, 24-25.
11 G. Phil., vii, 84.
" G. Phil., vii, 21.
13 G. Phil, vii, 157.
8 A Survey of Symbolic Logic
cepts is, indeed, real — vide the few primitives of Principia Mathematica.
But we should today recognize a certain arbitrariness in the selection of
these, though an arbitrariness limited by the nature of the subject. The
secret of Leibniz's faith that these primitive concepts are fixed in the nature
of things will be found in his conception of knowledge and of proof. He
believes that all predicates are contained in the (intension of the) subject
and may be discovered by analysis. Similarly, all truths which are not
absolutely primitive and self-evident admit of reduction by analysis into
such absolutely first truths. And finally, only one real definition of a
thing — "real" as opposed to "nominal" — is possible;14 that is, the result
of the correct analysis of any concept is unambiguously predetermined in
the concept itself.
The construction, from such primitives, of the complex concepts of
the various sciences, Leibniz speaks of as "synthesis" or "invention",
and he is concerned about the "art of invention". But while the result of
analysis is always determined, and only one analysis is finally correct,
synthesis, like inverse processes generally, has no such predetermined
character. In spite of the frequent mention of the subject, the only im
portant suggestions for this art have to do with the provision of a suitable
medium and of a calculus of reasoning. To be sure there are such obvious
counsels as to proceed from the simple to the complex, and in the early
essay, De Arte Combinatoria, there are studies of the possible permutations
and combinations or "syntheses" of fundamental concepts, but the author
later regarded this study as of little value. And in Initia et Specimina
Scientice novce Generalis, he says that the utmost which we can hope to
accomplish at present, toward the general art of invention, is a perfectly
orderly and finished reconstruction of existing science in terms of the
absolute primitives which analysis reveals.15 After two hundred years,
we are still without any general method by which logistic may be used in
fields as yet unexplored, and we have no confidence in any absolute primi
tives for such investigation.
The calculus of reasoning, or universal calculus, is to be the instrument
for the development and manipulation of systems in the universal language,
and it is to get its complete generality from the fact that all science will be
expressed in the ideographic symbols of that universal medium. The
calculus will consist of the general principles of operating with such ideo-
14 See G. Phil, vn, 194, footnote.
15 G. Phil, vii, 84.
The Development of Symbolic Logic 9
graphic symbols: "All our reasoning is nothing but the relating and sub
stituting of characters, whether these characters be words or marks or
images."16 Thus while the characteristica universalis is the project of the
logistic treatment of science in general, the universal calculus is the pre
cursor of symbolic logic.
The plan for this universal calculus changed considerably with the
development of Leibniz's thought, but he speaks of it always as a mathe
matical procedure, and always as more general than existing mathematical
methods.17 The earliest form suggested for it is one in which the simple
concepts are to be represented by numbers, and the operations are to be
merely those of arithmetical multiplication, division, and factoring. When,
later, he abandons this plan of procedure, he speaks of a general calculus
which will be concerned with what we should nowadays describe as "types
of order" — with combinations which are absolute or relative, symmetrical
or unsymmetrical, and so on.1* His latest studies toward such a calculus
form the earliest presentation of what we now call the "algebra of logic".
But it is doubtful if Leibniz ever thought of the universal calculus as
restricted to our algebra of logic: we can only say that it was intended to
be the science of mathematical and deductive form in general (it is doubtful
whether induction was included), and such as to make possible the appli
cation of the analytic method of mathematics to all subjects of which
scientific knowledge is possible.
Of the various studies to this end our chief interest will be in the early
essay, De Arte Combinatorial and in the fragments which attempt to
develop an algebra of logic.20
Leibniz wrote De Arte Combinatoria when he was, in his own words,
vix egressns ex Ephebis, and before he had any considerable knowledge of
mathematics. It was published, he tells us, without his knowledge or
consent. The intention of the work, as indicated by its title, is to serve the
general art of rational invention, as the author conceived it. As has been
mentioned, it seems that this end is to be accomplished by a complete
analysis of concepts of the topic under investigation and a general survey
of the possibilities of their combination. A large portion of the essay is
concerned with the calculation of the possible forms of this and that type
16 G. Phil, vii, 31.
17 See New Essays on the Human Understanding, Bk. iv, Chap, xvn, §§ 9-13.
18 See G. Phil, vn, 31, 198 jf., and 204.
19 G. Phil, iv, 35-104. Also Gerhardt, Leibnizens mathematische Schriften (1859), V,
1-79.
20 Scientia Generalis. Characteristica, xv-xx, G. Phil., vn.
10 A Survey of Symbolic Logic
of logical construct: the various dyadic, triadic, etc., complexes which
can be formed with a given number of elements; of the moods and figures
of the syllogism; of the possible predicates of a given subject (the com
plexity of the subject as a concept being itself the key to the predicates
which can be analyzed out of it); of the number of propositions from a
given number of subjects, given number of predicate relations, and given
number of quaestiones \ 21 of the variations of order with a given number of
terms, and so on. In fact so much space is occupied with the computation
of permutations and combinations that some of his contemporaries failed
to discover any more important meaning of the essay, and it is most fre
quently referred to simply as a contribution to combinatorial analysis.22
Beyond this the significance of the essay lies in the attempt to devise a
symbolism which will preserve the relation of analyzable concepts to their
primitive constituents. The particular device selected for this purpose —
representation of concepts by numbers — is unfortunate, but the attempt
itself is of interest. Leibniz makes application of this method to geometry
and suggests it for other sciences.23 In the geometrical illustration, the
concepts are divided into classes. Class 1 consists of concepts or terms
regarded as elementary and not further analyzable, each of which is given a
number. Thereafter, the number is the symbol of that concept. Class 2
consists of concepts analyzable into (definable in terms of) those of Class 1.
By the use of a fractional notation, both the class to which a concept
belongs and its place in that class can be indicated at once. The denomi
nator indicates the number of the class and the numerator is the number of
the concept in that class. Thus the concept numbered 7 in Class 2 is
represented by 7/2. Class 3 consists of concepts definable in terms of
those in Class 1 and Class 2, and so on. By this method, the complete
analysis of any concept is supposed to be indicated by its numerical symbol.24
21 Leibniz tells us that he takes this problem from the Ars Magna of Raymond Lully.
See G. Phil., v, 62.
22 See letter to Tschirnhaus, 1678, Gerhardt, Math., iv, 451-63. Cf. Cantor, Geschichte
d. Math., m, 39 ff.
23 See the Synopsis, G. Phil, iv, 30-31.
24 See Couturat, op. tit., appended Note vi, p. 554 ff.
The concepts are arranged as follows (G. Phil., iv, 70-72):
"Classis I; 1. Punctum, 2. Spatium, 3. intervallum, 4. adsitum seu contiguum, 5. dis-
situm seu distans, 6. Terminus seu quae distant, 7. Insitum, 8. inclusum (v.g. centrum est
insitum circulo, inclusum peripheriae), 9. Pars, 10. Totum, 11. idem, 12. diversum, 13. unum,
14. Numerus, etc. etc. [There are twenty-seven numbered concepts in this class.]
"Classis II; 1. Quantitas est 14 T&V 9 (15). [Numbers enclosed in parentheses have
their usual arithmetical significance, except that (15) signifies 'an indefinite number'.]
2. Indudens est 6.10. III. 1. Intervallum est 2.3.10. 2. Aequale A rijs ll.£. 3. Continuum
est A ad B, si TOV A j 9 est 4 et 7 TU> B.; etc. etc."
The Development of Symbolic Logic 1 1
In point of fact, the analysis (apart from any merely geometrical defects)
falls far short of being complete. Leibniz uses not only the inflected Greek
article to indicate various relations of concepts but also modal inflections
indicated by et, si, quod, quam faciunt, etc.
In later years Leibniz never mentions this work without apologizing for
it, yet he always insists that its main intention is sound. This method
of assuming primitive ideas which are arbitrarily symbolized, of introducing
other concepts by definition in terms of these primitives and, at the same
time, substituting a single symbol for the complex of defining symbols —
this is, in fact, the method of logistic in general. Modern logistic differs
from this attempt of Leibniz most notably in two respects: (1) modern
logistic would insist that the relations whereby two or more concepts are
united in a definition should be analyzed precisely as the substantives are
analyzed; (2) while Leibniz regards his set of primitive concepts as the
necessary result of any proper analysis, modern logistic would look upon
them as arbitrarily chosen. Leibniz's later work looks toward the elimina
tion of this first difference, but the second represents a conviction from
which he never departed.
At a much later date come various studies (not in Gerhardt), wrhich
attempt a more systematic use of number and of mathematical operations
in logic.25 Simple and primitive concepts, Leibniz now proposes, should be
symbolized by prime numbers, and the combination of two concepts (the
qualification of one term by another) is to be represented by their product.
Thus if 3 represent "rational" and 7 "animal", "man" will be 21. No
prime number will enter more than once into a given combination — a
rational rational animal, or a rational animal animal, is simply a rational
animal. Thus logical synthesis is represented by arithmetical multipli
cation: logical analysis by resolution into prime factors. The analysis of
"man", 21, would be accomplished by finding its prime factors, "rational",
3, and "animal", 7. In accordance with Leibniz's conviction that all
knowledge is analytic and all valid predicates are contained in the subject,
the proposition "All S is P" will be true if the number which represents
the concept S is divisible by that which represents P. Accordingly the
25 Dated April, 1679. Couturat (op. cit., p. 326, footnote) gives the titles of these
as follows: "Elementa Characteristicae Universalis (Collected manuscripts of Leibniz in
the Hanover Library, PHIL., v, 8 b); Calculi universalis Elementa (PHIL., v, 8 c); Calculi
universalis investigations (PHIL., v, 8 d); Modus examinandi consequentias per numeros
(PHIL., v, 8 e); Regulae ex quibus de bonitate consequentiarum formisque et modis syllogis-
morum categoricum judicari potest per numeros (PHIL., v, 8f)." These fragments, with
many others, are contained in Couturat's Opuscules et fragments inedits de Leibniz.
12 A Survey of Symbolic Logic
universal affirmative proposition may be symbolized by S/P = y or S = Py
(where y is a whole number) . By the plan of this notation, Py will represent
some species whose "difference", within the genus P, is y. Similarly Sx
will represent a species of S. Hence the particular affirmative, "Some
S is P," may be symbolized by Sx = Py, or S/P = y'x. Thus the uni
versal is a special case of the particular, and the particular will always be
true when the universal is true.
There are several objections to this scheme. In the first place, it
presumes that any part of a class is a species within the class as genus.
This is far-fetched, but perhaps theoretically defensible on the ground
that any part which can be specified by the use of language may be treated
as a logical species. A worse defect lies in the fact that Sx = Py will
always be true. For a given S and P, we can always find x and y which
will satisfy the equation Sx = Py. If no other choice avails, let x = P,
or some multiple of P, and y = S, or some multiple of S. "Angel-man"
= "man-angel" although no men are angels. "Spineless man" = "ra
tional invertebrate", but it is false that some men are invertebrates. A
third difficulty arises because of the existential import of the particular—
a difficulty which later drew Leibniz's attention. If the particular affirma
tive is true, then for some x and y, Sx = Py. The universal negative should,
then, be Sx 4= Py. And since the universal affirmative is S = Py, the
particular negative should be S 4= Py> But this symbolism would be
practically unworkable because the inequations would have to be verified
for all values of x and y. Also, as we have noted, the equality Sx = Py
will always hold and Sx 4= Py, where x and y are arbitrary, will never be
true.
Such difficulties led Leibniz to complicate his symbolism still further,
introducing negative numbers and finally using a pair of numbers, one
positive and one negative, for each concept. But this scheme also breaks
down, and the attempt to represent concepts by numbers is thereafter
abandoned.
Of more importance to symbolic logic are the later fragments included
in the plans for an encyclopedia which should collect and arrange all known
science as the proper foundation for future work.26 Leibniz cherished the
26 G. Phil., vii, xvi-xx. Of these, xvi, without title, states rules for inference in
terms of inclusion and exclusion; Difficultates quaedam logicae treats of subalternation
and conversion and of the symbolic expression for various types of propositions; xvm,
Specimen Calculi universalis with its addenda and marginal notes, gives the general prin
ciples of procedure for the universal calculus; xix, with the title Non inelegans specimen
The Development of Symbolic Logic 13
notion that this should be developed in terms of the universal characteristic.
In these fragments, the relations of equivalence, inclusion, and qualification
of one concept by another, or combination, are defined and used. These
relations are always considered in intension when it is a question of apply
ing the calculus to formal logic. "Equivalence" is the equivalence of
concepts, not simply of two classes which have the same members; "for A
to include B or B to be included in A is to affirm the predicate B universally
of the subject A".27 However, Leibniz evidently considers the calculus
to have many applications, and he thinks out the relations and illustrates
them frequently in terms of extensional diagrams, in which A, B, etc., are
represented by segments of a right line. Although he preferred to treat
logical relations in intension, he frequently states that relations of intension
are easily transformed into relations of extension. If A is included in B
in intension, B is included in A in extension; and a calculus may be inter
preted indifferently as representing relations of concepts in intension or
relations of individuals and classes in extension. Also, the inclusion rela
tion may be interpreted as the relation of an antecedent proposition to a
consequent proposition. The hypothesis A includes its consequence B,
just as the subject A includes the predicate B.~B This accords with his
frequently expressed conviction that all demonstration is analysis. Thus
these studies are by no means to be confined to the logic of intension. As
one title suggests, they are studies demonstrandi in abstracts.
demonsirandi in abstractis struck out, and xx, without title, are deductive developments
of theorems of symbolic logic, entirely comparable with later treatises.
The place of symbolic logic in Leibniz's plans for the Encyclopedia is sufficiently
indicated by the various outlines which he has left. In one of these (G. Phil., vu, 49),
divisions 1-6 are of an introductory nature, after which come :
"7. De scientiarum instauratione, ubi de Systematibus et Repertoriis, et de Encyclo
paedia demonstrativa codenda.
"8. Elementa veritatis aeternae, et de arte demonstrandi in omnibus disciplinis ut in
Mathesi.
"9. De novo quodam Calculo generali, cujus ope tollantur omnes disputationes inter
eos qui in ipsum consenserit; est Cabala sapientum.
"10. De Arte Inveniendi.
"11. De Synthesi seu Arte combinatoria.
"12. De Analysi.
"13. De Combinatoria speciali, seu scientia formarum, sive qualitatum in genere (de
Characterismis) sive de simili et dissimili.
"14. De Analysi speciali seu scientia quantitatum in genere seu de magno et parvo.
"15. De Mathesi generali ex duabus praecedentibus composita."
Then various branches of mathematics, astronomy, physics, biological science, medi
cine, psychology, political science, economics, military science, jurisprudence, and natural
theology, in the order named.
27 G. Phil, vii, 208.
28 "Generates Inquisitiones " (1686): see Couturat, Opuscules etc., pp. 356-99.
14 A Survey of Symbolic Logic
It is a frequent remark upon Leibniz's contributions to logic that he
failed to accomplish this or that, or erred in some respect, because he
chose the point of view of intension instead of that of extension. The
facts are these: Leibniz too hastily presumed a complete, or very close,
analogy between the various logical relations. It is a part of his sig
nificance for us that he sought such high generalizations and believed in
their validity. He preferred the point of view of intension, or connotation,
partly from habit and partly from rationalistic inclination. As a conse
quence, wherever there is a discrepancy between the intensional and ex-
tensional points of view, he is likely to overlook it, and to follow the former.
This led him into some difficulties which he might have avoided by an
opposite inclination and choice of example, but it also led him to make
some distinctions the importance of which has since been overlooked and
to avoid certain difficulties into which his commentators have fallen.29
In Difficultates quaedam logicae, Leibniz shows that at last he recognizes
the difficulty in connecting the universal and the corresponding particular.
He sees also that this difficulty is connected with the disparity between the
intensional point of view and the existential import of particular proposi
tions. In the course of this essay he formulates the symbolism for the four
propositions in two different ways. The first formulation is: 30
Univ. aff.; All A is B: AB = A, or A non-5 does not exist.
Part, neg.; Some A is not B; AB 4= A, or A non-5 exists.
Univ. neg.; No A is B; AB does not exist.
Part, aff.; Some A is 5; AB exists.
AB = A and AB =}= A may be interpreted as relations of intension or of
extension indifferently. If all men are mortal, the intension of "mortal
man" is the same as the intension of "man", and likewise the class of
mortal men is identical in extent with the class of men. The statements
concerning existence are obviously to be understood in extension only.
The interpretation here put upon the propositions is identically that of
contemporary symbolic logic. With these expressions, Leibniz infers the
subaltern and the converse of the subaltern, from a given universal, by
29 For example, it led him to distinguish the merely non-existent from the absurd, or
impossible, and the necessarily true from the contingent. See G. Phil, vn, 231, foot- ,C
n6te; and "Specimen certitudinis seu de conditionibus," Dutens, Leibnitii Opera, iv,
Part in, pp. 92 ff., also Couturat, La Logique de Leibniz, p. 348, footnote, and p. 353,
footnote.
. 30 G. Phil, vii, 212.
The Development of Symbolic Logic 15
means of the hypothesis that the subject, A, exists. Later in the essay, he
gives another set of expressions for the four propositions: 31
All A is B: AB = A.
Some A is not B: AB 4= A.
No A is B: AB does not exist, or AB 4= AB Ens.
Some A is B: AB exists, or AB = AB Ens.
In the last two of these, AB before the sign of equality represents the
possible AB's or the AB "in the region of ideas"; "AB Ens" represents
existing AB's, or actual members of the class AB. (Read AB Ens, "AB
which exists".) AB = AB Ens thus represents the fact that the class AB
has members; AB =f= AB Ens, that the class AB has no members. A
logical species of the genus A, "some A", may be represented by YA;
YA Ens will represent existing members of that species, or "some exist
ing A". Leibniz correctly reasons that if AB = A (All A is B), YAB
-- YA (Some A is B); but if AB 4= A, it does not follow that YAB 4= YA,
for if Y ••= B, YAB =-- YA. Again, if AB 4= AB Ens (No A is B), YAB
4= YAB Ens (It is false that some A is B); but if AB = AB Ens (Some
A is B), YAB = YAB Ens does not follow, because Y could assume values
incompatible with A and B. For example, some men are wise, but it does
not follow that foolish men are foolish wise persons, because "foolish" is
incompatible with "wise".32 The distinction here between AB, a logical
division of A or of B, and AB Ens, existing AB's, is ingenious. This is
our author's most successful treatment of the relations of extension and
intension, and of the particular to the universal.
In Specimen calculi universal™, the "principles of the calculus" are
announced as follows: 33
1) "Whatever is concluded in terms of certain variable letters may be
concluded in terms of any other letters which satisfy the same conditions;
for example, since it is true that [all] ab is a, it will also be true that [all]
be is b and that [all] bed is be. . . .
2) "Transposing letters in terms changes nothing; for example ab
coincides with ba, 'animal rational' with 'rational animal'.
3) "Repetition of a letter in the same term is useless. . . .
4) "One proposition can be made from any number by joining all the
subjects in one subject and all the predicates in one predicate: Thus, a is b
and c is d and e is /, become ace is bdf. . . .
31 G. Phil,, vii, 213-14.
32 G. Phil., vii, 215: the illustration is mine.
33 G. Phil, vii, 224-25.
16 A Survey of Symbolic Logic
5) "From any proposition whose predicate is composed of more than
one term, more than one proposition can be made; each derived proposition
having the subject the same as the given proposition but in place of the
given predicate some part of the given predicate. If [all] a is bed, then [all]
a is b and [all] a is c and [all] a is d." u
If we add to the number of these, two principles which are announced
under the head of "self-evident propositions" — (1) a is included in a;
and (2) ab is included in a — we have here the most important of the funda
mental principles of symbolic logic. Principle 1 is usually qualified by
some doctrine of the "universe of discourse" or of "range of significance",
but some form of it is indispensable to algorithms in general. The law
numbered 2 above is what we now call the "principle of permutation";
3, the "principle of tautology"; 4, the "principle of composition"; 5, the
"principle of division". And the two "self-evident propositions" are often
included in sets of postulates for the algebra of logic.
There remain for consideration the two fragments which are given in
translation in our Appendix, XIX and XX of Scientia Generalis: Char-
acteristica. The first of these, with the title Non inelegans specimen demon-
sir andi in abstractis, stricken out in the manuscript, is rather the more inter
esting. Here the relation previously symbolized by AB or ab is represented
by A+B. And A+B = L signifies that A is contained or included in
(est in} B. A scholium attached to the definition of this inclusion relation
distinguishes it from the part-whole relation. Comparison of this and
other passages shows that Leibniz uses the inclusion relation to cover
(1) the relation of a member of the class to the class itself; (2) the relation
of a species, or subclass, to its genus — a relation in extension; (3) the rela
tion of a genus to one of its species — a relation of intension. The first of
these is our e-relation; (2) is the inclusion relation of the algebra of logic;
and (3) is the analogous relation of intension. Throughout both these
fragments, it is clear that Leibniz thinks out his theorems in terms of
extensional diagrams, in which classes or concepts are represented by
segments of a line, and only incidently in terms of the intension of concepts.
The different interpretations of the symbols must be carefully dis
tinguished. If A is "rational" and B is "animal", and A and B are taken
m intension, then A + B will represent "rational animal". But if A and B
are classes taken in extension^ then A + B is the class made up of those
things which are either A or B (or both). Thus the inclusion relation,
34 4. and 5. are stated without qualification because this study is confined to the proper
ties of universal affirmative propositions. 4. is true also for universal negatives.
The Development of Symbolic Logic 17
A +B = L, may be interpreted either in intension or in extension as "A is
in L ". This is a little confusing to us, because we should nowadays invert
the inclusion relation when we pass from intension to extension; instead
of this, Leibniz changes the meaning of A +B from "both A and B" (in
intension) to "either A or B" (in extension). If A is "rational", B "ani
mal", and L "man", then A + B = L is true in intension, "rational animal"
= "man" or "rational" is contained in "man". If A, B, and L are classes
of points, or segments of a line, then A + B = L will mean that L is the
class of points comprising the points in A and the points in B (any points
common to A and B counted only once), or the segment made up of
segments A and B.
The relation A + B does not require that A and B should be mutually
exclusive. If L is a line, A and B may be overlapping segments; and, in
intension, A and B may be overlapping concepts, such as "triangle" and
"equilateral", each of which contains the component "figure".
Leibniz also introduces the relation L — A, which he calls detractio.
L — A = N signifies that L contains A and that if A be taken from L the
remainder is N. The relations [+] and [ — ] are not true inverses: if
A + B = L, it does not follow that L — A == B, because A and B may be
overlapping (in Leibniz's terms, communicantia) . If L — A = N, A and N
must be mutually exclusive (incommunicantia} . Hence if A+B = L and
A and B have a common part, M, L — A = B — M. (If the reader will
take a line, L, in which A and B are overlapping segments, this will be
clear.) This makes the relation of detractio somewhat confusing. In
extension, L — A may be interpreted "L which is not A ". In intension,
it is more difficult. Leibniz offers the example: "man" "rational"
= "brute", and calls our attention to the fact that "man" "rational"
is not "non-rational man" or "man" + "non-rational".35 In intension, the
relation seems to indicate an abstraction; not a negative qualification.
But there are difficulties, due to the overlapping of concepts. Say that 7
r * i w^~
" man" + "woodworking" = "carpenter" and "man" + "white-skinned" -^^
35 G. Phil., vn, 231, footnote. Couturat in commenting on this (op. cit., pp. 377-78) &£
says:
"Ailleurs Leibniz essaie de preciser cette opposition en disant:
'A — A est Nihilum. Sed A non-A est Absurdwn.' "Mais il oublie que le ne"ant
(non-Ens) n'est pas autre chose que ce qu'il appelle 1'absurde ou 1'impossible, c'est-a-dire
le contradictoire."
It may be that Couturat, not Leibniz, is confused on this point. Non-existence may
be contingent, as opposed to the necessary non-existence of the absurd. And the result of
abstracting A from the concept A seems to leave merely non-Ens, not absurdity.
3 , - - - ^
18 A Survey of Symbolic Logic
= "Caucasian". Then " Caucasian " + " carpenter " = "man" + "white-
skinned" + "woodworking". Hence ("Caucasian" + "carpenter") - "car
penter" = "white-skinned", because the common constituent "man" has
been abstracted in abstracting "carpenter". That is, the abstraction of
"carpenter" from "Caucasian carpenter" leaves, not "Caucasian" but
only that part of the concept "Caucasian" which is wholly absent in
"carpenter". We cannot here say "white-skinned man" because "man"
is abstracted, nor "white-skinned animal" because "animal" is contained
in "man": we can only say "white-skinned" as a pure abstraction. Such
abstraction is difficult to carry out and of very little use as an instrument
of logical analysis. Leibniz's illustration is scribbled in the margin of the
manuscript, and it seems clear that at this point he was not thinking out
his theorems in terms of intensions.
Fragment XX differs from XIX in that it lacks the relation symbolized
by [ — ]. This is a gain rather than a loss, both because of the difficulty of
interpretation and because [+ ] and [ — ] are not true inverses. Also XX
is more carefully developed : more of the simple theorems are proved, and
more illustrations are given. Otherwise the definitions, relations, and
methods of proof are the same. In both fragments the fundamental
operation by which theorems are proved is the substitution of equivalent
expressions.
If the successors of Leibniz had retained the breadth of view7 which
characterizes his studies and aimed to symbolize relations of a like generality,
these fragments might well have proved sufficient foundation for a satis
factory calculus of logic.
III. FROM LEIBNIZ TO DE MORGAN AND BOOLE
After Leibniz, various attempts wrere made to develop a calculus of
logic. Segner, Jacques Bernoulli, Ploucquet, Tonnies, Lambert, Holland,
Castillon, and others, all made studies toward this end. Of these, the
most important are those of Ploucquet, Lambert and Castillon, while one
of Holland's is of particular interest because it intends to be a calculus
in extension. But this attempt was not quite a success, and the net result
of the others is to illustrate the fact that a consistent calculus of logical
relations in intension is either most difficult or quite impossible.
Of Segner 's wrork and Ploucquet 's we can give no account, since no
copies of these writings are available.36 Venn makes it clear that Plouc-
36 There seem to be no copies of Ploucquet's books in this country, and attempts to
secure them from the continent have so far failed.
The Development of Symbolic Logic 19
quet's calculus was a calculus of intension and that it involved the quanti
fication of the predicate.
Lambert37 wrote voluminously on the subject of logic, but his most
important contribution to symbolic procedure is contained in the Seeks
Versuche einer Zeichenkunst in der' Vernunftlehre.*8 These essays are not
separate studies, made from different beginnings; later essays presuppose
those which precede and refer to their theorems; and yet the development
is not entirely continuous. Material given briefly in one will be found
set forth more at length in another. And discussion of more general prob
lems of the theory of knowledge and of scientific method are sometimes
introduced. But the important results can be presented as a continuous
development which follows in general the order of the essays.
Lambert gives the following list of his symbols:
The symbol of equality (Gleichgultigkeit) =
addition (Zmeizung) +
abstraction (Absonderung)
opposition (des Gegentheils) X
universality
particularity
copula
given concepts (Begriffe) a, b, c, d, etc.
undetermined concepts n, m, I, etc.
unknowns x, y, z.
the genus 7
the difference 5
The calculus is developed entirely from the point of view of intension:
the letters represent concepts, not classes, [ + ] indicates the union of two
concepts to form a third, [— ] represents the withdrawal or abstraction of
some part of the connotation of a concept, while the product of a and b
represents the common part of the two concepts. 7 and 8 qualify any
term "multiplied" into them. Thus ay represents the genus of a, ad the
difference of a. Much use is made of the well-known law of formal logic
that the concept (of a given species) equals the genus plus the difference.
(1) ay + ad = a(y + 6) = a
37 Johann Heinrich Lambert (1728-77), German physicist, mathematician, and astrono
mer. He is remembered chiefly for his development of the equation xn+px = q in an
infinite series, and his proof, in 1761, of the irrationality of TT.
38 In Logische und philosophische Abhandlungen; ed. Joh. Bernoulli (Berlin, 1782),
vol. i.
20 A Survey of Symbolic Logic
ay + a8 is the definition or explanation (Erkldrung) of a. As immediate
consequences of (1), we have also
(2) ay = a — ad (3) ad = a — ay
Lambert takes it for granted that [+ ] and [ — ] are strictly inverse opera
tions. We have already noted the difficulties of Leibniz on this point.
If two concepts, a and b, have any part of their connotation in common,
then (a + b) — b will not be a but only that part of a which does not belong
also to b. If "European" and "carpenter" have the common part "man",
then (" European "+ "carpenter") minus "carpenter" is not "European"
but "European" minus "man". And [+ ] and [— ] will not here be true
inverses. But this difficulty may be supposed to disappear where the
terms of the sum are the genus and difference of some concept, since genus
anddifference may be supposed to be mutually exclusive. We shall return
to this topic later.
More complex laws of genus and difference may be elicited from the
fact that the genus of any given a is also a concept and can be "explained,"
as can also the difference of a.
(4) a = a(y+ <5)2 = ay2 + ayd + ady + ad2
Proof: ay = ayy + ayd and ad = ady + add
But a = ay + ad. Hence Q.E.D.
That is to say: if one wish to define or explain a, one need not stop at
giving its genus and difference, but may define the genus in terms of its
genus and difference, and define the difference similarly. Thus a is equiva
lent to the genus of the genus of a plus the difference of the genus of a plus
the genus of the difference of a plus the difference of the difference of a.
This may be called a "higher" definition or "explanation" of a.
Obviously, this process of higher and higher "explanation" may be
carried to any length; the result is what Lambert calls his "Newtonian
formula". We shall best understand this if we take one more preliminary
step. Suppose the explanation carried one degree further and the resulting
terms arranged as follows:
a = a(73 + 776 + ydd + 63)
+ 767 + dyd
+ dyy + ddy
The three possible arrangements of two y's, and one d might be summarized
The Development of Symbolic Logic 21
by 3y2d; the three arrangements of two 6's and one 7 by 8752. With this
convention, the formula for an explanation carried to any degree, n, is:
This "Newtonian formula" is a rather pleasant mathematical conceit.
Two further interesting laws are given :
(6) a = ad + ayd + ay2d + ay*d + . . . etc.
Proof: a = ay + ad
But ay = ay2 + ayd
and ay2 = ay* + ay2d
ay* = ay4 + ay*d, etc. etc.
(7) a = ay + ady + ad2y + ad*y + . . . etc.
Proof: a = ay + ad
But ad = ady + ad2
and ad2 = ad2y + ad*
ad* = ad*y + ad*, etc. etc.
Just as the genus of a is represented by ay, the genus of the genus of
a by ay2, etc., so a species of which a is genus may be represented by ay~l,
and a species of which a is genus of the genus by ay~2, etc. In general, as
ayn represents a. genus above a, so a species below a may be represented by
a
ay~n or
yn
Similarly afny concept of which a is difference of the difference of the differ
ence . . . etc., may be represented by
ad~n or --
Also, just as a = a(y + d)n, where a is a concept and a(y + d)n its "explana
tion", so- — = a, where ^n'ls ^ne concept and a the "explanation"
of it.
Certain cautions in the transformation of expressions, both with respect
to "multiplication" and with respect to "division," need to be observed.40
39 Seeks Versuche, p. 5.
40 Ibid.j pp. 9-10.
22 A Survey of Symbolic Logic
The concept ay2 + ady is very different from the concept (ay + ad)y, because
(8) (ay + ad)y = a(y+d)y = ay(y+d) = ay
while «72 + ady is the genus of the genus of a plus the genus of the difference
of a. Also - y must be distinguished from — . - y is the genus of any
7 77
species x of which a is the genus, i. e.,
(9) - 7 = a
But ay/y is any species of which the genus of a is the genus, i. e., any
species x such that a and x belong to the same genus.
We turn now to consideration of the relation of concepts which have a
common part.
Similarity is identity of properties. Two concepts are similar if, and
in so far as, they comprehend identical properties. In respect to the
remaining properties, they are different.41
ab represents the common properties of a and b.
a — ab represents the peculiar properties of a.
a + b — ab — ab represents the peculiar properties of a together with
the peculiar properties of b.
It is evident from this last that Lambert does not wish to recognize in
his system the law a + a = a; else he need only have written a + b — ab.
If x and a are of the same genus, then
xy = ay and ax = ay = xy
If now we symbolize by a \ b that part of a which is different from 6,42 then
(10) a\b + b\a + ab + ab = a + b
Also
x — x
a = ay, or x = ay + x\a
ax = ad
a — ax = ad
a = ax + ad
ax = a — ad = ay = xy
41 Ibid., p. 10.
42 Lambert sometimes uses a \ b for this, sometimes a : b.
The Development of Symbolic Logic 23
And since
ay
x = -
y
ax + a
x = a ax + x\a = x
ax = a — a\x = x — x\a
a\x = a — ax x\a = x — ax
The fact that y is a property comprehended in x may be expressed by
y = xy or by y + x y = x. The manner in which Lambert deduces the
second of these expressions from the first is interesting.43 If y is a property
of x, then y x is null. But by (10),
2xy + x\y + y\x = x + y
Hence in this case, 2xy + x \ y = x + y
And since y = xy, 2y + x\y = x + y
Hence y+x\y = x
He has subtracted y from both sides, in the last step, and we observe that
2y — y = y> This is rather characteristic of his procedure; it follows,
throughout, arithmetical analogies which are quite invalid for logic.
With the complications of this calculus, the reader will probably be
little concerned. There is no general type of procedure for elimination or
solution. Formulae of solution for different types of equation are given.
They are highly ingenious, often complicated, and of dubious application.
It is difficult to judge of possible applications because in the whole course
of the development, so far as outlined, there is not a single illustration of a
solution which represents logical reasoning, and very few illustrations of
any kind.
The shortcomings of this calculus are fairly obvious. There is too
much reliance upon the analogy between the logical relations symbolized
and their arithmetical analogues. Some of the operations are logically
uninterpretable, as for example the use of numerical coefficients other than
0 and 1. These have a meaning in the "Newtonian formula", but 2y either
has no meaning or requires a conventional treatment which is not given.
And in any case, to subtract y from both sides of 2y = x + y and get y = x
represents no valid logical operation. Any adequate study of the properties
of the relations employed is lacking, x = a + b is transformed into a = x
— b, regardless of the fact that a and b may have a common part and that
43 Seeks Versuche, p. 12.
24 A Survey of Symbolic Logic
x — b represents the abstraction of the whole of b from x. Suppose, for
example, man = rational + animal. Then, by Lambert's procedure, we
should have also rational = man — animal. Since Leibniz had pointed
out this difficulty, — that addition and subtraction (with exactly these
meanings) are not true inverses, it is the more inexcusable that Lambert
should err in this.
There is a still deeper difficulty here. As Lambert himself remarks,44
no two concepts are so completely dissimilar that they do not have a common
part. One might say that the concept " thing" (Lambert's word) or " be
ing " is common to every pair of concepts. This being the case, [ + ] and [ — ]
are never really inverse operations. Hence the difficulty will not really
disappear even in the case of ay and a<5; and a — ay = ad, a — a8 = ay
will not be strictly valid. In fact this consideration vitiates altogether the
use of "subtraction" in a calculus based on intension. For the meaning
of a — b becomes wholly doubtful unless [ — ] be treated as a wholly con
ventional inverse of [ + ], and in that case it becomes wholly useless.
The method by which Lambert treats the traditional syllogism is only
remotely connected with what precedes, and its value does not entirely
depend upon the general validity of his calculus. He reconstructs the
whole of Aristotelian logic by the quantification of the predicate.45
The proposition "All A is B" has two cases:
(1) A = B, the case in which it has a universal converse, the concept
A is identical with the concept B.
(2) A > B, the case in which the converse is particular, the concept B
comprehended in the concept A.
The particular affirmative similarly has two cases :
(1) A < B, the case in which the converse is a universal, the subject A
comprehended within the predicate B.
(2) The case in which the converse is particular. In this case the
subject A is comprehended within a subsumed species of the predicate and
the predicate within a subsumed species of the subject. Lambert says
this may be expressed by the pair:
mA > B and A < nB
Those who are more accustomed to logical relations in extension must
not make the mistake here of supposing that A > mA, and mA < A.
mA is a species of A, and in intension the genus is contained in the species,
44 Ibid., p. 12.
45 7m, pp. 93 jf.
The Development of Symbolic Logic
25
not vice versa. Hence mA > B does not give A > B, as one might expect
at first glance. We see that Lambert here translates "Some A " by mA, a
species comprehended in A, making the same assumption which occurs in
Leibniz, that any subdivision or portion of a class is capable of being treated
as some species comprehended under that class as its genus.
In a universal negative proposition— Lambert says— the subject and
predicate each have peculiar properties by virtue of whose comprehension
neither is contained in the other. But if the peculiar properties of the
subject be taken away, then what remains is contained in the predicate;
and if the peculiar properties of the predicate be taken away, then what
remains is contained in the subject. Thus the universal negative is repre
sented by the pair
m
and
A>B-
n
The particular negative has two cases:
(1) When it has a universal affirmative converse, i. e., when some A
is not B but all B is A. This is expressed by
.4 <B
(2) When it has not a universal affirmative converse. In this case a
subsumed species of the subject is contained in the predicate, and a sub
sumed species of the predicate in the subject.
mA > B and A < nB
Either of the signs, < and >, may be reversed by transposing the
terms. And if P < Q, Q > P, then for some /, P = IQ. Also, " multi
plication" and "division" are strict inverses. Hence we can transform
these expressions as follows:
A > B is equivalent to A = mB
nA = B
mA = kB]
IA = nB ]
' A_ _B
m k
A = B_
I '" n J
It is evident from these transformations and from the prepositional equiva-
or pA = qB
or — = -
26 A Survey of Symbolic Logic
lents of the " inequalities" that the following is the full expression of these
equations:
(1) A = mB: All A is B and some B is not A.
(2) nA = B: Some A is not B and all B is A.
(3) mA = nB: Some, but not all, A is B, and some, but not all, B is A.
(4) - = - : No A is B.
m n
The first noticeable defect here is that A/m = B/n is transformable into
nA = mB and (4) can mean nothing different from (3). Lambert has, in
fact, only four different propositions, if he sticks to the laws of his calculus:
(1) A = B: All A is all B.
(2) A = mB: All A is some B.
(3) nA == B: Some A is all B.
(4) mA = nB: Some A is some B.
These are the four forms which become, in Hamilton's and De Morgan's
treatises, the four forms of the affirmative. A little scrutiny will show that
Lambert's treatment of negatives is a failure. For it to be consistent at
all, it is necessary that " fractions" should not be transformed. But
Lambert constantly makes such transformations, though he carefully re
frains from doing so in the case of expressions like A/m = B/n which are
supposed to represent universal negatives. His method further requires
that m and n should behave like positive coefficients which are always
greater than 0 and such that m 4= n. This is unfortunate. It makes it
impossible to represent a simple proposition without "entangling alliances".
If he had taken a leaf from Leibniz's book and treated negative propositions
as affirmatives with negative predicates, he might have anticipated the
calculus of De Morgan.
In symbolizing syllogisms, Lambert always uses A for the major term,
B for the middle term, and C for the minor. The perfectly general form of
proposition is:
mA nB
P (1
Hence the perfectly general syllogism will be : 46
AT . mA nB
Major
p q
46 Ibid., pp. 102-103.
47 Ibid., p. 107.
The Development of Symbolic Logic
nC vB
Minor
27
~ , fj.n mv
Conclusion - C = — A
irq pp
The indeterminates in the minor are always represented thus by Greek
letters.
The conclusion is delved from the premises as follows:
The major premise gives B = — A.
np
The minor gives B = — C.
irv
Hence A-C.
np irv
mv
and therefore -- C = -— A.
irq pp
The above being the general form of the syllogism, Lambert's scheme of
moods in the first figure is the following: it coincides with the traditional
classification only so far as indicated by the use of the traditional names:
B
= mA
nB = mA
I.
VII.
C
— yjj
nC = B
Barbara
Lilii
C
= mvB
nvC = mA
B
= A
II.
C
B
VIII.
B = A
Canerent
7T
C
P
_A
Magogos
MC = nB
uC = nA
7T
p
III.
B
_A
IX.
B_A
Decane
Q
C
P
Negligo
q p
C = vB
sive
sive
Celarent
C
vA
Ferio
i*C = vA
9
P
q p
28
A Survey of Symbolic Logic
IV.
Fideleo
nB = A
C B
7T
nC
X.
Pilosos
nB = A
V.C = B
= A
V.
B =
mA
Gabini
nC =
vB
sive
LiC =
mvA
Darii
nB =
mA
VI.
C =
B
Hilario
nC =
mA
XL
Romano
XII.
Somnio
nB = mA
C = B
nC = mA
nB = mA
»C = B
7ijjiC = mA
The difficulty about "division" does not particularly affect this scheme,
since it is only required that if one of the premises involve " fractions",
the conclusion must also. It will be noted that the mood Hilario is identi
cal in form with Romano, and Lilii with Somnio. The reason for this
lies in the fact that nB = mA has two partial meanings, one affirmative
and one negative (see above). Hilario and Lilii take the affirmative
interpretation, as their names indicate; Romano and Somnio, the negative.
Into the discussion of the other three figures, the reader will probably
not care to go, since the manner of treatment is substantially the same as
in the above.
There are various other attempts to devise a convenient symbolism and
method for formal logic;48 but these are of the same general type, and
they meet with about the same degree and kind of success.
Two brief passages in which there is an anticipation of the logic of
relatives possess some interest.49 Relations, Lambert says, are " external
attributes", by which he means that they do not belong to the object
an sich. " Metaphysical" (i. e., non-logical) relations are represented by
Greek letters. For example if / = fire, h = heat, and a = cause,
/ = a : : h
The symbol : : represents a relation which behaves like multiplication:
48 See in Seeks Versuche, v and vi. Also fragments "Uber die Vernunftlehre", in
Logische und Philosophische Abhandlungen, i, xix and xx; and Anlage zur Architektonik,
p. 190 jf.
49£ec/is Versuche, pp. 19, 27 ff.
The Development of Symbolic Logic 29
a : : h is in fact what Peirce and Schroder later called a "relative product".
Lambert transforms the above equation into:
Fire is to heat as cause to effect.
h
Fire is to cause as heat to effect.
a
-* — - Heat is to fire as effect to cause.
i <*
The dot here represents Wirkuny (it might be, Wirklichkeit, in consonance
with the metaphysical interpretation, suggestive of Aristotle, which he
gives to Ursache). It has the properties of 1, as is illustrated elsewhere50
by the fact that 7° may be replaced by this symbol.
Lambert also uses powers of a relation.
If a = <p : : b, and b = <p : : c,
a = <p : : (p : : c = (p- : : c
And if a = <p~ : : c,
o a j \a
<p- = - and V7 = \|-
c V
And more to the same effect.
No use is made of this symbolism; indeed it is difficult to see how
Lambert could have used it. Yet it is interesting that he should have felt
that the powers of a relation ought to be logically important, and that he
here hit upon exactly the concept by which the riddles of "mathematical
induction" were later to be solved.
Holland's attempt at a logical calculus is contained in a letter to Lam
bert.51 He himself calls it an "unripe thought", and in a letter some three
years later52 he expresses a doubt if logic is really a purely formal discipline
capable of mathematical treatment. But this study is of particular interest
because it treats the logical classes in extension — the only attempt at a
symbolic logic from the point of view of extension from the time of Leibniz
to the treatise of Solly in 1839.
Holland objects to Lambert's method of representing the relation of
concepts by the relation of lines, one under the other, and argues that the
50 Ibid., p. 21.
61 Joha'n. Lamberts deutscher Gelehrten Brief wechsel, Brief m, pp. 16 ff.
52 See Ibid., Brief xxvn, pp. 259 ff.
30 A Survey of Symbolic Logic
relation of "men" to " mortals" is not sub but inter. He is apparently not
aware that this means exchanging the point of view of intension for that
of extension, yet all his relations are consistently represented in extension,
as we shall see.
(1) If S represent the subject, P the predicate; and p, TT signify unde
termined variable numbers, S/p = P/TT will come to: A part of S is a part
of P, or certain of the S's are certain of the P's, or (at least) an S is a P.
This expression is the general formula of all possible judgments, as is
evident by the following:
(2) A member is either positive or negative, and in both cases, is either
finite or infinite. We shall see in what fashion p and IT can be understood.
(3) If p = 1 in S/p, then is S/p as many as all S, and in this way S/p
attains its logical maximum. Since, then, p cannot become less than 1,
it can still less disappear and consequently cannot become negative.
The same is true of TT.
(4) Therefore p and TT cannot but be positive and cannot be less than 1 .
If p or TT becomes infinite, the concept becomes negative.
(5) If /expresses a finite number > 1, then the possible forms of judg
ment are as follows:
(1) ^ --= ^ All S is all P.
(2) ^ --= j All S is some P.
Now 0 expresses negatively what l/oo expresses positively. To say that
an infinitely small part of a curved line is straight, means exactly: No part
of a curved line is straight.
(3) ~ = — All S is not P.
1 CO
(4) j- -. Some S is all P.
o p
(5) - = — Some S is some P.
S P
(6) - = - Some S is not- P.
'
(7) — = ^ All not-S is all P.
oo 1
53 See Ibid., Brief iv.
The Development of Symbolic Logic 31
S P
(8) — = j All not-S is some P.
S P
(9) — — All not-S is all not-P.
(1), (2), and (9) Holland says are universal affirmative propositions;
(3), (7), and (8), universal negatives; (4) and (5), particular affirmatives;
(6), a particular negative.
As Venn has said, this notation anticipates, in a way, the method of
Boole. If instead of the fraction we take the value of the numerator
indicated by it, the three values are
where 0 < v < 1, and £/co -.= Q-S. But the differences between this and
Boole's procedure are greater than the resemblances. The fractional form
is a little unfortunate in that it suggests that the equations may be cleared
of fractions, and this would give results which are logically uninterpretable.
But Holland's notation can be made the basis of a completely successful
calculus. That he did not make it such, is apparently due to the fact that
he did not give the matter sufficient attention to elaborate the extensional
point of view.
He gives the following examples:
Example 1. All men H are mortal M
All Europeans E are men H
P T
Ergo, E = [All Europeans are mortal]
Example 2. All plants are organisms P = -
P
All plants are no animals P — -
oo
0 A
Ergo, - = - [Some organisms are not animals]
32 A Survey of Symbolic Logic
Example 3. All men are rational // = -
P
T>
All plants are not rational P = -
CO
vH
Ergo, All plants are no men P = -
00
In this last example, Holland has evidently transformed // = R/p into
pH = R, which is not legitimate, as we have noted. pH = R would be
"Some men are all the rational beings". And the conclusion P = pH/ao
is also misinterpreted. It should be, "All plants are not some men". A
correct reading would have revealed the invalid operation.
Lambert replied vigorously to this letter, maintaining the superiority
of the intensional method, pointing out, correctly, that Holland's calculus
would not distinguish the merely non-existent from the impossible or
contradictory (no calculus in extension can), and objecting to the use of
°c in this connection. It is characteristic of their correspondence that each
pointed out the logical defects in the logical procedure of the other, and
neither profited by the criticism.
Castillon's essay toward a calculus of logic is contained in a paper
presented to the Berlin Academy in 1S03.54 The letters S, A, etc., represent
concepts taken in intension, M is an indeterminate, S + M represents the
'''synthesis" of S and M, S - M, the withdrawal or abstraction of M
from S. S — M thus represents a genus concept in which S is subsumed,
M being the logical "difference" of S in S - M. Consonantly S + M,
symbolizing the addition of some "further specification" to S, represents a
species concept which contains (in intension) the concept S.
The predicate of a universal affirmative proposition is contained in the
subject (in intension). Thus "All S is A" is represented by
S = A + M
The universal negative "No S is A" is symbolized by
S = - A + M = (- A) + M
The concept S is something, M, from which A is withdrawn— is no A.
Particular propositions are divided into two classes, "real" and "il
lusory". A real particular is the converse of a universal affirmative; the
"Memoire sur un nouvel algorithme logique", in Memoires de V Academic des Sciences
de Berlin, 1803, Classe de philosophie speculative, pp. 1-14. See also his paper, "Reflexions
sur la Logique", loc. cit., 1802.
The Development of Symbolic Logic 33
illusory particular, one whose converse also is particular. The real particu
lar affirmative is
A = S - M
since this is the converse of S = A + M. The illusory particular affirmative
is represented by
S = A=? M
Castillon's explanation of this is that the illusory particular judgment gives
us to understand that some S alone is A, or that S is got from A by ab
straction (S = A - M), when in reality it is A which is drawn from S by
abstraction (S == M + A). Thus this judgment puts ~ M where it should
put + M] one can, then, indicate it by S = A ^ M.
The fact is, of course, that "Some S is A " indicates nothing about
the relations of the concepts S and A except that they are not incompatible.
This means, in intension, that if one or both be further specified in proper
fashion, the results will coincide. It might well be symbolized by S + N
= A + M. We suspect that Castillon's choice of S = A =F M is really
governed by the consideration that S = A + M may be supposed to give
S = A ^F M, the universal to give its subaltern, and that A = S - M
will also give S = A =F M, that is to say, the real particular— which is
"All A is S"— will also give S = A =F M. Thus "Some S is A" may be
derived both from "All S is A" and from "All A is S", which is a de
sideratum.
The illusory negative particular is, correspondingly,
S = - A =F M
Immediate inference works out fairly well in this symbolism.
The universal affirmative and the real particular are converses.
S = A + M gives A = S — M, and vice versa. The universal negative
is directly convertible.
S = -- A + M gives A = — S + J/, and vice versa. The illusory par
ticular is also convertible.
S = A T J/ gives - A = - S =F M. Hence A = S =F M, which
comes back to S = A ^ M.
A universal gives its subaltern
S = A + M gives S = A =F J/, and
S = - A + M gives S = - A =T= M.
And a real particular gives also the converse illusory particular, for
A = S - M gives S = A + M,
4
34 A Survey of Symbolic Logic
which gives its subaltern, S = A ^ M,
which gives A = S ^ M.
All the traditional moods and figures of the syllogism may be symbolized
in this calculus, those which involve particular propositions being valid
both for the real particular and for the illusory particular. For example:
All Mis A M = A + N
All SisM S = M + P
All Sis ,4 .'. S = A + (N + P)
No M is A M = - A + N
All S is M S = M + P
No Sis A :. S = - A + (N + P)
All M is A M = A + N
Some S is M S = M =F P or S = M - P
.'. Some S is A .'. S = (A + N) =F P or S = (A + N) - P
This is the most successful attempt at a calculus of logic in intension.
The difficulty about "subtraction" in the XIX Fragment of Leibniz,
and in Lambert's calculus, arises because M — P does not mean "M but
not P " or " M which is not P ". If it mean this, then [ + ] and •[ — ] are not
true inverses. If, on the other hand, M - P indicates the abstraction from
the concept M of all that is involved in the concept P, then M - P is
difficult or impossible to interpret, and, in addition, the idea of negation
cannot be represented by [-]. How does it happen, then, that Castillon's
notation works out so well when he uses [-] both for abstraction and as
the sign of negation? It would seem that his calculus ought to involve
him in both kinds of difficulties.
The answer is that Castillon has, apparently by good luck, hit upon a
method in which nothing is ever added to or subtracted from a determined
concept, S or A, except an indeterminate, M or N or P, and this indeter
minate, just because it is indeterminate, conceals the fact that [+ ] and [-]
are not true inverses. And when the sign [ — ] appears before a determinate,
A, it may serve as the sign of negation, because no difficulty arises from
supposing the whole of what is negated to be absent, or abstracted.
Castillon's calculus is theoretically as unsound as Lambert's, or more
so if unsoundness admits of degree. It is quite possible that it was worked
out empirically and procedures which give invalid results avoided.
The Development of Symbolic Logic 35
Whoever studies Leibniz, Lambert and Castillon cannot fail to be con
vinced that a consistent calculus of concepts in intension is either immensely I
difficult or, as Couturat has said, impossible. Its main difficulty is not
the one which troubled Leibniz and which constitutes the main defect in
Lambert's system— the failure of [ + ] and [-] to behave like true inverses.
This can be avoided by treating negative propositions as affirmatives with
negative predicates, as Leibniz did. The more serious difficulty is that a
calculus of "concepts " is not a calculus of things in actu but only in possibile,
and in a rather loose sense of the latter at that. Holland pointed this out
admirably in a letter to Lambert.55 He gives the example according to
Lambert's method,
All triangles are figures. T = tF
All quadrangles are figures. Q = qF
T 0
Whence, F =- = •£, or qT = tQ
and he then proceeds: 56
"In general, if from A == mC and B = nC the conclusion nA = mB
be drawn, the calculus cannot determine whether the ideas nA and mB
consist of contradictory partial-ideas, as in the foregoing example, or not.
The thing must be judged according to the matter."
This example also calls attention to the fact that Lambert's calculus,
by operations which he continually uses, leads to the fallacy of the undis
tributed middle term. If "some A" is simply some further specification
of the concept A, then this mode is not fallacious. And this observation
brings down the whole treatment of logic as a calculus of concepts in in
tension like a house of cards. The relations of existent things cannot be
determined from the relations of concepts alone.
The calculus of Leibniz is more successful than any invented by his
continental successors — unless Ploucquet's is an exception. That the long
period between him and De Morgan and Boole did not produce a successful
system of symbolic logic is probably due to the predilection for this inten-
sional point of view. It is no accident that the English were so quickly
successful after the initial interest was aroused; they habitually think of
logical relations in extension, and when they speak of "intension" it is
usually clear that they do not mean those relations of concepts which the
"intension" of traditional logic signifies.
55 Deutscher Gelehrter Brief wechsel, I, Brief xxvu.
56 Ibid., pp. 262-63.
36 A Survey of Symbolic Logic
The beginning of thought upon this subject in England is marked by the
publication of numerous treatises, all proposing some modification of the
traditional logic by quantifying the predicate. As Sir William Hamilton
notes,57 the period from Locke to 1833 is singularly barren of any real con
tributions to logic. About that time, Hamilton himself proposed the
quantification of the predicate. As we now know, this idea was as old at
least as Leibniz. Ploucquet, Lambert, Holland, and Castillon also had
quantified the predicate. Both Hamilton and his student Thomson men
tion Ploucquet; but this new burst of logical study in England impresses
one as greatly concerned about its own innovations and sublimely indifferent
to its predecessors. Hamilton quarrelled at length with De Morgan to
establish his priority in the matter.58 This is the more surprising, since
George Bentham, in his Outline of a New System of Logic, published in 1827,
had quantified the predicate and given the following table of propositions:
1 . X in toto = Y ex parte ;
2. X in toto ! Y ex parte;
3. X in toto = Y in toto;
4. X in toto | Y in toto;
5. X ex parte = Y ex parte;
6. X ex parte | | Y ex parte;
7. X ex parte = Y in toto;
8. X ex parte | | 7 in toto.
(| | is here the sign of "diversity").
But Hamilton was certainly the center and inspirer of a new movement
in logic, the tendency of which was toward more precise analysis of logical
significances. Bayne's Essay on the New Analytic and Thomson's Laws of
Thought are the most considerable permanent record of the results, but
there was a continual fervid discussion of logical topics in various peri
odicals; logistic was in the air.
This movement produced nothing directly which belongs to the history of
symbolic logic. Hamilton's rather cumbersome notation is not made the
basis of operations, but is essentially only an abbreviation of language.
Solly's scheme of representing syllogisms was superior as a calculus. But
57 See Discussions on Philosophy, pp. 119 ff.
This controversy, begun in 1846, was continued for many years (see various articles
in the London Athenceum, from 1860 to 1867). It was concluded in the pages of the Con
temporary Review, 1873.
The Development of Symbolic Logic 3 7
this movement accomplished two things for symbolic logic: it emphasized
in fact — though not always in name — the point of view of extension, and
it aroused interest in the problem of a newer and more precise logic. These
may seem small, but whoever studies the history of logic in this period
will easily convince himself that without these things, symbolic logic might
never have been revived. Without Hamilton, we might not have had
Boole. The record of symbolic logic on the continent is a record of failure,
in England, a record of success. The continental students habitually
emphasized intension; the English, extension.
IV. DE MOP CAN
De Morgan59 is known to most students of symbolic logic only through
the theorem which bears his name. But he made other contributions of
permanent value — the idea of the "universe of discourse",60 the discovery
of certain new types of propositions, and a beginning of the logic of rela
tions. Also, his originality in the invention of new logical forms, his ready
wit, his pat illustrations," and the clarity and liveliness of his writing did
yeoman service in breaking down the prejudice against the introduction \
of "mathematical" methods in logic. His important writings on logic
are comprised in the Formal Logic, the Syllabus of a Proposed System of
Logic, and a series of articles in the Transactions of the Cambridge Philo
sophical Society?1
59 Augustus De Morgan (1806-78), A.B. (Cambridge, 1827), Professor of Mathematics
in the University of London 1828-31, reappointed 1835; writer of numerous mathematical
treatises which are characterized by exceptional accuracy, originality and clearness. Per
haps the most valuable of these is "Foundations of Algebra" (Camb. Phil. Trans., vu,
vm); the best known, the Budget of Paradoxes. For a list of his papers, see the Royal
Society Catalogue. For many years an active member of the Cambridge Philosophical
Society and the Royal Astronomical Society. Father of William F. De Morgan, the novelist
and poet. For a brief biography, see Monthly Notices of the Royal Astronomical Society,
xii, 112.
60 The idea is introduced with these words: "Let us take a pair of contrary names,
as man and not-man. It is plain that between them they represent everything, imaginable
or real, in the universe. But the contraries of common language embrace, not the whole
universe, but some one general idea. Thus, of men, Briton and alien are contraries:
every man must be one of the two, no man can be both. . . . The same may be said of
integer and fraction among numbers, peer and commoner among subjects of a realm,
male and female among animals, and so on. In order to express this, let us say that the
whole idea under consideration is the universe (meaning merely the whole of which we are
considering parts) and let names which have nothing in common, _but which between them
contain the whole of the idea under ^nsideratiqn, be called contraries in, or with respect to,
that universe." (Formal Logic, p. 37; see also Camb. Phil. Trans., vm, 380.)
61 Formal Logic: or, The Calculus of Inference, Necessary and Probable, 1847. Here
after to be cited as F. L.
38 A Survey of Symbolic Logic
Although the work of De IN [organ is strictly contemporary with that of
Boole, his methods and symbolism ally him rather more with his prede
cessors than with Boole and those who follow. Like Hamilton, he is bent
upon improving the traditional Aristotelian logic. His first step in this
direction is to enlarge the number of typical propositions by considering
all the combinations and distributions of two terms, X and F, and their
negatives. It is a feature of De Morgan's notation that the distribution of
each term,62 and the quality — affirmative or negative — of the proposition
are indicated, these being sufficient to determine completely the type of
the proposition.
That a term A" is distributed is indicated by writing half a parenthesis
^before or after it, with the horns turned toward the letter, thus: A"), or (X.
An undistributed term is marked by turning the half-parenthesis the other
way, thus: A"(, or )A". A"))}7, for example, indicates the proposition in
which the subject, X, is distributed and the predicate, F, is undistributed,
that is, "All A" is F". X()Y indicates a proposition with both terms un
distributed, that is, "Some A' is F".63 The negative of a term, X, is indi
cated by .r; of F by y, etc. A negative proposition is indicated by a dot
placed between the parenthetical curves; thus "Some X is not Y" will
be A"(-(F.64 Two dots, or none, indicates an affirmative proposition.
All the different forms of proposition which De Morgan uses can be
generated from two types, the universal, "All . . . is . . ./' and the
particular, "Some . . . is . . .," by using the four terms, X and its nega
tive, .r, F and y. For the universals we have:
Syllabus of a Proposed System of Logic, 1860. Hereafter to be cited as Syll.
Five papers (the first not numbered; various titles) in Camb. Phil. Trans., vm, ix, x.
The articles contain the most valuable material, but they are ill-arranged and inter
spersed with inapposite discussion. Accordingly, the best way to study De Morgan is to
get these articles and the Formal Logic, note in a general way the contents of each, and
then use the Syllabus as a point of departure for each item in which one is interested.
62 He does not speak of "distribution" but of terms which are "universally spoken of"
cr "particularly spoken of ", or of the "quantity" of a term.
63 This is the notation of Syll. and of the articles, after the first, in Camb. Phil. Trans.
For a table comparing the different symbolisms which he used, see Camb. Phil. Trans.,
ix, 91.
64 It is sometimes hard to determine by the conventional criteria whether De Morgan's
propositions should be classed as affirmative or negative. He gives the following ingenious
rule for distinguishing them (Syll., p. 13): "Let a proposition be affirmative which is true
of X and A', false of X and not-A" or x; negative, which is true of X and x, false of X and X.
Thus 'Every A is Y' is affirmative: 'Every A is A' is true; 'Every A is x' is false. But
'Some things are neither A's nor F's' is also affirmative, though in the form of a denial:
'Some things are neither A's nor A's' is true, though superfluous in expression; 'Some
things are neither A's nor z's' is false."
The Development of Symbolic Logic 39
(1) A)) 7 MIX is Y.
(2) x))y All not- A" is not- 7.
(3) X))y All X is not-7.
(4) x))Y Allnot-Zis 7.
and for particulars we have :
(5) X() Y Some X is 7.
(6) x()y Some not-X is not- 7.
(7) X()y Some X is not- 7.
(8) x()Y Some not-JT is 7.
The rule for transforming a proposition into other equivalent forms may
be stated as follows: Change the distribution of either term — that is, turn
its parenthetic curve the other way, — change that term into its negative,
and change the quality of the proposition. That this rule is valid will
appear if we remember that "two negatives make an affirmative", and note
that we introduce one negative by changing the term, another by changing
the quality of the proposition. That the distribution of the altered term
should be changed follows from the fact that whatever proposition distrib
utes a term leaves the negative of that term undistributed, and whatever
proposition leaves a term undistributed distributes the negative of that
term. Using this rule of transformation, we get the following table of
equivalents for our eight propositions:
(a) (b) (c) (d)
(1) X»Y = X)-(y --=x((y = x(-)Y
(2) x))y = x)-(Y = X((Y = X(-)y
(3) X))y - X).(Y = x((Y »*(•)?
(4) .r))7 = a-)-Q/ = X((y ••- X(-)Y
(5) T() 7 = X(-(y -=x)(y =x).)Y
(G) x()i/ *x(.(Y = Z)(7-Z)-)y
(7) XQy -.= X(.(Y = x)(Y = x)-)y
(8) a-()7 =x(.(y - X}(y -=X)-)Y
It will be observed that in each line there is one proposition with both
terms positive, A' and Y. Selecting these, we have the eight different types
of propositions:
40 A Survey of Symbolic Logic
(la) A))F All A" is Y.
(2c) A'((F Some X is all F; or, All Y is X.
(36) X)-(Y NoZis Y.
(4d) X(-)Y Everything is either X or Y. (See below.)
(5o) Ar()F Some A is 7.
(6c) X)(Y Some things are neither X nor F. (See below.)
(76) X(-(Y Some A" is not Y.
(Sd) X)-)Y All X is not some Y; or, Some 7 is not X.
Since the quantity of each term is indicated, any one of these propositions
may be read or written backwards — that is, writh Y subject and X predicate
—provided the distribution of terms is preserved. (4d) and (6c) are diffi
cult to understand. We might attempt to read X(-)Y "Some X is not
some F", but we hardly get from that the difference between X(-)Y and
A'(-(F, "Some X is not (any) F". Also, A"(-)F is equivalent to uni-
versals, and the reading, "Some X is not some F", would make it par
ticular. A"(-)F is equivalent to cc)) F, "All not-Z is F", and to x)-(y,
"Xo not-X is not-F". The only equivalent of these with the terms
A" and F is, "Everything (in the universe of discourse) is either X or F
(or both)". (6c), X)(Y, we should be likely to read "All X is all F", or
" X and F are equivalent "; but this would be an error,65 since its equivalents
are particular propositions. (6a), xQy, is "Some not-X is not-F".
The equivalent of this in terms of X and F is plainly, "Some things are
neither X nor F".
Contradictories66 of propositions in line (1) will be found in line (7);
of those in line (2), in line (8); of line (3), in line (5); of line (4), in line (6).
We give those with both terms positive :
(la) X))Y contradicts (76) X(-(Y
(2c) X((Y " (Sd) A»F
(36) A>(F " (5a) XQY
X(-)Y (6c) X)(Y
65 An error into which it might seem that De Morgan himself has fallen. See e. g.,
SylL, p. 25, and Camb. Phil. Trans., ix, 98, where he translates X)(Y by "All X is all F ",
or "Any one X is any one Y ". But this belongs to another interpretation, the "cumular",
which requires X and Y to be singular, and not-X and not- Y will then have common
members. However, as we shall note later, there is a real difficulty.
66 De Morgan calls contradictory propositions "contraries" (See F. L., p. 60; Sytt.,
p. 11), just as he calls terms which are negatives of one another "contraries".
The Development of Symbolic Logic 41
Thus the rule is that two propositions having the same terms contradict
one another when one is affirmative, the other negative, and the distribution
of terms is exactly opposite in the two cases.
The rule for transforming propositions which has been stated and
exemplified, together with the observation that any symbolized proposition
may be read or written backwards, provided the distribution of the terms
be preserved, gives us the principles for the immediate inference of uni-
versals from universals, particulars from particulars. For the rest, we have
the rule, "Each universal affirms the particulars of the same quality".67
For syllogistic reasoning, the test of validity and rule of inference are
as follows: 6i'
"There is inference: 1. When both the premises are universal; 2. When,
one premise only being particular, the middle term has different quantities
in the two premises.
"The conclusion is found by erasing the middle term and its quantities
[parenthetic curves]." This rule of inference is stated for the special
arrangement of the syllogism in which the minor premise is put first, and
the minor term first in the premise, the major term being the last in the
second premise. Since any proposition may be written backward, this
arrangement can always be made. According to the rule, X))Y, "All X
is F", and F) • (Z, "No Y is Z", give X)-(Z, "No X is Z". A>(F, "No
X is F", and F(-(Z, "Some F is not Z", give A") • - (Z, or A') (Z, which is
"Some things are neither A" nor Z." The reader may, by inventing other
examples, satisfy himself that the rule given is sufficient for all syllogistic
reasoning, with any of De Morgan's eight forms of propositions.
De Morgan also invents certain compound propositions which give com
pound syllogisms in a fashion somewhat analogous to the preceding: 69
"1. X)0)Forboth A))Fand A»F All X's and some things be
sides are F's.
2. X For both Ar))Fand A'((F All X's are F's, and all F's
are X's.
3. X(0 (F or both X((Y and X(-(Y Among X's are all the F's and
some things besides.
4. A') O (For both X)-(Y and X)(Y Nothing both A' and F and
some things neither.
67 Sytt., p. 16.
**Sytt.,p. 19.
69 SyU., p. 22.
42 A Survey of Symbolic Logic
5. Z|- | For both A>(Fand X(-)Y Nothing both X and F and
everything one or the other.
6. X(O)Y or both X(-)Y and XQY Everything either X or Y and
some things both."
Each of these propositions may, with due regard for the meaning of the
sign O, be read or written backward, just as the simple propositions. The
rule of transformation into other equivalent forms is slightly different:
Change the quantity, or distribution, of any term and replace that term
by its negative. We are not required, as with the simple propositions, to
change at the same time the quality of the proposition. This difference
is due to the manner in which the propositions are compounded.
The rules for mediate, or " syllogistic", inference for these compound
propositions are as follows: 70
"If any two be joined, each of which is [of the form of] 1, 3, 4, or 6,
with the middle term of different quantities, these premises yield a con
clusion of the same kind, obtained by erasing the symbols of the middle
term and one of the symbols [O]. Thus X)O(Y(O)Z gives X)O)Z: or
if nothing be both X and 7 and some things neither, and if everything be
either Y or Z and some things both, it follows that all X and tivo lots of
other things are Z's.
" In any one of these syllogisms, it follows that may be written for
)O) or )O( in one place, without any alteration of the conclusion, except
reducing the two lots to one. But if this be done in both places, the con
clusion is reduced to | | or | • | , and both lots disappear. Let the reader
examine for himself the cases in which one of the premises is cut down to a
simple universal.
"The following exercises will exemplify what precedes. Letters written
under one another are names of the same object. Here is a universe of 12
instances of which 3 are X's and the remainder P's; 5 are F's and the
remainder Q's; 7 are Z's and the remainder R's.
XXX PP PP PPPPP
YYY YY QQ QQQQQ
Z Z Z Z Z Z Z RRRRR
We can thus verify the eight complex syllogisms
X)0)Y)0)Z P(0)Y)0)Z P(0(Q(O)Z P(O(Q(O(R
P(0)Y)Q(R X)0)Y)0(R X)Q(Q(0(R X)O(Q(O)Z
70 Sytt., p. 23.
The Development of Symbolic Logic 45f
In every case it will be seen that the two lots in the middle form the quantity
of the particular proposition of the conclusion."
In so much of his work as we have thus far reviewed, De Morgan is still
too much tied to his starting point in Aristotelian logic. He somewhat
simplifies traditional methods and makes new generalizations which include
old rules, but it is still distinctly the old logic. He does not question the
inference from universals to particulars nor observe the problems there
involved.71 He does not seek a method by which any number of terms
may be dealt with but accepts the limitation to the traditional two. And
his symbolism has several defects. The dot introduced between the
parenthetic curves is not the sign of negation, so as to make it possible to
read (•) as, "It is false that ()". The negative of () is ) • (, so that this
simplest of all relations of propositions is represented by a complex trans
formation applicable only when no more than two terms are involved in the
prepositional relation. Also, there are two distinct senses in which a
term in a proposition may be distributed or "mentioned universally", and
De Morgan, following the scholastic tradition, fails to distinguish them and
symbolizes both the same way. This is the secret of the difficulty in reading
X)(Y, which looks like "All A" is all 7", and really is "Some things are
neither A" nor Y ".72 Mathematical symbols are introduced but without any
corresponding mathematical operations. The sign of equality is used both
for the symmetrical relation of equivalent propositions and for the un-
symmetrical relation of premises to their conclusion.73
His investigation of the logic of relations, however, is more successful,
and he laid the foundation for later researches in that field. This topic
is suggested to him by consideration of the formal and material elements
in logic. He says: 7l
71 But he does make the assumption upon which all inference (in extension) of a
particular from a universal is necessarily based: the assumption that a class denoted by a
simple term has members. He says (F. L., pp. 110), "Existence as objects, or existence as
ideas, is tacitly claimed for the terms of every syllogism".
72 A universal affirmative distributes its subject in the sense that it indicates the class
to which every member of the subject belongs, i. e., the class denoted by the predicate.
Similarly, the universal negative, No X is F, indicates that every A' is not- Y, every Y is
not- A'. No particular proposition distributes a term in that sense. The particular nega
tive tells us only that the predicate is excluded from some unspecified portion of the class
denoted by the subject. A)(F distributes X and Y in this sense only. Comparison with
its equivalents shows us that it can tell us, of A", only that it is excluded from some un
specified portion of not-F; and of Y, only that it is excluded from some unspecified portion
of not-A. We cannot infer that X is wholly included in Y, or Y in X, or get any other
relation of inclusion out of it.
73 In one passage (Camb. Phil. Trans., x, 183) he suggests that the relation of two
premises to their conclusion should be symbolized by A B < C.
74 Camb. Phil. Trans., x, 177, footnote.
44 A Survey of Symbolic Logic
"Is there any consequence without form? Is not consequence an action
of the machinery? Is not logic the science of the action of the machinery?
Consequence is always an act of the mind : on every consequence logic ought
to ask, What kind of act? What is the act, as distinguished from the acted
on, and from any inessential concomitants of the action ? For these are of
the form, as distinguished from the matter.
". . . The copula performs certain functions; it is competent to those
functions . . . because it has certain properties, which are sufficient to
validate its use. . . . The word 'is,' which identifies, does not do its work
because it identifies, except insofar as identification is a transitive and
convertible motion: 'A is that which is B' means 'A is B'; and 'A is B'
means 'B is A'. Hence every transitive and convertible relation is as fit
to validate the syllogism as the copula 'is', and by the same proof in each
case. Some forms are valid when the relation is only transitive and not
convertible; as in 'give'. Thus if X— Y represent X and 7 connected
by a transitive copula, Camestres in the second figure is valid, as in
EveryZ— 7, No X— Y, therefore No X~ Z.
... In the following chain of propositions, there is exclusion of matter,
form being preserved at every step :
Hypothesis
(Positively true) Every man is animal
Every man is Y Y has existence.
Every X is 7 X has existence.
Every X — 7 - is a transitive relation.
a of X — 7 a is a fraction < or = 1.
(Probability (3) a of .Y— 7 /? is a fraction < or - 1.
The last is nearly the purely formal judgment, with not a single material
point about it, except the transit! veness of the copula.75
"... I hold the supreme form of the syllogism of one middle term to
be as follows: There is the probability a that X is in the relation L to 7;
there is the probability 0 that 7 is in the relation M to Z; whence there is
the probability a/3 that X is in the relation L of M to Z.76
"... The copula of cause and effect, of motive and action, of all which
post hoc is of the form and propter hoc (perhaps) of the matter, will one day
be carefully considered in a more complete system of logic." 77
75 Ibid., pp. 177-78.
76 Ibid., p. 339.
77 Ibid., pp. 179-80.
The Development of Symbolic Logic 45
De Morgan is thus led to a study of the categories of exact thinking in
general, and to consideration of the types and properties of relations.
His division of categories into logico-mathematical, logico-physical, logico-
metaphysical, and logico-contraphysical,78 is inauspicious, and nothing
much comes of it. But in connection with this, and an attempt to rebuild
logic in the light of it, he propounds the well-known theorem: "The con
trary [negative] of an aggregate [logical sum] is the compound [logical
product] of the contraries of the aggregants: the contrary of a compound
is the aggregate of the contraries of the components." 79
For the logic of relations, X, Y, and Z will represent the class names;
L, M, N, relations. X . . LY will signify that A" is some one of the objects
of thought which stand to Y in the relation L, or is one of the L's of 7.80
X . LY will signify that X is not any one of the L's of Y. X . . (LM) Y or
X . . LM Y will express the fact that AT is one of the L's of one of the M's
of y, or that X has the relation L to some Z which has the relation M to Y.
X . LM Y will mean that X is not an L of any M of Y.
It should be noted that the union of the two relations L and M is what
we should call today their " relative product " ; that is, A" . . LY and Y . . MZ
together give A^ . . LM Z, but A' . . LY and A' . . MY do not give A" . . LM Y.
If L is the relation "brother of" and M is the relation "aunt of ", A' . . LM Y
will mean " X is a brother of an aunt of Y". (Do not say hastily, " X is
uncle of 7". "Brother of an aunt" is not equivalent to "uncle" since
some uncles have no sisters.) L, or M, written by itself, will represent
that which has the relation L, or M, that is, a brother, or an aunt, and LY
stands for any X which has the relation L to F, that is, a brother of T.81
In order to reduce ordinary syllogisms to the form in which the copula
has that abstractness which he seeks, that is, to the form in which the
copula may be any relation, or any relation of a certain type, it is necessary
to introduce symbols of quantity. Accordingly LM* is to signify an L of
every M, that is, something which has the relation L to every member of
the class M (say, a lover of every man). L*M is to indicate an L of none
but M's (a lover of none but men). The mark of quantity, * or *, always
78 See ibid., p. 190.
79 Ibid., p. 208. See also Sytt., p. 41. Pp. 39-60 of Syll. present in summary the ideas
of the paper, "On the Syllogism, No. 3, and on Logic in General."
80 Camb. Phil. Trans., x, 341. We follow the order of the paper from this point on.
81 1 tried at first to make De Morgan's symbolism more readily intelligible by intro
ducing the current equivalents of his characters. But his systematic ambiguities, such
as the use of the same letter for the relation and for that which has the relation, made
this impossible. For typographical reasons, I use the asterisk where he has a small accent.
46 A Survey of Symbolic Logic
goes with the letter which precedes it, but L*M is read as if '*] modified
the letter which follows. To obviate this difficulty, De Morgan suggests
that L*M be read, "An every-!, of If; an L of M in every way in which
it is an Z, " but we shall stick to the simpler reading, "An L of none but
M's".
LM*X means an L of every M of X: L*MX, an L of none but M's of X:
L*M*, an L of every M and of none but 3f's: LMX*, an L of an M of
every X, and so on.
Two more symbols are needed. The converse of L is symbolized by L~l.
If L is "lover of", L-1 is "beloved of"; if L is "aunt", L-1 is "niece or
nephew ". The contrary (or as we should say, the negative) of L is symbol
ized by 1; the contrary of M by m.
In terms of these relations, the following theorems can be stated :
(1) Contraries of converses are themselves contraries.
(2) Converses of contraries are contraries.
(3) The contrary of the converse is the converse of the contrary.
(4) If the relation L be contained in, or imply, the relation M, then (a) the
converse of L, L~l, is contained in the converse of M, M~l\ and (6) the
contrary of M, m, is contained in the contrary of L, I.
For example, if "parent of" is contained in "ancestor of", (a) "child of"
is contained in "descendent of", and (b) "not ancestor of" is contained in
"not parent of".
(5) The conversion of a compound relation is accomplished by converting
both components and inverting their order; thus, (LM)'1 = M~lL~l.
If X be teacher of the child of Y, Y is parent of the pupil of X.
When a sign of quantity is involved in the conversion of a compound
relation, the sign of quantity changes its place on the letter; thus,
If X be teacher of every child of F, 7 is parent of none but pupils of X.
(6) When, in a compound relation, there is a sign of quantity, if each
component be changed into its contrary, and the sign of quantity be shifted
from one component to the other and its position on the letter changed,
the resulting relation is equivalent to the original; thus LM* = l*m and
L*M = lm*.
A lover of every man is a non-lover of none but non-men; and a lover
of none but men is a non-lover of everv non-man.
The Development of Symbolic Logic 47
(7) When a compound relation involves no sign of quantity, its contrary
is found by taking the contrary of either component and giving quantity
to the other. The contrary of LM is IM* or L*m.
"Not (lover of a man) " is "non-lover of every man" or "lover of none
but non-men"; and there are two equivalents, by ((>).
But if there be a sign of quantity in one component, the contrary is
taken by dropping that sign and taking the contrary of the other component.
The contrary of LM* is IM; of L*M is Lm.
"Not (lover of every man)" is "non-lover of a man"; and "not (lover
of none but men) " is "lover of a non-man ".
So far as they do not involve quantifications, these theorems are familiar
to us today, though it seems not generally known that they are due to
De Morgan. The following table contains all of them:
Converse of Contrary
Combination Converse Contrary Contrary of Converse
LM M-lL~l /J/* or L*M J/*-1/-1 or m~lL-1*
LM*mhm M*~lL~l or m~H-1* IM M~ll~l
L*Morlm* M^L-1* or m*-}l~l Lm m~lL~l
The sense in which one relation is said to be "contained in" or to
"imply" another should be noted: L is contained in M in case every X
which has the relation L to any Y has also the relation M to that Y. This
must not be confused with the relation of class inclusion between two rela
tive terms. Every grandfather is also a father, the class of grandfathers is
contained in the class of fathers, but "grandfather of" is not contained in
"father of", because the grandfather of Y is not also the father of Y. The
relation "grandfather of" is contained in "ancestor of", since the grand
father of F is also the ancestor of Y. But De Morgan appropriately uses
the same symbol for the relation " L contained in M" that he uses for "All
L is M", where L and M are class terms, that is, L))M.
In terms of this relation of relations, the following theorems can be
stated :
(8) If L))M, then the contrary of M is contained in the contrary of L,—
that is, L))M gives m))l.
Applying this theorem to compound relations, we have:
(8'-) LM))N gives w))/J/* and n))L*m.
(8") If LM))N, then L~ln))m and wJ/-1))/.
Proof: If LM))N, then n))lM*. Whence nM-l))lM*M~l. But an / of
48 A Survey of Symbolic Logic
every M of an M~l of Z must be an Z of Z. Hence nM~l))l. Again; if
LM))N, then ri))L*m. Whence L~ln))L-lL*m. But . whatever has the
relation converse-of-Z to an L of none but m's must be itself an m. Hence
L~ln))m.
De Morgan calls this "theorem K" from its use in Baroko and Bokardo.
(9) If LM = N, then L))NM~l and M))L~1N.
Proof: If LM = N, then LMM~l = NM~l and L^LM = Z~W. Now
for any X, MM~1X and L~1LX are classes which contain X; hence the
theorem.
We do not have L = NM~l and M == L~1N, because it is not generally
true that MM~1X == X and LrlLX --= X. For example, the child of the
parent of X may not be X but A"'s brother : but the class " children of the
parent of X" will contain X. The relation MM~l or M~1M will not always
cancel out. MM~l and M~1M are always symmetrical relations ; if XMM~l Y
then YMM~1X. If X is child of a parent of Y, then Y is child of a parent
of X. But MM-1 and M~1M are nctf exclusively reflexive. XMM^X does
not always hold. If we know that a child of the parent of X is a celebrated
linguist we may not hastily assume that X is the linguist in question.
With reference to transitive relations, we may quote : 82
"A relation is transitive when a relative of a relative is a relative of
the same kind; as symbolized in ZZ))Z, whence ZZZ))ZZ))Z; and so on.
"A transitive relation has a transitive converse, but not necessarily a
transitive contrary: for L~lL~l is the converse of LL, so that ZZ))Z gives
L 1L *))L l. From these, by contraposition, and also by theorem K and
its contrapositions, we obtain the following results:
L is contained in LL-1*, Z*H, HZ*, L*~1L
I
ZZ-1, Z-!Z ...... Z-1
" I omit demonstration, but to prevent any doubt about correctness of
printing, I subjoin instances in words: Z signifies ancestor and L~l descendent.
82 Camb. Phil. Trans., x, 346. For this discussion of transitive relations, De Morgan
treats all reciprocal relations, such as XLL^Y, as also reflexive, though not necessarily
exclusively reflexive.
The Development of Symbolic Logic 49
"An ancestor is always an ancestor of all descendents, a non-ancestor
of none but non-descendents, a non-descendent of all non-ancestors, and a
descendent of none but ancestors. A descendent is always an ancestor of
none but descendents, a non-ancestor of all non-descendents, a non-descend
ent of none but non-ancestors, and a descendent of all ancestors. A non-
ancestor is always a non-ancestor of all ancestors, and an ancestor of none
but non-ancestors. A non-descendent is a descendent of none but non-
descendents, and a non-descendent of all descendents. Among non-
ancestors are contained all descendents of non-ancestors, and all non-
ancestors of descendents. Among non-descendents are contained all
ancestors of non-descendents, and all non-descendents of ancestors."
In terms of the general relation, L, or M, representing any relation, the
syllogisms of traditional logic may be tabulated as follows: 83
1
2
3
4
X
..LY
X
.LY
X
..LY
X
.LY
I
Y
..LZ
Y
. . MZ
Y
.MZ
Y
.MZ
X
. . LMZ
X
. . LMZ
X
. . LmZ
X
. . ImZ
X
.LY
X
..LY
X
..LY
X
.LY
II
Z
. . MY
Z
. . MY
Z
. MY
Z
. MY
X
. . IM-W
X
. . LM~1Z
X
. . Lm~lZ
X
. . lm-y
Y
. . LX
Y
. LX •
Y
. . LX
Y
.LX
III
Y
.MZ
Y
. . MZ
Y
. . MZ
Y
.MZ
X
. . L~lmZ
X
. . 1-iMZ
X
. . L-^MZ
X
. . l~lrnZ
Y
. LX
Y
. . LX
Y
.LX
Y
. . LX
IV
Z
.MY
Z
. MY
Z
..MY
Z
..MY
X
. l-lm~lZ
X
.L-lm~lZ
X
. 1~1M~1Z
X
.L-1M~1Z"
The Roman numerals here indicate the traditional figures. All the con
clusions are given in the affirmative form; but for each affirmative con
clusion, there are two negative conclusions, got by negating the relation and
replacing it by one or the other of its contraries. Thus A" . . LMZ gives
A' . IH*Z and A' . L*mZ; X . . 1M~1Z gives A' . LM~l*Z and A' . hm-lZ,
and so on for each of the others.
63 Ibid., p. 350.
5
50 A Survey of Symbolic Logic
When the copula of all three propositions is limited to the same transitive
relation, L, or its converse, the table of syllogisms will be: 84
X..LY X.LY X..LY
I Y..LZ Y..L-1Z Y.L~1Z
X ..LZ X .LZ X . L~1Z
X .
LY
X .
.LY
X
..LY
II
z .
.LY
Z .
.LrlY
Z
.LY
X .
LZ
X .
.LZ
X
.L~1Z
Y . . LX
Y .LX
Y ..LX
III Y . LZ
Y ..LZ
Y . . L~\
X . LZ
X . L-^Z
X . . L~\
Y ..LX Y .LX Y ..LX
IV Z.L-1Y Z..L~1Y Z..LY
X.LZ X.L~1Z X..L~1Z"
Here, again, in the logic of relations, De Morgan would very likely have
done better if he had left the traditional syllogism to shift for itself. The
introduction of quantifications and the systematic ambiguity of L, M,
etc., which are used to indicate both the relation and that which has the
relation, hurry him into complications before the simple analysis of rela
tions, and types of relations, is ready for them. This logic of relations was
destined to find its importance in the logistic of mathematics, not in any
applications to, or modifications of, Aristotelian logic. And these compli
cations of De Morgan's, due largely to his following the clues of formal logic,
had to be discarded later, after Peirce discovered the connection between
Boole's algebra and relation theory. The logic of relative terms has been
reintroduced by the work of Frege and Peano, and more especially of
Whitehead and Russell, in the logistic development of mathematics. But
it is there separated — and has to be separated — from the simpler analysis
of the relations themselves. Nevertheless, it should always be remembered
that it was De Morgan who laid the foundation; and if some part of his
work had to be discarded, still his contribution was indispensable and of
permanent value. In concluding his paper on relations, he justly remarks : 85
84 Ibid., p. 354.
85 Ibid., p. 358.
The Development of Symbolic Logic 51
"And here the general idea of relation emerges, and for the first time
in the history of knowledge, the notions of relation and relation of relation
are symbolized. And here again is seen the scale of graduation of forms,
the manner in which what is difference of form at one step of the ascent is
difference of matter at the next. But the relation of algebra to the higher
developments of logic is a subject of far too great extent to be treated here.
It will hereafter be acknowledged that, though the geometer did not think
it necessary to throw his ever-recurring principium et exemplum into imita
tion of Omnis homo est animal, Sortes est homo, etc., yet the algebraist was
living in the higher atmosphere of syllogism, the unceasing composition of
relation, before it was admitted that such an atmosphere existed." M
V. BOOLE
The beginning from which symbolic logic has had a continuous develop
ment is that made by George Boole.87 His significant and vital contribution
was the introduction, in a fashion more general and systematic than before,
of mathematical operations. Indeed Boole allows operations which have
no direct logical interpretation, and is obviously more at home in mathe
matics than in logic. It is probably the great advantage of Boole's work
that he either neglected or was ignorant of those refinements of logical
theory which hampered his predecessors. The precise mathematical
development of logic needed to make its own conventions and interpreta
tions; and this could not be done without sweeping aside the accumulated
traditions of the non-symbolic Aristotelian logic. As we shall see, all the
nice problems of intension and extension, of the existential import of uni-
versals and particulars, of empty classes, and so on, return later and demand
consideration. It is well that, with Boole, they are given a vacation long
enough to get the subject started in terms of a simple and general procedure.
Boole's first book, The Mathematical Analysis of Logic, being an Essay
toward a Calculm of Deductive Reasoning, was published in 1847, on the
86 1 omit, with some misgivings, any account of De Morgan's contributions to prob
ability theory as applied to questions of authority and judgment. (See SylL, pp. 67-72;
F. L., Chap, ix, x; and Camb. Phil. Trans., vm, 384-87, and 393-405.) His work on this
topic is less closely connected with symbolic logic than was Boole's. The allied subject of
the "numerically definite syllogism" (see Syll., pp. 27-30; F. L., Chap, vm; and Camb.
Phil. Trans., x, *355-*358) is also omitted.
87 George Boole (1815-1864) appointed Professor of Mathematics in Queen's College,
Cork, 1849; LL.D. (Dublin, 1852), F.R.S. (1857), D.C.L. (Oxford, 1859). For a biographi
cal sketch, by Harley, see Brit. Quart. Rev., XLIV (1866), 141-81. See also Proc. Roy.
Soc., xv (1867), vi-xi.
52 A Survey of Symbolic Logic
o
~*same day as De Morgan's Formal Logic.88 The next year, his article, "The
Calculus of Logic," appeared in the Cambridge Mathematical Journal. This
article summarizes very briefly and clearly the important innovations pro
posed by Boole. But the authoritative statement of his system is found
in An Investigation of the Laws of Thought, on which are founded the Mathe
matical Theories of Logic and Probability, published in 1854. 89
Boole's algebra, unlike the systems of his predecessors, is based squarely
upon the relations of extension. The three fundamental ideas upon which
his method depends are: (1) the conception of " elective symbols"; (2) the
Jaws of thought expressed as rules for operations upon these symbols; (3)
the observation that these rules of operation are the same which would
hold for an algebra of the numbers 0 and I.90
For reasons which will appear shortly, the " universe of conceivable
objects" is represented by 1. All other classes or aggregates are supposed
to be formed from this by selection or limitation. This operation of electing,
in 1, all the A"s, is represented by l-.r or x; the operation of electing all
the T's is similarly represented by 1-y or y, and so on. Since Boole does
not distinguish between this operation of election represented by x, and
the result of performing that operation — an ambiguity common in mathe
matics— .r becomes, in practice, the symbol for the class of all the A^'s.
Thus x, y, z, etc., representing ambiguously operations of election or classes,
are the variables of the algebra. Boole speaks of them as "elective symbols"
to distinguish them from coefficients.
This operation of election suggests arithmetical multiplication: the
suggestion becomes stronger when we note that it is not confined to 1.
1 -x -y or xy will represent the operation of electing, first, all the Ar's in the
" universe", and from this class by a second operation, all the Fs. The
result of these two operations will be the class whose members are both
A"s and F's. Thus xy is the class of the common members of x and y;
xyz, the class of those things which belong at once to x, to y, and to z,
and so on. And for any x, l-x = x.
The operation of "aggregating parts into a whole" is represented by + .
x + y symbolizes the class formed by combining the two distinct classes,
x and y. It is a distinctive feature of Boole's algebra that x and y in x + y
must have no common members. The relation may be read, "that which
88 See De Morgan's note to the article "On Propositions Numerically Definite", Camb.
Phil Trans., xi (1871), 396.
89 London, Walton and Maberly.
90 This principle appears for the first time in the Laws of Thought. See pp. 37-3g.
Work hereafter cited as L. of T.
The Development of Symbolic Logic 53
is either x or y but not both". Although Boole does not remark it, x + y
cannot be as completely analogous to the corresponding operation of
ordinary algebra as xy is to the ordinary algebraic product. In numerical
algebras a number may be added to itself: but since Boole conceives the
terms of any logical sum to be " quite distinct ",91 mutually exclusive classes,
x + x cannot have a meaning in his system. As we shall see, this is very
awkward, because such expressions still occur in his algebra and have to be
dealt with by troublesome devices.
But making the relation x + y completely disjunctive has one advantage
—it makes possible the inverse relation of "subtraction". The "separa
tion of a part, x, from a whole, ?/", is represented by y — x. If x + z = y,
then since x and z have nothing in common, y — x = z and y — z = x.
Hence [ + ] and [ — ] are strict inverses.
x + y, then, symbolizes the class of those things which are either members
of x or members of y, but not of both, x-y or xy symbolizes the class of
those things which are both members of x and members of y. x — y repre
sents the class of the members of x which are not members of y — the x's
except the y's. [ = ] represents the relation of two classes which have the
same members, i. e., have the same extension. These are the fundamental
relations of the algebra.
The entity (1 — .r) is of especial importance. This represents the
universe except the .r's, or all things which are not x's. It is, then, the
supplement or negative of x.
With the use of this symbolism for the negative of a class, the sum of two
classes, x and y, which have members in common, can be represented by
xy + x(l -#) + (! - x)y.
The first term of this sum is the class which are both x's and y's; the second,
those which are x's but not y's; the third, those which are y's but not x's.
Thus the three terms represent classes which are all mutually exclusive,
and the sum satisfies the meaning of + . In a similar fashion, x + y may
be expanded to
x(l - y) + (l - x)y.
Consideration of the laws of thought and of the meaning of these sym
bols will show us that the following principles hold :
(1) xy = yx What is both x and y is both y and x.
(2) x + y = y + x What is either x or y is either y or .r.
91 See L. of T., pp. 32-33.
54 A Survey of Symbolic Logic
(3) z(x + y) = zx + zy That which is both z and (either .r or y)
is either both z and x or both z and y.
(4) z(.r — y) = zx — zy That which is both z and (a; but not y)
is both z and x but not both 2 and y.
(5) If .T = y, then Z.T = zy
z + x = z + y
x-z = y - z
(6) .r - y = - y + .r
This last is an arbitrary convention: the first half of the expression gives
the meaning of the last half.
It is a peculiarity of "logical symbols" that if the operation .r, upon 1,
be repeated, the result is not altered by the repetition:
l-x = l-x-x = 1 • x • x • x . . . . Hence we have :
(7) .T2 = X
Boole calls this the "index law".92
All these laws, except (7), hold for numerical algebra. It may be
noted that, in logic, "If .r = y, then zx = zy" is not reversible. At first
glance, this may seem to be another difference between numerical algebra
and the system in question. But "If zx = zy, then x = y" does not hold
in numerical algebra when z = 0. Law (7) is, then, the distinguishing
principle of this algebra. The only finite numbers for which it holds are
0 and 1. All the above laws hold for an algebra of the numbers 0 and 1. With
this observation, Boole adopts the entire procedure of ordinary algebra,
modified by the law x2 = x, introduces numerical coefficients other than 0
and 1, and makes use, on occasion, of the operation of division, of the
properties of functions, and of any algebraic transformations which happen
to serve his purpose.93
This borrowing of algebraic operations which often have no logical
interpretation is at first confusing to the student of logic; and commen
tators have seemed to smile indulgently upon it. An example will help:
the derivation of the "law of contradiction" or, as Boole calls it, the "law
of duality", from the "index law".94
92 In Mathematical Analysis of Logic he gives it also in the form xn = x, but in L. of T.
he avoids this, probably because the factors of xn — x (e. g., z3 — x) are not always logically
iriterpretable.
93 This procedure characterizes L. of T. Only 0 and 1, and the fractions which can
be formed from them appear in Math. An. of Logic, and the use of division and of fractional
coefficients is not successfully explained in that book.
94 L. of T., p. 49.
The Development of Symbolic Logic 55
Since x2 = .r, x — a-2 = 0.
Hence, factoring, .r(l — x) = 0.
This transformation hardly represents any process of logical deduction.
Whoever says "What is both x and a-, a*2, is equivalent to x; therefore what
is both x and not-z is nothing" may well be asked for the steps of his reason
ing. Nor should we be satisfied if he reply by interpreting in logical terms
the intermediate expression, .r — .r2 = 0.
Nevertheless, this apparently arbitrary way of using uninterpretable
algebraic processes is thoroughly sound. Boole's algebra may be viewed
as an abstract mathematical system, generated by the laws we have noted,
which has two interpretations. On the one hand, the "logical" or "elec
tive" symbols may be interpreted as variables whose value is either numeri
cal 0 or numerical 1, although numerical coefficients other than 0 and 1 are
admissible, provided it be remembered that such coefficients do not obey
the "index law" which holds for "elective" symbols. All the usual alge
braic transformations will have an interpretation in these terms. On the
other hand, the "logical" or "elective" symbols may be interpreted as
logical classes. For this interpretation, some of the algebraical processes
of the system and some resultant expressions will not be expressible in terms
of logic. But whenever they are interpretable, they will be valid conse
quences of the premises, and even when they are not interpretable, any
further results, derived from them, which are interpretable, will also be
valid consequences of the premises.
It must be admitted that Boole himself does not observe the proprieties
of his procedure. His consistent course would have been to develop this al
gebra without reference to logical meanings, and then to discuss in a thorough
fashion the interpretation, and the limits of that interpretation, for logical
classes. By such a method, he would have avoided, for example, the
difficulty about .1- + .r. We should have .1- + x = 2.r, the interpretation of
which for the numbers 0 and 1 is obvious, and its interpretation for logical
classes would depend upon certain conventions which Boole made and
which will be explained shortly. The point is that the two interpretations
should be kept separate, although the processes of the system need not be
limited by the narrower interpretation — that for logical classes. Instead
of making this separation of the abstract algebra and its two interpretations,
Boole takes himself to be developing a calculus of logic; he observes that
its "axioms" are identical with those of an algebra of the numbers 0 and 1; 95
95 L. of T., pp. 37-38.
56 A Survey of Symbolic Logic
hence he applies the whole machinery of that algebra, yet arbitrarily rejects
from it any expressions which are not finally interpretable in terms of logical
relations. The retaining of non-interpretable expressions which can be
transformed into interpretable expressions he compares to "the employ
ment of the uninterpretable symbol V — 1 in the intermediate processes
of trigonometry. "96 It would be a pretty piece of research to take Boole's
algebra, find independent postulates for it (his laws are entirely insufficient
as a basis for the operations he uses), complete it, and systematically investi
gate its interpretations.
But neglecting these problems of method, the expression of the simple
logical relations in Boole's symbolism will now be entirely clear. Classes
will be represented by x, y, z, etc.; their negatives, by (1 — x), (1 — y),
etc. That which is both x and y will be xy\ that which is x but not y will
be .r(l — y), etc. That which is x or y but not both, will be x + y, or
;r(l — y) + (1 — x}y. That which is x or y or both will be x + (1 — x)y—
i. e., that which is x or not x but y — or
xy + x(l - y) + (1 - x)y
—that which is both x and y or x but not y or y but not x. 1 represents the
"universe" or "everything". The logical significance of 0 is determined
by the fact that, for any y, Oy = 0: the only class which remains unaltered
by any operation of electing from it whatever is the class "nothing".
Since Boole's algebra is the basis of the classic algebra of logic — which
is the topic of the next chapter — it will be unnecessary to comment upon
those parts of Boole's procedure which were taken over into the classic
algebra. These will be clear to any who understand the algebra of logic
in its current form or who acquaint themselves with the content of Chapter
II. We shall, then, turn our attention chiefly to those parts of his method
which are peculiar to him.
Boole does not symbolize the relation "x is included in ?/". Conse
quently the only copula by which the relation of terms in a proposition can
be represented is the relation =. And since all relations are taken in
extension, x = y symbolizes the fact that x and y are classes with identical
membership. Propositions must be represented by equations in which
something is put = 0 or == 1, or else the predicate must be quantified.
Boole uses both methods, but mainly relies upon quantification of the
predicate. This involves an awkward procedure, though one which still
survives — the introduction of a symbol v or w, to represent an indefinite
96 L. of T., p. 69.
The Development of Symbolic Logic 57
class and symbolize "Some". Thus "All x is (some) i/" is represented by
x = vy: "Some x is (some) #", by wx = vy. If v, or w. were here "the
indefinite class" or "any class", this method would be less objectionable.
But in such cases v, or w, must be very definitely specified: it must be a
class "indefinite in all respects but this, that it contains some members of
the class to whose expression it is prefixed".97 The universal affirmative
can also be expressed, without this symbol for the indeterminate, as .r(l — y)
= 0; "All xisy" means "That which is x but not y is nothing". Negative
propositions are treated as affirmative propositions with a negative predi
cate. So the four typical propositions of traditional logic are expressed as
follows: 98
All x is y: x = vy, or, x(l — y) = 0.
Xo x is y: .r = v(l — y), or xy = 0.
Some x is y: v.r = w(\ — y), or, v = xy.
Some .r is not y: vx = w(l — y), or, v = x(l — y).
Each of these has various other equivalents which may be readily deter
mined by the laws of the algebra.
To reason by the aid of this symbolism, one has only to express his
premises explicitly in the proper manner and then operate upon the resultant
equation according to the laws of the algebra. Or, as Boole more explicitly
puts it, valid reasoning requires: "
" 1st, That a fixed interpretation be assigned to the symbols employed
in the expression of the data; and that the laws of the combination of these
symbols be correctly determined from that interpretation.
"2nd, That the formal processes of solution or demonstration be con
ducted throughout in obedience to all the laws determined as above, with
out regard to the question of the interpretation of the particular results
obtained.
"3rd, That the final result be interpretable in form, and that it be
actually interpreted in accordance with that system of interpretation which
has been employed in the expression of the data."
As we shall see, Boole's methods of solution sometimes involve an
uninterpretable stage, sometimes not, but there is provided a machinery by
97 L. of T., p. 63. This translation of the arbitrary v by "Some" is unwarranted, and
the above statement is inconsistent with Boole's later treatment of the arbitrary coefficient.
There is no reason why such an arbitrary coefficient may not be null.
98 See Math. An. of Logic, pp. 21-22; L. of T., Chap. iv.
99 L. of T., p. 68.
58 A Survey of Symbolic Logic
which any equation may be reduced to a form which is interpretable. To
comprehend this we must first understand the process known as the develop
ment of a function. With regard to this, we can be brief, because Boole's
method of development belongs also to the classic algebra and is essentially
the process explained in the next chapter.100
Any expression in the algebra which involves x or (1 — x) may be
called a function of x. A function of x is said to be developed when it has
the form Ax + B(l — .r). It is here required that x be a "logical symbol",
susceptible only of the values 0 and 1. But the coefficients, A and B, are
not so limited: A, or B, may be such a "logical symbol" which obeys the
"law of duality", or it may be some number other than 0 or 1, or involve
such a number. If the function, as given, does not have the form Ax
+ B(1 — .r), it may be put into that form by observing certain interesting
laws which govern coefficients.
Let /(.r) = Ax + B(l - x)
Then /(I) = A-1 + B(1 ~1) = A
And /(O) - A-Q + B(l - 0) = B
Hence f(x) = /(I) -x +/(0) • (1 - x)
Thus if f(x) = ~,
Hence f(x) = 2x + ; (1 - x)
A developed function of two variables, x and y} will have the form:
x)(l - y)
And for any function, /(.r, y), the coefficients are determined by the law:
f(*,y) = /(I, !)•.?# +/(!,())• x(l -y) +/((), !)•(! -x)y
[°°See Math. An. of Logic, pp. 60-69; L. of T., pp. 71-79; "The Calculus of Logic,"
Cambridge and Dublin Math. Jour., m, 188-89. That this same method of development
should belong both to Boole's algebra and to the remodeled algebra of logic, in which +
is not completely disjunctive, is at first surprising. But a completely developed function,
m either algebra, is always a sum of terms any two of which have nothing in common.
This accounts for the identity of form where there is a real and important difference in the
meaning of the symbols.
The Development of Symbolic Logic 59
Thus if f(x, y) = ax + 2by,
/(I, 1) = a-1 + 26-1 = a + 26
/(I, 0) = a- 1 + 26-0 = a
/((), 1) = a-0 + 26-1 = 26
/((), 0) = tt-0 + 26-0 = 0
Hence /(.r, y) = (a + 21) xy + ay (I - y) + 26(1 - .r)y
An exactly similar law governs the expansion and the determination of
coefficients, for functions of any number of variables. In the words of
Boole:101
"The general rule of development will . . . consist of two parts, the
first of which will relate to the formation of the constituents of the expansion,
the second to the determination of their respective coefficients. It is as
follows :
"1st. To expand any function of the symbols x, y, z — Form a series
of constituents in the following manner: Let the first constituent be the
product of the symbols: change in this product any symbol z into 1 — z,
for the second constituent. Then in both these change any other symbol
y into 1 — y, for two more constituents. Then in the four constituents
thus obtained change any other symbol .r into 1 — .r, for four new constit
uents, and so on until the number of possible changes has been exhausted.
"2ndly. To find the coefficient of any constituent — If that constituent
involves x as a factor, change in the original function .r into 1 ; but if it
involves 1 — x as a factor, change in the original function .r into 0. Apply
the same rule with reference to the symbols y, z, etc.: the final calculated
value of the function thus transformed will be the coefficient sought."
Two further properties of developed functions, which are useful in
solutions and interpretations, are: (1) The product of any two constituents
is 0. If one constituent be, for example, xyz, any other constituent will
have as a factor one or more of the negatives, 1 — .r, 1 — y, 1 — z.
Thus the product of the two will have a factor of the form x(l — .r). And
where .r is a "logical symbol ", susceptible only of the values 0 and 1, x(l — x)
is always 0. And (2) if each constituent of any expansion have the coef
ficient 1, the sum of all the constituents is 1.
All information which it may be desired to obtain from a given set of
premises, represented by equations, will be got either (1) by a solution, to
determine the equivalent, in other terms, of some "logical symbol" .r, or
101L. o/T7., pp. 75-76.
60 A Survey of Symbolic Logic
(2) by an elimination, to discover what statements (equations), which are
independent of some term x, are warranted by given equations which in
volve x, or (3) by a combination of these two, to determine the equivalent
of x in terms of /, u, v, from equations which involve x, t, u, v, and some
other ''logical" symbol or symbols which must be eliminated in the desired
result. " Formal " reasoning is accomplished by the elimination of "middle"
terms.
The student of symbolic logic in its current form knows that any set
of equations may be combined into a single equation, that any equation
involving a term x may be given the form Ax + B(l — x) = 0, and that
the result of eliminating x from such an equation is AB = 0. Also, the
solution of any such equation, provided the condition AB = 0 be satisfied,
will be x = B + v(l —A), where v is undetermined. Boole's methods
achieve these same results, but the presence of numerical coefficients other
than 0 and 1, as well as the inverse operations of subtraction and division,
necessarily complicates his procedure. And he does not present the matter,
of solutions in the form in which we should expect to find it but in a more
complicated fashion which nevertheless gives equivalent results. We have
now to trace the procedures of interpretation, reduction, etc. by which
Boole obviates the difficulties of his algebra which have been mentioned.
The simplest form of equation is that in which a developed function,
of any number of variables, is equated to 0, as:
Ax + B(l -x) = 0, or,
Axy + B.r(l - y) + C(l - x)y + D(l - .r)(l - y) = 0, etc.
It is an important property of such equations that, since the product of
any two constituents in a developed function is 0, any such equation gives
any one of its constituents, whose coefficient does not vanish in the develop
ment, = 0. For example, if we multiply the second of the equations given
by xy, all constituents after the first will vanish, giving Axy = 0. Whence
we shall have xy = 0.
Any equation in which a developed function is equated to 1 may be
reduced to the form in which one member is 0 by the law; If V = 1,
1 -- V = 0.
The more general form of equation is that in which some " logical
symbol", w, is equated to some function of such symbols. For example,
suppose x = yz, and it be desired to interpret z as a function of x and y.
= yz gives z = x/y- but this form is not in terpre table. We shall, then,
x
The Development of Symbolic Logic 61
develop x/y by the law
).(! - x)y
+/(0,0)-(1 -x)(l -y)
By this law:
3*
If z = -, then
» = x+-x(l 7) + 0(1 + ?
0 0
These fractional coefficients represent the sole necessary difference of Boole's
methods from those at present familiar— a difference due to the presence
of division in his system. Because any function can always be de
veloped, and the difference between any two developed functions, of the
same variables, is always confined to the coefficients. If, then, we can
interpret and successfully deal with such fractional coefficients, one of the
difficulties of Boole's system is removed.
The fraction 0/0 is indeterminate, and this suggests that a proper inter
pretation of the coefficient 0/0 would be to regard it as indicating an unde
termined portion of the class whose coefficient it is. This interpretation
may be corroborated by considering the symbolic interpretation of "All
x is ?/", which is x(l — y) = 0.
If x(l — y) = 0, then x — xy = 0 and x = xy.
Whence y = x/x.
Developing x/x, we have y = x + - (1 — x).
If "All x is ?/", the class y is made up of the class x plus an undetermined
portion of the class not-.r. Whence 0/0 is equivalent to an arbitrary
parameter v, which should be interpreted as "an undetermined portion of"
or as "All, some, or none of".
The coefficient 1/0 belongs to the general class of symbols which do not
obey the "index law", x~ ••= .r, or its equivalent, the "law of duality",
.r(l — x) = 0. At least Boole says it belongs to this class, though the
numerical properties of 1/0 would, in fact, depend upon laws which do not
belong to Boole's system. But in any case, 1/0 belongs with the class of
such coefficients so far as its logical interpretation goes. Any constituent of a
developed function which does not satisfy the index law must be separately
equated to 0. Suppose that in any equation
w = At + P
62 A Survey of Symbolic Logic
w be a " logical symbol", and t be a constituent of a developed function
whose coefficient A does not satisfy the index law, A2 = A. And let P
be the sum of the remaining constituents whose coefficients do satisfy this
law. Then
w2 = w, t* = t, and P2 = P
Since the product of any two constituents of a development is 0,
w2 = (At+P)2 = A2t2 + P2
Hence w = A2t+P
Subtracting this from the original equation,
(A - A2)t = 0 = A(l - A)t
Hence since A(l - A) 4= 0, t = 0
Hence any equation of the form
w = P+OQ+ IR+IS
is equivalent to the two equations
w = P + vR and S = 0
which together represent its complete solution.
It will be noted that a fraction, in Boole's algebra, is always an am
biguous function. Hence the division operation must never be performed:
the value of a fraction is to be determined by the law of development only,
except for the numerical coefficients, which are elsewhere discussed. We
have already remarked that ax = bx does not give a = b, because .r may
have the value 0. But we may transform ax = bx into a = bx/x and
determine this fraction by the law
f(b,x) = /(!, l).fcr+/(l,0).&(l -*)+/<) !•! - bx
We shall then have
bx 0
a= -== bx * 5 (1 ~ ^
and this is not, in general, equivalent to 6. Replacing 0/0 by indeterminate
coefficients, v and w, this gives us,
If ax = bx, then
a =
- x)
The Development of Symbolic Logic 63
And this result is always valid. Suppose, for example, the logical equation
rational men = married men
and suppose we wish to discover who are the rational beings. Our equation
will not give us
rational = married
but instead
rational = married men + v married non-men + w - non-married non-men
That is, our hypothesis is satisfied if the class "rational beings" consist of
the married men together with any portion (which may be null) of the
class "married women" and a similarly undetermined portion of the class
"unmarried women".
If we consider Boole's system as an algebra of 0 and 1, and the fact that
for any fraction, x/y,
x 1 0
--a*+6*(l-»)+6(l -*)(!-„)
we shall find, by investigating the cases
(1) x = 1 and y = 1; (2) x = 1, y - 0; (3) x = 0, y = 1;
and (4) x = 0, y = 0,
that it requires these three possible cases:
. 0
5 - «
Or, to speak more accurately, it requires that 0/0 be an ambiguous function
susceptible of the values 0 and 1.
Since there are, in general, only four possible coefficients, 1, 0, 0/0, and
such as do not obey the index law, of which 1/0 is a special case, this means
that any equation can be interpreted, and the difficulty due to the presence
of an uninterpretable division operation in the system has disappeared.
And any equation can be solved for any "logical symbol" .r, by trans
ferring all other terms to the opposite side of the equation, by subtraction
or division or both, and developing that side of the equation.
Any equation may be put in the form in which one member is 0 by
64 A Survey of Symbolic Logic
bringing all the terms to one side. When this is done, and the equation
fully expanded, all the coefficients which do not vanish may be changed to
unity, except such as already have that value. Boole calls this a "rule of
interpretation".102 Its validity follows from two considerations: (1) Any
constituent of an equation with one member 0, whose coefficient does not
vanish in development, may be separately equated to 0; (2) the sum of
the constituents thus separately equated to 0 will be an equation with one
member 0 in which each coefficient will be unity.
Negative coefficients may be eliminated by squaring both sides of any
equation in which they appear. The " logical symbols" in any function
are not altered by squaring, and any expression of the form (1 — x), where
x is a "logical symbol", is not altered, since it can have only the values
0 and 1. Hence no constituent is altered, except that its coefficient may be
altered. And any negative coefficient will be made positive. Xo new
terms will be introduced by squaring, since the product of any two terms
of a developed function is always null. Hence the only change effected
by squaring any developed function is the alteration of any negative coef
ficients into positive. Their actual numerical value is of no consequence,
because coefficients other than 1 can be dealt with by the method described
above.
For reducing any two or more equations to a single equation, Boole
first proposed the "method of indeterminate multipliers",103 by which
each equation, after the first, is multiplied by an arbitrary constant and the
equations then added. But these indeterminate multipliers complicate the
process of elimination, and the method is, as he afterward recognized, an
inferior one. More simply, such equations may be reduced, by the methods
just described, to the form in which one member is 0, and each coefficient
is 1. They may then be simply added; the resulting equation will combine
the logical significance of the equations added.
Any "logical symbol" which is not wanted in an equation may be
eliminated by the method which is familiar to all students of symbolic
logic. To eliminate x, the equation is reduced to the form
Ax + B(l - a-) = 0
The result of elimination will be104
AB = 0
102 L. of T., p. 90.
103 See Math. An. of Logic, pp. 78-81; L. of T., pp. 115-120.
104 See L. of T., p. 101. We do not pause upon this or other matters which will be
entirely clear to those who understand current theory.
The Development of Symbolic Logic 65
By these methods, the difference between Boole's algebra and the classic
algebra of logic which grew out of it is reduced to a minimum. Any logical
results obtainable by the use of the classic algebra may also be got by
Boole's procedures. The difference is solely one of ease and mathematical
neatness in the method. Two important laws of the classic algebra which
do not appear among Boole's principles are:
(1) x + x = x, and (2) x = x + xy
These seem to be inconsistent with the Boolean meaning of + ; the first of
them does not hold for x =-- 1; the second does not hold for x = 1, y = 1.
But although they do not belong to Boole's system as an abstract algebra,
the methods of reduction which have been discussed will always give x in
place of x + x or of x + xy, in any equation in which these appear. The
expansion of x + x gives 2x-, the expansion of x + xy gives 2xy + x(l — y).
By the method for dealing with coefficients other than unity, 2x may be
replaced in the equation by x, and 2xy + x(l - y) by xy + x(l - y), which
is equal to x.
The methods of applying the algebra to the relations of logical classes
should now be sufficiently clear. The application to propositions is made
by the familiar device of correlating the " logical symbol", x, with the
times when some proposition, A', is true, xy will represent the times when
X and r are both true; x(l — y), the times when A" is true and 1" is false,
and so on. Congruent with the meaning of + , x + y will represent the
times when either A" or 1' (but not both) is true. In ord-r to symbolize
the times when A' or Y or both are true, we must write x + (1 — x)y, or
xy + x(l - y)+ (1 - x)y. 1, the " universe", will represent "all times" or
"always"; and 0 will be "no time" or "never", x = 1 will represent
"A' is always true"; x = 0 or (i - .r) == 1, "A" is never true, is always
false".
Just as there is, with Boole, no symbol for the inclusion relation of
classes, so there is no symbol for the implication relation of propositions.
For classes, "All A" is T" or "A" is contained in Y" becomes x = vy. Cor
respondingly, "All times when A" is true are times when Y is true" or "If
A' then ]'" or "A' implies F" is x = vy. x = y will mean, "The times
when A" is true and the times when Y is true are the same" or "A' implies
Y and Y implies X'\
The entire procedure for "secondary propositions" is summarized as
follows: 105
105 L. of T., p. 178.
G
66 A Survey of Symbolic Logic
"Rule. — Express symbolically the given propositions. . . .
"Eliminate separately from each equation in which it is found the
indefinite symbol ».
"Eliminate the remaining symbols which it is desired to banish from
the final solution: always before elimination, reducing to a single equation
those equations in which the symbol or symbols to be eliminated are found.
Collect the resulting equations into a single equation [one member of which
isO], V = 0.
"Then proceed according to the particular form in which it is desired
to express the final relation, as
1st. If in the form of a denial, or system of denials, develop the
function V, and equate to 0 all those constituents whose coefficients do
not vanish.
2ndly. If in the form of a disjunctive proposition, equate to 1 the
sum of those constituents whose coefficients vanish.
3rdly. If in the form of a conditional proposition having a simple
element, as x or 1 — x, for its antecedent, determine the algebraic
expression of that element, and develop that expression.
4thly. If in the form of a conditional proposition having a com
pound expression, as xy, xy + (1 — .r)(l — y), etc., for its antecedent,
equate that expression to a new symbol t, and determine t as a developed
function of the symbols which are to appear in the consequent. . . .
5thly. ... If it only be desired to ascertain whether a particular
elementary proposition x is true or false, we must eliminate all the
symbols but x\ then the equation x = 1 will indicate that the proposi
tion is true, x = 0 that it is false, 0 = 0 that the premises are insuf
ficient to determine whether it is true or false."
It is a curious fact that the one obvious law of an algebra of 0 and 1
which Boole does not assume is exactly the law which would have limited
the logical interpretation of his algebra to propositions. The law
If x 4= 1, x = 0 and if x 4= 0, x = 1
is exactly the principle which his successors added to his system when it
is to be considered as a calculus of propositions. This principle would have
made his system completely inapplicable to logical classes.
For propositions, this principle means, " If x is not true, then x is false,
and if x is not false, it is true". But careful attention to Boole's interpre
tation for "propositions" shows that in his system x = 0 should be inter-
The Development of Symbolic Logic 67
preted "x is false at all times (or in all cases)", and x = 1 should be in
terpreted "x is true at all times". This reveals that fact that what Boole
calls " propositions" are what we should now call " prepositional functions ",
that is, statements which may be true under some circumstances and false
under others. The limitation put upon what we now call "propositions"—
namely that they must be absolutely determinate, and hence simply true
or false— does not belong to Boole's system. And his treatment of "prepo
sitional symbols" in the application of the algebra to probability theory
gives them the character of " prepositional functions" rather than of our
absolutely determinate propositions.
The last one hundred and seventy-five pages of the Laws of Thought
are devoted to an application of the algebra to the solution of problems in
probabilities.106 This application amounts to the inyention of a new
method— a method whereby any logical analysis involved in the problem
is performed as automatically as the purely mathematical operations.
We can make this provisionally clear by a single illustration :
All the objects belonging to a certain collection are classified in three
ways— as ^4's or not, as B's or not, and as C's or not. It is then found
that (1) a fraction m/n of the .4's are also B's and (2) the C's consist of the
A's which are not B's together with the B's which are not A's.
Required: the probability that if one of the A's be taken at random,
it will also be a C.
By premise (2)
C = A(l - B) + (l - A)B
Since A, B, and C are "logical symbols", A2 = A and A (I - A) = 0.
Hence, AC = A* (I - B) + .1(1 - A)B = A(l - B).
The A's which are also C's are identical with the A's which are not B's.
Thus the probability that a given A is also a C is exactly the probability
that it is not a B ; or by premise (1 ), 1 — m/n, which is the required solution.
In any problem concerning probabilities, there are usually two sorts of
difficulties, the purely mathematical ones, and those involved in the logical
analysis of the situation upon which the probability in question depends.
The methods of Boole's algebra provide a means for expressing the relations
of classes, or events, given in the data, and then transforming these logical
106 Chap. 16 ff. See also the Keith Prize essay "On the Application of the Theory of
Probabilities to the Question of the Combination of Testimonies or Judgments", Trans.
Roy. Soc. Edinburgh, xxi, 597 ff. Also a series of articles in Phil. Mag., 1851-54 (see
Bibl). An article on the related topic "Of Propositions Numerically Definite" appeared
posthumously; Carafe. Phil. Trans., xi, 396-411.
68 A Survey of Symbolic Logic
equations so as to express the class which the quaesitum concerns as a func
tion of the other classes involved. It thus affords a method for untangling
the problem — often the most difficult part of the solution.
The parallelism between the logical relations of classes as expressed in
Boole's algebra and the corresponding probabilities, numerically expressed,
is striking. Suppose x represent the class of cases (in a given total) in which
the event X occurs — or those which "are favorable to" the occurrence of
A".107 And let p be the probability, numerically expressed, of the event X.
The total class of cases will constitute the logical "universe", or 1; the
null class will be 0. Thus, if x = 1 — if all the cases are favorable to X—
then p = 1 — the probability of X is "certainty". If x = 0, then p — 0.
Further, the class of cases in which X does not occur, will be expressed by
1 — x] the probability that X will not occur is the numerical 1 — p. Also,
x + (1 — x) = 1 and p + (1 — p) = 1.
This parallelism extends likewise to the combinations of two or more
events. If x represent the class of cases in which X occurs, and y the class
of cases in which Y occurs, then xy will be the class of cases in which X
and 1' both occur; x(l — y), the cases in which X occurs without Y;
(1 — x)y, the cases in which Y occurs without X; (1 — x)(l — y), the
cases in which neither occurs; x(l — y~) + y(l — x), the cases in which
X or Y occurs but not both, and so on. Suppose that X and Y are " simple "
and "independent" events, and let p be the probability of X, q the prob
ability of Y. Then we have:
Combination of events Corresponding probabilities
expressed in Boole's algebra numerically expressed
xy pq
x(l - y) p(l - q)
(1 - x)y (1 - q)p
(1 -*)(! -y) (1 -p)(l -q)
x(l - y) + (1 - x)y p(l - q) + (1 - p)q
Etc. etc.
In fact, this parallelism is complete, and the following rule can be
formulated: 108
107 Boole prefers to consider x as representing the times when a certain proposition,
asserting an occurrence, will be true. But this interpretation comes to exactly the same
thing.
108 L. ofT., p. 258.
The Development of Symbolic Logic 69
"If p, q, r, . . . are the respective probabilities of unconditioned simple
events, x, y, z, . . . , the probability of any compound event V will be [V],
this function [V] being formed by changing, in the function F, the symbols
x, y, z, . . . into p, q, r, . . . .
"According to the well-known law of Pascal, the probability that if
the event V occur, the event V will occur with it, is expressed by a fraction
whose numerator is the probability of the joint occurrence of V and V,
and whose denominator is the probability of the occurrence of V. We can
then extend the rule just given to such cases:
"The probability that if the event V occur, any other event V will
[V V]
also occur, will be -— - , where [V V] denotes the result obtained by
multiplying together the logical functions V and V, and changing in the
result x, y, z, . . . into p, q, r, . . . ."
The inverse problem of finding the absolute probability of an event
when its probability upon a given condition is known can also be solved.
Given : The probabilities of simple events x, y, z, . . . are respectively
p, q, r, . . . when a certain condition V is satisfied.
To determine: the absolute probabilities /, m, n, . . . of x, y, z, . . . .
By the rule just given,
U'f] [yV] [zV]
[''I > W\ >
And the number of such equations will be equal to the number of unknowns,
I, m, 7i, ... to be determined.109 The determination of any logical expres
sion of the form xV is peculiarly simple since the product of x into any
developed function V is the sum of those constituents of V which contain x
as a factor. For example:
if V = xyz + x(l - y)z + (1 - x)y(l - z) + (1 - *)(! - y)z,
xV = xyz + x(l - y)z
yV = xyz+(l -x)y(l - z)
zV = xyz + x(l - y)z + (1 - .r)(l - y)z
Thus any equation of the form
109 On certain difficulties in this connection, and their solution, see Cayley, "On a
Question in the Theory of Probability" (with discussion by Boole), Phil. Mag., Ser. iv,
xxm (1862), 352-65, and Boole, "On a General Method in the Theory of Probabilities",
ibid., xxv (1863), 313-17.
70 A Survey of Symbolic Logic
is readily determined as a numerical equation. Boole gives the following
example in illustration: no
" Suppose that in the drawings of balls from an urn, attention had only
been paid to those cases in which the balls drawn were either of a particular
color, 'white/ or of a particular composition, ' marble/ or were marked by
both of these characters, no record having been kept of those cases in which
a ball which was neither white nor of marble had been drawn. Let it then
have been found, that whenever the supposed condition was satisfied, there
was a probability p that a white ball would be drawn, and a probability q
that a marble ball would be drawn: and from these data alone let it be
required to find the probability m that in the next drawing, without refer
ence at all to the condition above mentioned, a white ball will be drawn;
also a probability n that a marble ball will be drawn.
"Here if x represent the drawing of a white ball, y that of a marble
ball, the condition F will be represented by the logical function
xy + x(l - y) + (1 - x)y
Hence we have
xV = xy + x(l — y) = or
yV = xy + (1 - x)y = y
Whence
[xV] = m, [yV] = n
and the final equations of the problem are
m
= P
mn + m(l — n) + (1 — m)n
n
mn + m(l — n) + (1 — m)n
= q
from which we find
p + q — p + q —
m = - n = -
q p
... To meet a possible objection, I here remark that the above reasoning
does not require that the drawings of a white and a marble ball should be
independent, in virtue of the physical constitution of the balls.
"In general, the probabilities of any system of independent events
being given, the probability of any event X may be determined by finding a
logical equation of the form
x = A+OB + ^C + ID
110 L. of T., p. 262. I have slightly altered the illustration by a change of letters.
The Development of Symbolic Logic 71
where A, B, C, and D are functions of the symbols of the other events.
As has already been shown, this is the general type of the logical equation,
and its interpretation is given by
x = A + vC, where v is arbitrary and
D = 0
By the properties of constituents, we have also the equation,
A+B+C+D = 1
and, since D = 0,
A+B + C = 1
A + B + C thus gives the 'universe ' of the events in question, and the prob
abilities given in the data are to be interpreted as conditioned by A + B + C
= 1, since D = 0 is the condition of the solution x = A + vC. If the given
probability of some event S is p, of T is q, etc., then the supposed 'absolute '
probabilities of S, T, etc., may be determined by the method which has
been described. Let V = A+B+C, then
[sV] [IV]
W]-=P, M-?,
where [sV], [tV], etc. are the "absolute probabilities" sought. These,
being determined, may be substituted in the equation
[A + vC]
Prob. w = — \yr~
which will furnish the required solution.
"The term vC will appear only in cases where the data are insufficient
to determine the probability sought. Where it does appear, the limits of
this probability may be determined by giving v the limiting values, 0 and 1.
Thus
\A]
Lower limit of Prob. w = -—
[A + cr
Upper limit
With the detail of this method, and with the theoretical difficulties of
its application and interpretation, we need not here concern ourselves.
Suffice it to say that, with certain modifications, it is an entirely workable
method and seems to possess certain marked advantages over those more
generally in use. It is a matter of surprise that this immediately useful
application of symbolic logic has been so generally overlooked.
72 A Survey of Symbolic Logic
VI. JEVONS
It has been shown that Boole's "calculus of logic" is not so much a
system of logic as an algebra of the numbers 0 and 1, some of whose ex
pressions are capable of simple interpretation as relations of logical classes,
or propositions, and some of whose transformations represent processes of
reasoning. If the entire algebra can, with sufficient ingenuity, be inter
preted as a system of logic, still Boole himself failed to recognize this fact,
and this indicates the difficulty and unnaturalness of some parts of this
interpretation.
Jevons111 pointed a way to the simplification of Boole's algebra, dis
carding those expressions which have no obvious interpretation in logic,
and laying down a procedure which is just as general and is, in important
respects, superior. In his first book on this subject, Jevons says: l12
"So long as Professor Boole's system of mathematical logic was capable
of giving results beyond the power of any other system, it had in this fact
an impregnable stronghold. Those who were not prepared to draw the
same inferences in some other manner could not quarrel with the manner
of Professor Boole. But if it be true that the system of the foregoing
chapters is of equal power with Professor Boole's system, the case is altered.
There are now two systems of notation, giving the same formal results, one
of which gives them with self-evident force and meaning, the other by dark
and symbolic processes. The burden of proof is shifted, and it must be
for the author or supporters of the dark system to show that it is in some
way superior to the evident system."
He sums up the advantages of his system, compared with Boole's, as
follows:113
"1. Even- process is of self-evident nature and force, and governed by
laws as simple and primary as those of Euclid's axioms.
"2. The process is infallible, and gives us no uninterpretable or anom
alous results.
"3. The inferences may be drawn with far less labor than in Professor
Boole's system, which generally requires a separate computation and
development for each inference."
111 William Stanley Jevons (1835-1882), B.A., M.A. (London), logician and economist;
professor of logic and mental and moral philosophy and Cobden professor of political
economy in Owens College, Manchester, 1866-76; professor of political economy, Uni
versity College, London, 1876-80.
112 Pure Logic, or the Logic of Quality apart from Quantity, p. 75.
113 Ibid., p. 74.
The Development of Symbolic Logic 73
The third of these observations is not entirely warranted. Jevons
unduly restricts the operations and methods of Boole in such wise that
his own procedure is often cumbersome and tedious where Boole's would
be expeditious. Yet the honor of first pointing out the simplifications
which have since been generally adopted in the algebra of logic belongs to
Jevons.
He discards Boole's inverse operations, a — b and a/6, and he interprets
the sum of a and b as " either a or 6, where a and b are not necessarily
exclusive classes". (We shall symbolize this relation by a + b: Jevons has
A + B or A • I -J^.)114 Thus, for Jevons, a + a = a, whereas for Boole a + a
is not interpretable as any relation of logical classes, and if it be taken as
an expression in the algebra of 0 and 1, it obeys the usual arithmetical laws,
so that a + a = 2a. As has been indicated, this is a source of much awk
ward procedure in Boole's system. The law a + a = a eliminates numerical
coefficients, other than 0 and 1, and this is a most important simplification.
Jevons supposes that the fundamental difference between himself and
Boole is that Boole's system, being mathematical, is a calculus of things
taken in their logical extension, while his own system, being "pure logic",
is a calculus of terms in intension. It is true that mathematics requires
that classes be taken in extension, but it is not true that the calculus of
logic either requires or derives important advantage from the point of view
of intension. Since Jevons's system can be interpreted in extension without
the slightest difficulty, we shall ignore this supposed difference.
The fundamental ideas of the system are as follows :
(1) a b denotes that which is both a and b, or (in intension) the sum
of the meanings of the two terms combined.
(2) a + b denotes that which is either a or b or both, or (in intension) a
term with one of two meanings.115
(3) a = b a is identical with 6, or (in intension) a means the same as b.
(4) -b Not-6, the negative of b, symbolized in Boole's system by
1 - 6.116
(o) 0 According to Jevons, 0 indicates that which is contradictory or
"excluded from thought". He prefers it to appear as a factor rather than
114 A + B in Pure Logic; A' \ ' B in the other papers. (See Bibl.)
115 Jevons would add "but it is not known which". (See Pure Logic, p. 25.) But
this is hardly correct; it makes no difference if it is known which, since the meaning of
a + 6 does not depend on the state of our knowledge. Perhaps a better qualification would
be "but it is not specified which".
113 Jevons uses capital roman letters for positive terms and the corresponding small
italics for their negatives. Following De Morgan, he calls A and a "contrary" terms.
74 A Survey of Symbolic Logic
by itself.117 The meaning given is a proper interpretation of the symbol in
intension. Its meaning in extension is the null-class or "nothing", as with
Boole.
Jevons does not use any symbol for the "universe", but writes out the
"logical alphabet". This "logical alphabet", for any number n of ele
ments, a, 6, c, . . . , consists of the 2n terms which, in Boole's system, form
the constituents of the expansion of 1. Thus, for two elements, a and 6,
the "logical alphabet" consists of a b, a-b, -ab, and -a -b. For three
terms, xy y, z, it consists of x y z, x y -z, x -y z, x -y -z, x y -z, -x -y z, and
-x -y -Zf Jevons usually writes these in a column instead of adding them
and putting the sum == 1. Thus the absence of 1 from his system is simply
a whim and represents no real difference from Boole's procedure.
The fundamental laws of the system of Jevons are as follows:
(1) If a = b and b = c} then a = c.
(2) a b = b a.
(3) a a = a.
(4) a -a = 0.
(5) a + b = b + a.
(6) a + a = a.
(7) a + 0 = a. This law is made use of but is not stated.
(8) a(b + c) = a b + a c and (a + b)(c + d) =ac + ad + bc + bd.
(9) a + a b = a. This law, since called the "law of absorption", allows
a direct simplification which is not possible in Boole. Its analogue for
multiplication
a(a+ b) = a
follows from (8), (3), and (9). The law of absorption extends to any
number of terms, so that we have also
a + ab + ac + ab d+ . . . = a
(10) a = a(b + -b)(c + -c) .... This is the rule for the expansion of
any term, a, with reference to any other terms, b, c, etc. For three terms
it gives us
a = a(b + -b) (c + -c) = abc + ab-c + a-bc + a-b-c
This expansion is identical with that which appears in Boole's system, except
for the different meaning of + . But the product of any two terms of such
an expansion will always have a factor of the form a -a, and hence, by (4),
will be null. Thus the terms of any expansion will always represent classes
117 See Pure Logic, pp. 31-33.
The Development of Symbolic Logic 75
which are mutually exclusive. This accounts for the fact that, in spite of
the different meaning of + , developed functions in Boole's system and in
Jevons's always have the same form.
(11) The "logical alphabet" is made up of any term plus its negative,
a + -a. It follows immediately from this and law (10) that the logical
alphabet for any number of terms, a, b, c, . . . , will be
and will have the character which we have described. It corresponds to
the expansion of 1 in Boole's system because it is a developed function and
its terms are mutually exclusive.
A procedure by which Jevons sets great store is the "substitution of
similars ", of a for b or b for a when a = b. Not only is this procedure valid
when the expressions in which a and b occur belong to the system, but it
holds good whatever the rational complex in which a and b stand. He
considers this the first principle of reasoning, more fundamental than
Aristotle's dictum de omni et nullo.11* In this he is undoubtedly correct,
and yet there is another principle, which underlies Aristotle's dictum, which
is equally fundamental— the substitution for variables of values of these
variables. And this procedure is not reducible to any substitution of
equivalents.
The only copulative relation in the system is [ = ]; hence the expression
of simple logical propositions is substantially the same as with Boole:
All a is b: a = a b
No a is b : a = a -b
Some a is b: c a = c ab or U a = V a b
"U" is used to suggest " Unknown".
The methods of working with this calculus are in some respects simpler
than Boole's, in some respects more cumbersome. But, as Jevons claims,
they are obvious while Boole's are not. Eliminations are of two sorts,
"intrinsic" and " extrinsic". Intrinsic eliminations may be performed by
substituting for any part of one member of an equation the whole of the
other. Thus from a = b c d, we get
a = a c d = ab d = a c = a d
This rule follows from the principles a a = a, a b = b a, and if a = b,
ac = be. For example
If a = b c d
a-a = bed-bed = bb-cc-dd = bcd-d = ad.
118 See Substitution of Similars, passim.
76 A Survey of Symbolic Logic
Also, in cases where a factor or a term of the form a(b + -6), or of the form
a -a, is involved, eliminations may be performed by the rules a(b + -b) = a
and a -a = 0.
Extrinsic elimination is that simplification or "solution" of equations
which may occur when two or more are united. Jevons does not add or
multiply such equations but uses them as a basis for striking out terms in
the same "logical alphabet".
This method is equivalent, in terms of current procedures, to first
forming the expansion of 1 (which contains the terms of the logical alphabet)
and then putting any equations given in the form in which one member
is 0 and "subtracting" them from the expansion of 1. But Jevons did not
hit upon the current procedures. His own is described thus: 119
"1. Any premises being given, form a combination containing every
term involved therein. Change successively each simple term of this
into its contrary [negative], so as to form all the possible combinations of
the simple terms and their contraries. [E. g., if a, b, and c are involved,
form the "logical alphabet" of all the terms in the expansion of
(a + -a)(6 + -&)(c + -c).]
"2. Combine successively each such combination [or term, as a be,]
with both members of a premise. When the combination forms a con
tradiction [an expression having a factor of the form (a -a)] with neither
side of a premise, call it an included subject of the premise; when it forms a
contradiction with both sides, call it an excluded subject of the premise;
when it forms a contradiction with one side only, call it a contradictory com
bination or subject, and strike it out.
"We may call an included or excluded subject a possible subject as
distinguished from a contradictory combination or impossible subject.
"3. Perform the same process with each premise. Then a combination
is an included subject of a series of premises, when it is an included subject
of any one; it is a contradictory subject when it is a contradictory subject
of any one; it is an excluded subject when it is an excluded subject of
every premise.
"4. The expression of any term [as a or b] involved in the premises
consists of all the included and excluded subjects containing the term,
treated as alternatives [in the relation + ].
"5. Such expressions may be simplified by reducing all dual terms [of
119 Pure Logic, pp. 44-46.
The Development of Symbolic Logic 77
the form a(b + -b)], and by intrinsic elimination of all terms not required
in the expression.
" 6. When it is observed that the expression of a term contains a com
bination which would not occur in the expression of any contrary of that
term, we may eliminate the part of the combination common to the term
and its expression. . . .
"7. Unless each term of the premises and the contrary of each appear
in one or other of the possible subjects, the premises must be deemed in
consistent or contradictory. Hence there must always remain at least two
possible subjects.
" Required by the above process the inferences of the premise a = b c.
"The possible combinations of the terms a, b, c, and their contraries
are as given [in the column at the left, which is, for this case, the 'logical
alphabet']. Each of these being combined with both sides of the premise,
we have the following results:
abc ale = ab c abc included subject
ab-c ab-c = a b c -c = 0 ab -c contradiction
a-b c a-b c = ab-bc = 0 a -b c contradiction
a -b -c a-b -c = a b -b c -c = 0 a -b -c contradiction
-abc 0 = a -a b c = -a b c -abc contradiction
-ab-c 0 = a -a b -c = -ab c -c = 0 -a b -c excluded subject
-a -be 0 = a -a -b c = -ab-b c =0 -a -b c excluded subject
-a -b -c 0 = a -a -b -c = -ab -b c -c = 0 -a -b -c excluded subject
"It appears, then, that the four combinations ab-c to -abc are to
be struck out, and only the rest retained as possible subjects. Suppose we
now require an expression for the term -b as inferred from the premise
a = b c. Select from the included and excluded subjects such as contain -6,
namely -a -b c and -a -b -c.
"Then -b = -a -b c + -a -b -c, but as -a c occurs only with -b, and
not with 6, its contrary, we may, by Rule 0, eliminate -b from -a-b c',
hence -b = -a c + -a -b -c."
This method resembles nothing so much as solution by means of the
Venn diagrams (to be explained in Chapter III). The "logical alphabet"
is a list of the different compartments in such a diagram ; those marked
"contradiction" are the ones which would be struck out in the diagram by
transforming the equations given into the form in which one member is 0.
78 A Survey of Symbolic Logic
The advantage which Jevons claims for his method, apart from its obvious
ness, — namely, that the solutions for different terms do not require to be
separately performed, — is also an advantage of the diagram, which exhibits
all the possibilities at once.
If any problem be worked out by this method of Jevons and also that
of Boole, it will be found that the comparison is as follows: The " logical
alphabet" consists of the terms which when added give 1, or the universe.
Any term marked " contradiction " will, by Boole's method, have the coef
ficient 0 or 1/0; any term marked "included subject" will have the coef
ficient 1; any marked "excluded subject" will have the coefficient 0/0, or v
where v is arbitrary. If, then, we remember that, according to Boole,
terms with the coefficient 1/0 are equated to 0 and thus eliminated, we
see that the two methods give substantially the same results. The single
important difference is in Boole's favor: the method of Jevons does not
distinguish decisively between the coefficients 1 and v. If, for example,
the procedure of Jevons gives x = x -y z, Boole's will give either x = -y z
or x = v -y z.
One further, rather obvious, principle may be mentioned : 12°
Any subject of a proposition remains an included, excluded, or con
tradictory subject, after combination with any unrelated terms. This
means simply that, in any problem, the value of a term remains its value
as a factor when the term is multiplied by any new terms which may be
introduced into the problem. In a problem involving a, 6, and c, let
a -be be a "contradictory" term. Then if x be introduced, a-bcx and
a -b c -x will be "contradictory".
On the whole Jevons 's methods are likely to be tedious and have little
of mathematical nicety about them. Suppose, for example, we have three
equations involving altogether six terms. The "logical alphabet" will
consist of sixty-four members, each of which will have to be investigated
separately for each equation, making one hundred and ninety-two separate
operations. Jevons has emphasized his difference from Boole to the extent
of rejecting much that would better have been retained. It remained for
others, notably Mrs. Ladd-Franklin and Schroder, to accept Jevons 's
amended meaning of addition and its attendant advantages, yet retain
Boole's methods of development and similar methods of elimination and
solution. But Jevons should have credit for first noting the main clue to
this simplification — the laws a + a = a and a + a b = a.
120 Pure Logic, p. 48.
The Development of Symbolic Logic 79
VII. PEIRCE
The contributions of C. S. Peirce121 to symbolic logic are more numerous
and varied than those of any other writer— at least in the nineteenth
century. He understood how to profit by the work of his predecessors,
Boole and De Morgan, and built upon their foundations, and he anticipated
the most important procedures of his successors even when he did not
work them out himself. Again and again, one finds the clue to the most
recent developments in the writings of Peirce. These contributions may
be summed up under three heads: (1) He improved the algebra of Boole
by distinguishing the relations which are more characteristic of logical
classes (such as multiplication in Boole's algebra) from the relations which
are more closely related to arithmetical operations (such as subtraction and
division in Boole). The resulting algebra has certain advantages over the
system of Jevons because it retains the mathematical methods of develop
ment, transformation, elimination, and solution, and certain advantages
over the algebra of Boole because it distinguishes those operations and
relations which are always interpretable for logical classes. Also Peirce
introduced the "illative" relation, "is contained in", or "implies", into
symbolic logic. (2) Following the researches of De Morgan, he made
marked advance in the treatment of relations and relative terms. The
method of dealing with these is made more precise and "mathematical",
and the laws which govern them are related to those of Boole's algebra of
classes. Also the method of treating "some" and "all" propositions as
sums (2) and products (II) respectively of "propositions" containing
variables was here first introduced. This is the historic origin of "formal
implication" and all that has been built upon it in the more recent develop
ment of the logic of mathematics. (3) Like Leibniz, he conceived symbolic
logic to be the science of mathematical form in general, and did much to
revive the sense of logistic proper, as we have used that term. He worked
out in detail the derivation of various multiple algebras from the calculus
of relatives, and he improved Boole's method of applying symbolic logic to
problems in probability.
121 Charles Saylnders Peirce (1839-1914), son of Benjamin Peirce, the celebrated
mathematician, A.B. (Harvard, 1859), B.S. (Harvard, 1863), lecturer in logic at Johns
Hopkins, ^596- ?. For a number of years, Peirce was engaged in statistical researches
for the U. S. Coast Survey, and was at one time head of the Department of Weights and
Measures. His writings cover a wide variety of topics in the history of science, meta
physics, mathematics, astronomy, and chemistry. According to William James, his
articles on "Some Illustrations of the Science of Logic", Pop. Sri. Mo., 1877-78, are the
source of pragmatism.
80 A Survey of Symbolic Logic
We shall take up these contributions in the order named.
The improvement of the Boolian algebra is set forth mainly in the
brief article, "On an Improvement in Boole's Calculus of Logic",122 and in
two papers, "On the Algebra .of Logic".123
It will be remembered that Boole's calculus has four operations, or rela
tions : a + b indicates the class made up of the two mutually exclusive classes,
a and 6; [ — ] is the strict inverse of [ + ]t so that if x + b = a, then x = a — b;
ax b or a b denotes the class of those things which are common to a and 6;
and division is the strict inverse of multiplication, so that if x b = a, then
x = afb. These relations are not homogeneous in type. Boole's [+]
and [ — ] have properties which approximate closely those of arithmetical
addition and subtraction. If [n]x indicate the number of members of the
class x,
[n]a+ [n]b = [n](a + b)
because a and b are mutually exclusive classes, and every member of a
is a member of (a + b) and every member of b is a member of (a + b). This
relation, then, differs from arithmetical addition only by the fact that
a and b are not necessarily to be regarded as numbers or quantities. Simi
larly,
[n]a - [n]b = [n](a - b)
But in contrast to this, for Boole's a x b or a b,
[n]ax [n]b = [n] (a b)
will not hold except for 0 and 1 : this relation is not of the type of its arith
metical counterpart. And the same is true of its inverse, a/6. Thus, in
Boole's calculus, addition and subtraction are relations of the same type
as arithmetical addition and subtraction; but multiplication and division
are different in type from arithmetical multiplication and division.
Peirce rounds out the calculus of Boole by completing both sets of these
relations, adding multiplication and division of the arithmetical type, and
addition and subtraction of the non-arithmetical type.124 The general
character of these relations is as follows :
122 Proc. Amer. Acad., vn, 250-61. This paper will be referred to hereafter as "Boole's
Calculus ".
123 Amer. Jour. Math., in (1880), 15-57, and vn (1885), 180-202. These two papers
will be referred to hereafter as Alg. Log. 1880, and Alg. Log. 1885, respectively.
124 "Boole's Calculus," pp. 250-54.
The Development of Symbolic Logic 81
A. The " N on- Arithmetical" or Logical Relations
(1) a+ b denotes the class of those things which are either a's or 6's or
both.125
(2) The inverse of the above, a \-b, is such that if x + b = a, then
x — a \-b.
Since x and 6, in x + b, need not be mutually exclusive classes, a h b is
an ambiguous function. Suppose x + b = a and all b is x. Then
a \- b = x, and a h b = a
Thus a h b has an upper limit, a. But suppose that x + b = a and no b
isx. Then a \- b coincides with a — b (a which is not 6) — i. e.,
a f- b = x, and a \- b = a — b
Thus a f- b has a lower limit, a — 6, or (as we elsewhere symbolize it)
a -b. And in any case, a (- b is not interpretable unless all b is a, the
class b contained in the class a. We may summarize all these facts by
a h b = a -6 + v a b + [0] -« b
where v is undetermined, and [0] indicates that the term to which it is
prefixed must be null.
(3) a b denotes the class of those things which are both a's and 6's.
This is Boole's a b.
(4) The inverse of the preceding, a/b such that if b x = o, then x = a/6.
This is Boole's a/b.
a/b is an ambiguous function. Its upper limit is a + -b; its lower
limit, a.126 It is uninterpretable unless b is contained in a — i. e.,
a/b = a b + v -a -b + [0] a -b
B. The "Arithmetical" Relations
(5) a + b denotes the class of those things which are either a's or 6's,
where a and 6 are mutually exclusive classes. This is Boole's a + 6.
a + 6 = a -6 + -a 6 + [0] a 6
(6) The inverse of the preceding, a — b signifies the class "a which
is not 6". As has been mentioned, it coincides with the lower limit of a [-6.
(7) a X 6 and a -f- 6 are strictly analogous to the corresponding relations
125 peirce indicates the logical relations by putting a comma underneath the sign of
the relation: that which is both a and b is a, b.
126 Peirce indicates the upper limit by a : 6, the lower limit by a -5- 6.- These occur
only in the paper " Boole's Calculus".
7
82 A Survey of Symbolic Logic
of arithmetic. They have no such connection with the corresponding
" logical" relations as do a + b and a — b. Peirce does not use them
except in applying this system to probability theory.
For the "logical" relations, the following familiar laws are stated: m
a + a = a a a = a
a + b = b + a ab = b a
(a + b) + c = a + (b + c) (a b)c = a(b c)
(a + b)c = ac + bc a b + c = (a + c) (b + c)
The last two are derived from those which precede.
Peirce 's discussion of transformations and solutions in this system is
inadequate. Any sufficient account would carry us quite beyond what
he has given or suggested, and require our report to be longer than the
original paper. We shall be content to suggest ways in which the methods
of Boole's calculus can be extended to functions involving those relations
which do not appear in Boole. As has been pointed out, if any function
be developed by Boole's laws,
/(*) =/(!)- x +/(0).-z,
<f>(*> y} = <p(l, l)-xy + <p(l, Q)-x-y + ^(0, !)• -xy + <p(Q, 0)- -x-y,
Etc., etc.,
the terms on the right-hand side of these equations will always represent
mutually exclusive classes. That is to say, the difference between the
"logical" relation, +, and the "arithmetical" relation, +, here vanishes.
Thus any relation in this system of Peirce 's can be interpreted by developing
it according to the above laws, provided that we can interpret these rela
tions when they appear in the coefficients. And the correct interpretation
of these coefficients can always be discovered.
Developing the "logical" sum, x + y, we have,
x + y = (l + l)-xy+(l + Q)-x-y+(0+l)- -zy+(0 + 0)- -x-y
Comparing this with the meaning of x + y given above, we find that (1 + 1)
= 1, (1 + 0) == 1, (0+1) == 1, and (0+0) = 0.
Developing the "logical" difference, a \-b, we have
x \-y = (1 \-l)-xy+(l hO)-z-2/+(0 hi)- -x y + (0 f-0) . -x -y
Comparing this with the discussion of x \-y above, we see that (1 hi) is
equivalent to the undetermined coefficient 0; that (1 hO) = 1; that
127 "Boole's Calculus," pp. 250-53.
The Development of Symbolic Logic 83
(0 hi) is equivalent to [0], which indicates that the term to which it is
prefixed must be null, and that (0 |-0) = 0.
The interpretation of the "arithmetical" relations, X and ^, in coef
ficients of class-symbols is not to be attempted. These are of service only
in probability theory, where the related symbols are numerical in their
significance.
The reader does not require to be told that this system is too complicated
to be entirely satisfactory. In the " Description of a Notation for the
Logic of Relatives", all these relations except -f- are retained, but in later
papers we find only the " logical" relations, a + b and a b.
The relation of "inclusion in" or "being as small as" (which we shall
symbolize by c)128 appears for the first time in the "Description of a
Notation for the Logic of Relatives".129 Aside from its treatment of
relative terms and the use of the "arithmetical" relations, this monograph
gives the laws of the logic of classes almost identically as they stand in the
algebra of logic today. The following principles are stated.130
(1) If x cy and y cz, then x cz.
(2) If acb, then there is such a term x that a + x = b.
(3) If a c b, then there is such a term y that b y = a.
(4) If b x = a, then a c b.
(5) If a c 6, (c + a) c (c + b) .
(6) If a cb, c a cc b.
(7) If a cb, a c cb c.
(8) abca.
(9) xc(x + y).
(10) x + y = y + x.
(11) (x + y) + z = x + (y + z).
(12) x(y + z) = xy + xz.
(13) x y = y x.
(14) (xy)z = x(yz).
(15) x x = x.
(16) x -x = O.131
(17) x + -x = 1.
128 Peirce's symbol is — < which he explains as meaning the same as < but being sim
pler to write.
129 Memoirs of the Amer. Acad., n. s., ix (1867), 317-78.
130 "Description of a Notation for the Logic of Relatives," loc. cit., pp. 334-35, 338-39,
342.
131 In this paper, not-x is symbolized by n1, "different from every x, " or by <r~z.
84 A Survey of Symbolic Logic
(18) z + 0 = x.
(19) x + l = 1.
(20) <p(x) = ?(1) •*+*>(())• -a.
(21) <K.r) - [*(!)
(22) If <p(x) = 0,
(23) If <p(x) = 1,
The last of these gives the equation of condition and the elimination re
sultant for equations with one member 1. Boole had stated (22), which
is the corresponding law for equations with one member 0, but not (23).
Most of the above laws, beyond (9), had been stated either by Boole or
by Jevons. (1) to (9) are, of course, novel, since the relation c appears
here for the same time since Lambert.
Later papers state further properties of the relation c , notably, —
If x c y, then -y c -x.
And the methods of elimination and solution are given in terms of this
relation.132 Also, these papers extend the relation to propositions. In this
interpretation, Peirce reads x cy, "If a; is true, y is true," but he is well
aware of the difference between the meaning of x cy and usual significance
of "a; implies y". He says: 133
"It is stated above that this means 'if a: is true, y is true'. But this
meaning is greatly modified by the circumstance that only the actual state
of things is referred to. ... Now the peculiarity of the hypothetical
proposition [ordinarily expressed by 'if x is true, y is true'] is that it goes
out beyond the actual state of things and declares what would happen were
things other than they are or may be. The utility of this is that it puts
us in possession of a rule, say that 'if A is true, B is true', such that should
we afterward learn something of which we are now ignorant, namely that
A is true, then, by virtue of this rule, we shall find that we know something
else, namely, that B is true. [In contrast to this] . . . the proposition,
a c b, is true if a is false or if b is true, but is false if a is true while b is false.
. . . For example, we shall see that from -(x cy) [the negation of x cy]
we can infer z ex. This does not mean that because in the actual state of
things x is true and y false, therefore in every state of things either z is
false or x true ; but it does mean that in whatever state of things we find x
true and y false, in that state of things either z is false or x is true [since,
ex hypothesi, x is true anyway]."
132 Alg. Log. 1880, see esp. § 2.
133 Alg. Log. 1885, pp. 186-87.
The Development of Symbolic Logic 85
We now call this relation, x cy, " material implication," and the peculiar
theorems which are true of it are pretty well known. Peirce gives a number
of them. They will be intelligible if the reader remember that x c y means,
"The actual state of things is not one in which x is true and y false".
(1) arc (yea:). This is the familiar theorem: "A true proposition is
implied by any proposition".
(2) [(xcy) ex] ex. If "x implies y" implies that x is true, then x is
true.
(3) [(a: cy) ca] ex, where a is used in such a sense that (ar cy) c a
means that from x cy every proposition follows.
The difference between "material implication" and the more usual
meaning of "implies" is a difficult topic into which we need not go at this
time.134 But it is interesting to note that Peirce, who introduced the
relation, understood its limitations as some of his successors have not.
Other theorems in terms of this relation are:
(4) ar c x.
(5) [xc(ycz)] c [yc(xcz)].
(6) xc[(xcy)cy].
(7) (ar cy)c[(ycz)c (xcz)].. This is a fundamental law, since called
the "Principle of the Syllogism".
Peirce worked most extensively with the logic of relatives. His interest
here reflects a sense of the importance of relative terms in the analyses of
mathematics, and he anticipates to some extent the methods of such later
researches as those of Peano and of Principia Mathematica. To follow
his successive papers on this topic would probably result in complete con
fusion for the reader. Instead, we shall make three divisions of this entire
subject as treated by Peirce: (1) the modification and extension of De
Morgan's calculus of relatives by the introduction of a more "mathe
matical" symbolism — for the most part contained in the early paper,
"Description of a Notation for the Logic of Relatives"; (2) the calculus
of relations, expressed without the use of exponents and in a form which
makes it an extension of the Boolean algebra — a later development which
may be seen at its best in "The Logic of Relatives", Note B in the Studies
in Logic by members of Johns Hopkins University; and (3) the systematic
consideration of the theory of relatives, which is scattered throughout the
papers, but has almost complete continuity.
134 But see below, Chap, iv, Sect, i, and Chap, v, Sect. v.
86 A Survey of Symbolic Logic
The terms of the algebra of relatives may usually be regarded as simple
relative terms, such as "ancestor", "lover," etc. Since they are also class
names, they will obey all the laws of the logic of classes, which may be
taken for granted without further discussion. But relative terms have
additional properties which do not belong to non-relatives; and it is to
these that our attention must be given. If w signifies "woman" and s,
"servant," logic is concerned not only with such relations as s iv, the
"logical product" "servant woman", s + w, the "logical sum" "either
servant or woman (or both) ", and s cw, "the class 'servants' is contained
in the class ' women', — relations which belong to class-terms in general—
but also with the relations first symbolized by De Morgan, "servant of a
woman," "servant of every woman, " and "servant of none but women".
We may represent "servant of a woman" by s\w.U5 This is a kind of
"multiplication" relation. It is associative,
s\(l\w) = (s\l)\w
"Servant of a lover-of-a-woman " is " servant-of-a-lover of a woman".
Also, it is distributive with respect to the non-relative "addition" symbol
ized by + ,
s\(m + w) = s\m + s w
"Servant of either a man or a woman" is "servant of a man or servant of
a woman ". But it is not commutative : s\lis not l\s, " servant of a lover "
is not equivalent to "lover of a servant". To distinguish s w from s w,
or s x w — the class of those who are both servants and women — we shall
call s | w the relative product of s and w.
For "servant of every woman" Peirce proposed sw, and for "servant
of none but women" sw. As we shall see, this notation is suggested by
certain mathematical analogies. We may represent individual members
of the class w as Wly W2, W3, etc., and the class of all the Ws as Wl + W2
+ W3 + .... Remembering the interpretation of + , we may write
w = W1 + W2 + W, + . . .
and this means, "The class-term, w, denotes Wl or W2 or W3 or ...,"
that is, w denotes an unspecified member of the class of PF's. The servant
of a (some, any) woman is, then, s w.
sw =
"A woman" is either W} or W2 or W3, etc.; "servant of a woman" is either
135 Peirce's notation for this is s w, he uses s, w for the simple logical product.
The Development of Symbolic Logic 87
servant of Wl or servant of W2 or servant of PF3, etc. Similarly, "servant
of every woman" is servant of Wl and servant of Wz and servant of Wz,
etc. ; or remembering the interpretation of x ,
where, of course, s\ Wn represents the relative product, "s of Wn," and x
represents the non-relative logical product translated by "and". The
above can be more briefly symbolized, following the obvious mathematical
analogies,
w = 2 W
s\w = 2w(s | W)
sw = Hw(s\W)
Unless w represent a null class, we shall have
or j»c*w
The class "servants of every woman" is contained in the class "servants
of a woman". This law has numerous consequences, some of which are:
A lover of a servant of all women is a lover of a servant of a woman.
Kc (/]*)«
A lover of every servant-of-all-women stands to every woman in the rela
tion of lover-of-a-servant of hers (unless the class sw be null).
Is ' w c Is | w
A lover of every servant-of-a-woman stands to a (some) woman in the
relation of lover-of-a -servant of hers.
From the general principle,136
m\[Uxf(x)]cUf(m\f(x)]
136 The proof of this theorem is as follows :
a = abc ... +ab -c ... +a -b c ... + ...,
or a = a b c . . . + P, where P is the sum of the remaining terms.
Whence, if O represent any relation distributive with respect to + ,
mOa = mQabc . . . +mQP
Similarly, mOb = mQa be . . . + mOQ
mOc = mQa b c . . . + rnQR
Etc., etc.
Now let a, 6, c, etc., be respectively /(.Ci), f(x*), fM, etc., and multiply together all
the above equations. On the left side, we have
88 A Survey of Symbolic Logic
we have also,
^c*, or 8»cs»
A lover of a (some) servant-of-every-woman stands to every woman in the
relation of lover-of-a-servant of hers.
We have also the general formulae of inclusion,
If / c s, then lw c sw
and, If s c w, then lw c Is
The first of these means: If all lovers are servants, then a lover of every
woman is also a servant of every woman. The second means : If all servants
are women, then a lover of every woman is also a lover of every servant.
These laws are, of course, general. We have also: m
(l\s)\w = I (s\w)
— s x
The last of these is read: A lover of every person who is either a servant
or a woman is a lover of every servant and a lover of every woman. An
interesting law which remainds us of Lambert's "Newtonian formula" is,
(I + s) w = lw+ 2q(lw-<* x s9) + sw
One who is either-lover-or-servant of every woman, is either lover of every
woman or, for some portion q of the class women, is lover of every woman
except members of q and servant of every member of q, or, finally, is servant
of every woman. Peirce also gives this law in a form which approximates
even more closely the binomial theorem. The corresponding law for the
product is simpler,
(I xs)w = lw xsw
which is
On the right side, we have
(mOabc ...) + (raOP) + (mQQ) + (mOR) + . .., or (mQabc...)+K
where K is a sum of other terms.
But (mOabc ...} is ?nQ[f(x1) x f(x2) x /(z3) . . .], which is
mOHxf(x)
Hence [mOHf(x)]+K = Ux[mOf(x)].
Hence m O n /Or) c nx [m Of(x) ] .
Peirce does not prove this theorem, but illustrates it briefly for logical multiplication (see
"Description of a Notation", p. 346).
137 "Description of a Notation, p. 334.
The Development of Symbolic Logic 89
One who is both-lover-and-servant of every woman, is both a lover of every
woman and a servant of every woman.
Peirce introduces a fourth term, and summarizes in a diagram the inclu
sion relations obtained by extending the formulae already given.138 The
number of such inclusions, for four relatives, is somewhat more than one
hundred eighty. He challenges the reader to accomplish the precise
formulation of these by means of ordinary language and formal logic.
An s of none but members of w, Peirce symbolizes by 8w. He calls this
operation "backward involution", and relatives of the type aw he refers to
as " infinitesimal relatives", on account of an extended and difficult mathe
matical analogy which he presents.139 The laws of this relation are analo
gous to those of sw.
If s c w, then ls c lw
If all servants are women, then a lover of none but servants is lover of none
but women.
If / c s, then 8w c lw
If all lovers are servants, then a servant of none but women is a lover of
none but women.
i(*w) = ("OH;
The lovers of none but servants-of-none-but-women are the lovers-of-
servants of none but women.
i+*w = lw x sw
Those who are either-lovers-or-servants of none but women are those who
are lovers of none but women and servants of none but women.
•(w xv) = sw x sv
The servants of none but those who are both women and violinists are
those who are servants of none but women and servants of none but vio
linists.
<"«>MJC «')&'
Whoever is lover-of-a-servant of none but women is a lover-of-every-
servant of none but women.
l\'w c <*«>w
A lover of one who is servant to none but women is a lover-of-none-but-
servants to none but women.
ls w c l(s | w)
138 Ibid., p. 347.
139 Ibid., pp. 348 jf.
90 A Survey of Symbolic Logic
Whoever stands to a woman in the relation of lover-of-nothing-but-servants
of hers is a lover of nothing but servants of women.
The two kinds of involution are connected by the laws:
A lover of none but those who are servants of every woman is the same as
one who stands to every woman in the relation of a lover of none but
servants of hers.
is = -/ -» HO
Lover of none but servants is non-lover of every non-servant. It appears
from this last that x and xy are connected through negation :
-(Is) = -l\s, Not a lover of every servant is non-lover of a servant.
-O) = l\-s, Not a lover of none but servants is lover of a non-
servant.
l-s = ~(l\s} = -Is, A lover of none but non-servants is one who is
not lover-of-a-servant, a non-lover of every servant.
-is = -(-/| -s) = l~s, A non-lover of none but servants is one who is
not a non-lover-of-a-non-servant, a lover of every non-servant.
We have the further laws governing negatives: 141
In the early paper, "On the Description of a Notation for the Logic
of Relatives", negatives are treated in a curious fashion. A symbol is
used for "different from " and the negative of s is represented by ns, " differ
ent from every s". Converses are barely mentioned in this study. In the
paper of 1880, converses and negatives appear in their usual notation,
"relative addition" is brought in to balance "relative multiplication", and
the two kinds of involution are retained. But in " The Logic of Relatives "
in the Johns Hopkins Studies in Logic, published in 1882, involution has
disappeared, converses and negatives and "relative addition" are retained.
This last represents the final form of Peirce's calculus of relatives. We
have here,
(1) Relative terms, a, b, ... x, y, z.
(2) The negative of x, -x.
140 See ibid., p. 353. Not-z is here symbolized by (1 - z)
141 Alg. Log. 1880, p. 55.
The Development of Symbolic Logic 91
(3) The converse of x, ^x. If x is "lover", ^.r is "beloved"; if KC is
"lover", a; is "beloved".
(4) Non-relative addition, a + b, "either a or 6".
(5) Non-relative multiplication, a x6, or a b, "both a and 6".
(6) Relative multiplication, a\b, "a of a 6".
(7) Relative addition, a t b, "a of everything but 6's, a of every non-6 ".
(8) The relations = and c , as before.
(9) The universal relation, 1, "consistent with," which pairs every
term with itself and with every other.
(10) The null-relation, 0, the negative of 1.
(11) The relation "identical with", 7, which pairs every term with
itself.
(12) The relation "different from", N, which pairs any term with
every other term which is distinct.142
In terms of these, the fundamental laws of the calculus, in addition to
those which hold for class-terms in general, are as follows:
(1) .(.a) - a
(2) -(wo) = w(-0)
(3) (a c b) = (-6 c w0)
(4) If acb, then (a\x)c(b\x) and (x\a)c(x\b).
(5) If acb, then (a t x) c (6 t x) and (x t a) c (x t 6).
(6) x\(a\b) = (x a)|6
(7) x t (a t b) = (x t a) t b
(8) x\(a-tb)c(x a) t6
(9) (a t b) x c a t (6 x)
(10) (a\x) + (b\x)c(a+b)\x
(11) z|(at&)c(3ta)(a:t&)
(12) (a+b)xc(a\x) + (b\x)
(13) (at*)(6tz)c(a|6) to:
(14) -(at 6) - -a | -6
(15) -(a 1 6) = -at -6
(16) ^(a+ 6) = ^a+ ^b
(17) «(a b) = «a ^6
(18) w(a t 6) = ^a t ^6
(19) w(0|6) = va|w6
For the relations 1, 0, /, and N, the following additional formulae are
given :
142 1 have altered Peirce's notation, as the reader may see by comparison.
92 A Survey of Symbolic Logic
(20) Ocx (21) xcl
(22) x + 0 = x (23) x-l = x
(24) .r + 1 = 1 (25) z-0 = 0
(26) ztl = 1 (27) z|0 = 0
(28) 1 t.T = 1 (29) 0\x = 0
(30) sttf = x (31) z|/-cg
(32) Art* = x (33) /|z = x
(34) or + -x = 1 (35) x -x = 0
(36) 7c[a;1X-aO] (37) fcr| «(-*)] ctf
This calculus is, as Peirce says, highly multiform, and no general prin
ciples of solution and elimination can be laid down.143 Not only the variety
of relations, but the lack of symmetry between relative multiplication and
relative addition, e. g., in (10)-(13) above, contributes to this multiformity.
But, as we now know, the chief value of any calculus of relatives is not in
any elimination or solution of the algebraic type, but in deductions to be
made directly from its formulae. Peirce's devices for solution are, there
fore, of much less importance than is the theoretic foundation upon which
his calculus of relatives is built. It is this which has proved useful in later
research and has been made the basis of valuable additions to logistic
development.
This theory is practically unmodified throughout the papers dealing
with relatives, as a comparison of "Description of a Notation for the Logic
of Relatives" with "The Logic of Relatives" in the Johns Hopkins studies
and with the paper of 1884 will indicate.
"Individual" or "elementary" relatives are the pairs (or triads, etc.)
of individual things. If the objects in the universe of discourse be A, B, C:
etc., then the individual relatives will constitute the two-dimensional array,
A:A,A\B,A:e,A:D, ...
B : A, B : B, B : C, B : D, ...
C:A, C:B, C:C, C:D, ...
. . . Etc., etc.
It will be noted that any individual thing coupled with itself is an individual
relative but that in general A : B differs from B : A— individual relatives
are ordered couples.
A general relative is conceived as an aggregate or logical sum of such
143 "Logic of Relatives" in Studies in Logic by members of Johns Hopkins University,
p. 193.
The Development of Symbolic Logic 93
individual relatives. If b represent "benefactor", then
b = Z,Sy(6)t.,.(/ : ,/),
where (&)»•/ is a numerical coefficient whose value is 1 in case / is a bene
factor of J, and otherwise 0, and where the sums are to be taken for all the
individuals in the universe. That is to say, b is the logical sum of all the
benefactor-benefitted pairs in the universe. This is the first formulation
of "definition in extension", now widely used in logistic, though seldom in
exactly this form. By this definition, b is the aggregate of all the individual
relatives in our two-dimensional array which do not drop out through having
the coefficient 0. It is some expression of the form,
b = (X: y)!+(Z: Y)2+(X : }r)3+...
If, now, we consider the logical meaning of + , we see that this may be read,
"b is either (X : F), or (X : F)2 or (X : F)8 or ... ". To say that 6 repre
sents the class of benefactor-benefitted couples is, then, inexact: b repre
sents an unspecified individual relative, any one of this class. (That it
should represent "some" in a sense which denotes more than one at once—
which the meaning of + in the general case admits — is precluded by the
fact that any two distinct individual relatives are ipso facto mutually
exclusive.) A genera) relative, so defined, is what Mr. Russell calls a
"real variable". Peirce discusses the idea of such a variable in a most
illuminating fashion.144
"Demonstration of the sort called mathematical is founded on suppo
sition of particular cases. The geometrician draws a figure; the algebraist
assumes a letter to signify a certain quantity fulfilling the required condi
tions. But while the mathematician supposes a particular case, his hypoth
esis is yet perfectly general, because he considers no characters of the
individual case but those which must belong to every such case. The ad
vantage of his procedure lies in the fact that the logical laws of individual
terms are simpler than those which relate to general terms, because indi
viduals are either identical or mutually exclusive, and cannot intersect or
be subordinated to one another as classes can. . . .
"The old logics distinguish between indimduum signatum and indi
mduum vagum. ' Julius Caesar' is an example of the former; 'a certain
man', of the latter. The indimduum vagum, in the days when such con
ceptions were exactly investigated, occasioned great difficulty from its
having a certain generality, being capable, apparently, of logical division.
144 "Description of a Notation, pp. 342-44.
94 A Survey of Symbolic Logic
If we include under indimduum vagum such a term as 'any individual
man', these difficulties appear in a strong light, for what is true of any
individual man is true of all men. Such a term is in one sense not an
individual term; for it represents every man. But it represents each man
as capable of being denoted by a term which is individual; and so, though
it is not itself an individual term, it stands for any one of a class of such
terms. . . . The letters which the mathematician uses (whether in algebra
or in geometry) are such individuals by second intention. . . . All the
formal logical laws relating to individuals will hold good of such individuals
by second intention, and at the same time a universal proposition may be
substituted for a proposition about such an individual, for nothing can be
predicated of such an individual which cannot be predicated of the whole
class."
The relative b, denoting ambiguously any one of the benefactor-bene-
fitted pairs in the universe, is such an individual by second intention.
It is defined by means of the " prepositional function", "I benefits J",
as the logical sum of the (7 : J) couples for which "I benefits J" is true.
The compound relations of the calculus can be similarly defined.
If a = S,-Sy(a)iy(Z : J), and b = SiSy(&)<y(/ : J),
then a + b = S,-Sy[(a),-y + (&),-,•](/ : J)
That is, if "agent" is the logical sum of all the (7 : J) couples for which
"7 is agent of J" is true, and "benefactor" is the sum of all the (7 : J)
couples for which "7 benefits J" is true, then "either agent or benefactor"
is the logical sum of all the (7 : J) couples for which " Either 7 is agent of
J or 7 benefits J" is true. We might indicate the same facts more simply
by defining only the "prepositional function", (a + 6)t-,-.145
The definition of a + b given above, follows immediately from this simpler
equation. The definitions of the other compound relations are similar:
(ax6)0- = (a)t-yx(6)iy
or a xb = SiSy[(a)tV x (6)iy](7 : J)
"Both agent and benefactor" is the logical sum of the (7 : J) couples for
which "7 is agent of J and 7 is benefactor of J" is true.
(a I &).-/ = S*{(a)*x(6)Ay}
or a\b = S,-Sy[S*{(a)<Ax (b)hi]](I : J)
145 See "Logic of Relatives ", loc. cit., p. 188.
The Development of Symbolic Logic 95
"Agent of a benefactor" is the logical sum of all the (7 : J) couples such
that, for some //, "I is agent of 77 and // is benefactor of J" is true.
There are two difficulties in the comprehension of this last. The first
concerns the meaning of "agent of a benefactor". Peirce, like De Morgan,
treats his relatives as denoting ambiguously either the relation itself or
the things which have the relation — either relations or relative terms.
a is either the relation "agent of" or the class name "agent". Now note
that the class name denotes the first term in the pairs which have the
relation. With this in mind, the compound relation, a\b, will become
clear. "Agent of a benefactor" names the 7's in the / : J pairs which
make up the field of the relation, " agent of a benefactor of ". Any reference
to the J's at the other end of the relation is gone, just as "agent" omits
any reference to the J's in the field of the relation "agent of". The second
difficulty concerns the operator, SA, which we have read, "For some H".
Consider any statement involving a "prepositional function", <pz, where z
is the variable representing the individual of which <p is asserted.
That is, 2z<pz symbolizes "Either <p is true of Zi or <p is true of Z2 or (p is
true of Z3 or ... ", and this is most simply expressed by "For some z (some
2 or other), <pz". In the particular case in hand, <pz is (o)t*x(6)Ay, "/ is
agent of // and // is benefactor of J". The terms, I and J, which stand in
the relation "7 is agent of a benefactor of J", are those for which there is
some H or other such that 7 is agent of H and H is benefactor of J.
Suppose we consider any " propositional function", <pz with the oper
ator n.
Tl2(pz = <f>Zi x (pZ/z x <pZs x ...
That is, Uz<pz symbolizes " <p is true of Z\ and <p is true of Z2 and <p is true
of Z3 and . . . ", or " <p is true for every z". This operator is needed in the
definition of a t b.
(ot&),/- nA{(a)t-A+(6)A,-}
" 7 is agent of everyone but benefactors of J" is equivalent to " For every 77,
either 7 is agent of 77 or H is benefactor of J".
ot6- 2&[Ilh{(a)ik+(b)hj}](I:J)
"Agent of all non-benefactors" is the logical sum of all the (7 : J) couples
such that, for every H, either 7 is agent of H or 77 is benefactor of J. The
same considerations about the ambiguity of relatives — denoting either the
96 A Survey of Symbolic Logic
relation itself or those things which are first terms of the relation — applies
in this case also. We need not, for the relations still to be discussed, con
sider the step from the definition of the compound "prepositional func
tions", (a t b)a in the above, to the definition of the corresponding relation,
a t b. This step is always taken in exactly the same way.
The converse, converse of the negative, and negative of the converse,
are very simply defined.
That the negative of the converse is the converse of the negative follows
from the obvious fact that -(6).-/ = (-6)/f.
All the formulae of the calculus of relatives, beyond those which belong
also to the calculus of non-relative terms,146 may be proved from such
definitions. For example:
To prove, v(a + b) = ^a + ^b
«(a + &),-/ = (a + &),-,- = (a)/t-+ (&)/<
But (a)ji = («a)t-/, and (6)yf = (w6)tV
Hence «(a + &)»•/ = (wo)t-/+ (W>)»v
Hence S<S,Ka + 6),-,-} (7 : J) = S;Zy{(-a)t7 + («&)»/}(/ : J) Q.E.D.
For the complete development of this theory, there must be a discussion
of the laws which govern such expressions as (a) ,-,-, or in general, expressions
of the form (px, where <px is a statement which involves a variable, x, and (px
is either true or false whenever any individual value of the variable is
specified. Such expressions are now called " propositional functions".147
(a)*, or in the more convenient notation, <px, is a propositional function of
one variable; (a),-/, or <p(x, y), may be regarded as a propositional function
of two variables, or as a function of the single variable, the individual rela
tive (7 : J), or (X : Y).
This theory of propositional functions is stated in the paper of 1885,
'On the Algebra of Logic". It is assumed, as also in earlier papers, that
4 he laws of the algebra of classes hold for propositions as well.148 The
Additional law which propositions obey is stated here for the first time.
46 The formulae of the calculus of classes can also be derived from these, considered as
themselves laws of the calculus of propositions (see below, Chap, vi, Sec. iv).
147 Peirce has no name for such expressions, though he discusses their properties acutely
(see Alg. Log. 1880, §2).
148 This assumption first appears in Alg. Log. 1880.
The Development of Symbolic Logic 97
The current form of this law is "If x 4= 0, then x = 1",— which gives
immediately "If & * 1, then x = 0"— "If x is not false, then x is true,
and if x is not true, then x is false". Peirce uses v and / for "true" and
"false", instead of 1 and 0, and the law is stated in the form
(x -f)(v-x) =0
But the calculus of propositional functions, though derived from the
algebra for propositions, is not identical with it. "x is a man" is neither
true nor false. A propositional function may be true in some cases, false
in some cases. "If a: is a man, then x is a mortal" is true in all cases, or
true of any x; "x is a man" is true in some cases, or true for some values
of x. For reasons already suggested,
2x<px represents " <?x is true for some value of the variable, x — that is,
either <px} is true or <px2 is true or <px3 is true or . . ." Similarly,
TLx<px = (pxi x <px2 x <px3 x ...
TLt<px represents tf tpx is true for all values of the variable, x — that is,
<pxi is true and <px2 is true and #r3 is true and ..."
If (a)xy, or more conveniently, <p(x, y), represent "x is agent of y",
and (b)xy, or more conveniently, \f/(x, y), mean "x is benefactor of y", then
ILx2y[<p(x,y) *t(x, y)]
will mean that for all values of x and some values of y, "x is agent of y
and x is benefactor of y" is true — that is, it represents the proposition
"Everyone is both agent and benefactor of someone". This will appear
if we expand nx2y[<f>(x, y) x^(.r, y)]:
, 2/0 xiKzi, 2/i)] + [<?Oi, 2/2)
x {[<p(x2, 2/j) x^(.r2, yi)] + [<f>(xz, yz) x^(zz, y2)] + . ..)
x U</>(>3, 2/i) x^(.r3, 2/i)] + [^(.r3, 2/2) x^(.r3, 2/2)]+ . - . !
x ... Etc., etc.
This expression reads directly " {Either [xi is agent of y\ and x\ is bene
factor of yi] or [xi is agent of y2 and Xi is benefactor of yz] or . . .[ and {either
[xz is agent of 2/1 and a:2 is benefactor of y\] or [x2 is agent of y2 and x2 is bene
factor of y2] or . . .} and {either [xs is agent of y\ and x3 is benefactor of
2/i] or [#3 is agent of y2 and .r3 is benefactor of yz] or . . f] and . . . Etc., etc".
8
98 A Survey of Symbolic Logic
The operator Z, which is nearer the argument, or "Boolian" as Peirce calls
it, indicates the operation, + , within the lines. The outside operator, II,
indicates the operation, x, between the lines — i. e., in the columns; and
the subscript of the operator nearer the Boolian indicates the letter which
varies within the lines, the subscript of the outside operator, the letter
which varies from line to line. Three operators would give a three-dimen
sional array. With a little patience, the reader may learn to interpret
any such expression directly from the meaning of simple logical sums and
logical products. For example, with the same meanings of <p(x, y) and
t(x, y),
lUZj^Cr, y) xifr(y, x)]
will mean "Everyone (x) is agent of some (y) benefactor of himself".
(Note the order of the variables in the Boolian.) And
will symbolize " There is some x and some y such that, for every z, either
x is agent of z or z is benefactor of y"; or, more simply, "There is some
pair, x and y, such that x is agent of all non-benefactors of y".
The laws for the manipulation of such Boolians with n and 2 operators
are given as follows : 149
"1st. The different premises having been written with distinct indices
(the same index not being used in two propositions) are written together,
and all the n's and Z's are to be brought to the left. This can evidently be
done, for
[Or in the more convenient, and probably more familiar, notation,
Ilx<f>x Xlly<py = UxUy(<px x <py)
x <py)]
"2d. Without deranging the order of the indices of any one premise,
the n's and Z's belonging to different premises may be moved relatively
to one another, and as far as possible the Z's should be carried to the left
149 Alg. Log. 1885, pp.- 196-98.
The Development of Symbolic Logic 99
of the II's. We have
n,-ny.Tt-y = n;itey [Or, mix*, y) = nji^or, y)]
S.-SyZiy = SyZtf.-y [Or, S.S^fo y) = SySz^(a-, y)]
and also Silly&.-yy = n/Z^y [Or, SJIy(^r x ^) = nySx(^r x^y)J
But this formula does not hold when i and j are not separated. We do
have, however,
SillyZiy -< IlyZtf.v [Or, ZJI^ar, y) c n,2x^(.r, y)]
It will, therefore, be well to begin by putting the 2's to the left as far as
possible, because at a later stage of the work they can be carried to the
right but not [always] to the left. For example, if the operators of two
premises are II ^II* and 2 JItf2z, we can unite them in either of the two
orders
and shall usually obtain different conclusions accordingly. There will
often be room for skill in choosing the most suitable arrangement.
. . . "oth. The next step consists in multiplying the whole Boolian
part, by the modification of itself produced by substituting for the index
of any n any other index standing to the left of it in the Quantifier. Thus,
for
Silly/,-,- [Or, for 2JItf*>(.r, y),
we can write 2 JI//^ 2JItf( <p(x, y) x ^(.r, x)}]
"6th. The next step consists in the re-manipulation of the Boolian
part, consisting, 1st, in adding to any part any term we like; 2d, in dropping
from any part any factor we like, and 3d, in observing that
xx = /, x + x = v,
so that xxy + z = z (x + x + y)z = z
"7th. n's and S's in the Quantifier whose indices no longer appear in
the Boolian are dropped.
"The fifth step will, in practice, be combined with part of the sixth
and seventh. Thus, from 2 »•!! ,-/,-/ we shall at once proceed to 2t/lt if we like."
We may say, in general, that the procedures which are valid in this
calculus are those which can be performed by treating 2xv?.r as a sum,
<pxi + <£.r2 + <px3 + . . . , and Ilx<px as a product, tpx\ x <^.r2 x <^.r3 x . . . ;
2 JI,,^(,r, y) as a sum, for the various values of x, of products, each for
100 A Survey of Symbolic Logic
the various values of y, and so on. Thus this calculus may be derived from
the calculus of propositions. But Peirce does not carry out any proofs
of the principles of the system, and he notes that this method of proof
would be theoretically unsound.150 "It is to be remarked that SjX,- and
IliXi are only similar to a sum and a product; they are not strictly of that
nature, because the individuals of the universe may be innumerable."
Another way of saying the same thing would be this: The laws of the
calculus of propositions cannot extend to 2iXt- and Ito, because the extension
of these laws to aggregates in general, by the method which the mathemati
cal analogies of sum and product suggest, would require the principle of
mathematical induction, which is not sufficient for proof in case the aggre
gate is infinite.
The whole of the calculus of relatives may be derived from this calculus
of propositional functions by the methods which have been exemplified —
that is, by representing any relation, 6, as 2*2/(&){/(7 : J), and defining the
relations, such as "converse of", "relative-product," etc., which dis
tinguish the calculus, as II and 2 functions of the elementary relatives.
We need not enter into the detail of this matter, since Sections II and III
of Chapter IV will develop the calculus of propositional functions by a
modification of Peirce's method, while Section IV of that chapter will show
how the calculus of classes can be derived from this calculus of propositional
functions, Section V will indicate the manner in which the calculus of rela
tions may be similarly derived, and Section VI will suggest how, by a
further important modification of Peirce's method, a theoretically adequate
logic of mathematics may be obtained.
It remains to consider briefly Peirce's studies toward the derivation of
other mathematical relations, operations, and systems from symbolic logic.
The most important paper, in this connection, is "Upon the Logic of
Mathematics".151 Certain portions of the paper, "On an Improvement in
Boole's Calculus of Logic", and of the monograph, "Description of a Nota
tion for the Logic of Relatives", are also of interest.
The first-mentioned of these is concerned to show how the relations
+ , =, etc., of arithmetic can be defined in terms of the corresponding logi
cal relations, and the properties of arithmetical relations deduced from
theorems concerning their logical analogues.152
"Imagine ... a particular case under Boole's calculus, in which the
150 Alg. Log. 1885, p. 195.
151 Proc. Amer. Acad., vn, 402-12.
152 Loc. tit., pp. 410-11.
The Development of Symbolic Logic 101
letters are no longer terms of first intention, but terms of second intention,
and that of a special kind. ... Let the letters . . . relate exclusively to
the extension of first intensions. Let the differences of the characters of
things and events be disregarded, and let the letters signify only the differ
ences of classes as wider or narrower. In other words, the only logical
comprehension which the letters considered as terms will have is the greater
or less divisibility of the class. Thus, n in another case of Boole's calculus
might, for example, denote 'New England States'; but in the case now
supposed, all the characters which make these states what they are, being
neglected, it would signify only what essentially belongs to a class which
has the same relation to higher and lower classes which the class of New
England States has, — that is, a collection of six.
11 In this case, the sign of identity will receive a special meaning. For,
if m denotes what essentially belongs to a class of the rank of 'sides of a
cube', then [the logical] m = n will imply, not that every New England
State is the side of a cube, and conversely, but that whatever essentially
belongs to a class of the numerical rank of ' New England States ' essentially
belongs to a class of the rank of 'sides of a cube', and conversely. Identity
of this particular sort may be termed equality. ..."
If a, b, c, etc. represent thus the number of the classes, a, b, c, etc.,
then the arithmetical relations can be defined as logical relations. The
logical relation a + 6, already defined, will represent arithmetical addition:
And from the fact that the logical + is commutative and associative, it
will follow that the arithmetical + is so also. Arithmetical multiplication
is more difficult to deal with but may be defined as follows: 153
a X b represents an event when a and b are events only if these events
are independent of each other, in which case a X b = a b [where a b is the
logical product]. By the events being independent is meant that it is
possible to take two series of terms, A\, A2, A3, etc., and B\, B2, B3, etc.,
such that the following conditions are satisfied. (Here x denotes any
individual or class, not nothing; Am, An, Bm, Bn, any members of the
two series of terms, and 2 A, 2 B, 2 (.4 B) logical sums of some of the
An's, the Bn's, and the (An BJ's respectively.)
Condition 1. No Am is An
2. No£mis£n
3. x = 2 (.4 B)
4. a = 2 A
153 Loc. cit., p. 403.
102 A Survey of 'Symbolic Logic
Condition 5. b = 2 B
6. Some Am is Bn
This definition is somewhat involved: the crux of the matter is that
a b will, in the case described, have as many members as there are combina
tions of a member of a with a member of b. Where the members of a are
distinct (condition 1) and the members of 6 are distinct (condition 2), these
combinations will be of the same multitude as the arithmetical a X b.
It is worthy of remark that, in respect both to addition and to multi
plication, Peirce has here hit upon the same fundamental ideas by means
of which arithmetical relations are defined in Principia Mathematical
The "second intention" of a class term is, in Principia, Nc'«; a + b, in
Peirce's discussion, corresponds to what is there called the "arithmetical
sum" of two logical classes, and a X b to what is called the "arithmetical
product". But Peirce's discussion does not meet all the difficulties — that
could hardly be expected in a short paper. In particular, it does not
define the arithmetical sum in case the classes summed have members in
common, and it does not indicate the manner of defining the number of a
class, though it does suggest exactly the mode of attack adopted in Prin
cipia, namely, that number be considered as a property of cardinally similar
classes taken in extension.
The method suggested for the derivation of the laws of various numerical
algebras from those of the logic of relatives is more comprehensive, though
here it is only the order of the systems which is derived from the order of
the logic of relatives; there is no attempt to define the number or multitude
of a class in terms of logical relations.155
We are here to take a closed system of elementary relatives, every
individual in which is either a T or a P and none is both.
Let c = (T : T)
s = (P:P)
p = (P:T)
t = (T :P)
Suppose T here represent an Individual teacher, and P an individual pupil :
the system will then be comparable to a school in which every person is
either teacher or pupil, and none is both and every teacher teaches every
pupil. The relative term, c, will then be defined as the relation of one
354 See Vol. n, Section A.
155 "Description of a Notation, pp. 359 jf.
The Development of Symbolic Logic 103
teacher to another, that is, " colleague". Similarly, s is (P : P), the rela
tion of one pupil to another, that is, "schoolmate". The relative term, p,
is (P : T), the relation of any pupil to any teacher, that is, " pupil". And
the relative term, t, is (T : P), the relation of any teacher to any pupil,
that is, "teacher". Thus from the two non-relative terms, T and P, are
generated the four elementary relatives, c, s, t, and p.
The properties of this system will be clearer if we venture upon certain
explanations of the properties of elementary relatives — which Peirce does
not give and to the form of which he might object. For any such relative
(7 : J), where the /'s and the J's are distinct, we shall have three laws:
(1) (I:J)\J = 1
Whatever has the (7 : J) relation to a J must be an 7: whoever has the
teacher-pupil relation to a pupil must be a teacher.
(2) (7:./)l/ = 0
Whatever has the teacher-pupil relation to a teacher (where teachers and
pupils are distinct) does not exist.
(3) (I :J)\(II:K) = [(I : J)\H] : K
The relation of those which have the (7 : J) relation to those which have
the (77 : K) relation is the relation of those-which-have-the-(7 : J)-relation-
to-an-77 to a K.
It is this third law which is the source of the important properties of
the system. For example:
t\p = (T :P)|(P : T) = [(T : P) |P] : T = (T : 71) = c
The teachers of any person's pupils are that person's colleagues. (Our
illustration, to fit the system, requires that one may be his own colleague
or his own schoolmate.)
c\c = (T : T)\(T : T) = [(T : T)\ T] : T = (T : T) = c
The colleagues of one's colleagues are one's colleagues.
t t = (T : P) | (T : P) = [(T : P) T] : P = (0 : P) = 0
There are no teachers of teachers in the system.
p s = (P : T) | (P : P) = [(P : T) \P] : P = (0 : P) = 0
There are no pupils of anyone's schoolmates in the system.
The results may be summarized in the following multiplication table,
in which the multipliers are in the column at the right and the multiplicands
104 A Survey of Symbolic Logic
at the top (relative multiplication not being commutative) : 156
t p
c t 0 0
0 0 c t
p s 0 0
0 0 p s
The symmetry of the table should be noted. The reader may easily in
terpret the sixteen propositions which it gives.
To the algebra thus constituted may be added modifiers of the terms,
symbolized by small roman letters. If f is "French", f will be a modifier
of the system in case French teachers have only French pupils, and vice
versa. Such modifiers are "scalars" of the system, and any expression of
the form
a c + bt+ c p + d s
where c, t, p, and s are the relatives, as above, and a, b, c, d are scalars,
Peirce calls a "logical quaternion". The product of a scalar with a term
is commutative,
bt = tb
since this relation is that of the non-relative logical product. Inasmuch as
any (dyadic, triadic, etc.) relative is resolvable into a logical sum of (pairs,
triads, etc.) elementary relatives, it is plain that any general relative what
ever is resolvable into a sum of logical quaternions.
If we consider a system of relatives, each of which is of the form
az + bj + ck + dl+ ...
where i, j, k, I, etc. are each of the form
mu + nv + oiv+ ...
where m, n, o, etc. are scalars, and u, v, w, etc. are elementary relatives,
we shall have a more complex algebra. By such processes of complication,
multiple algebras of various types can be generated. In fact, Peirce says: 157
"I can assert, upon reasonable inductive evidence, that all such [linear
associative] algebras can be interpreted on the principles of the present
notation in the same way as those given above. In other words, all such
algebras are complications and modifications of the algebra of (156) [for
which the multiplication table has been given]. It is very likely that this
155 Ibid., p. 361.
157 Ibid., pp. 363-64.
The Development of Symbolic Logic 105
is true of all algebras whatever. The algebra of (156), which is of such a
fundamental character in reference to pure algebra and our logical nota
tion, has been shown by Professor [Benjamin] Peirce to be the algebra of
Hamilton's quaternions."
Peirce gives the form of the four fundamental factors of quaternions and
of scalars, tensors, vectors, etc., with their logical interpretations as relative
terms with modifiers such as were described above.
One more item of importance is Peirce 's modification of Boole's calculus
of probabilities. This is set forth with extreme brevity in the paper, "On
an Improvement in Boole's Calculus of Logic".158 For the expression of
the relations involved, we shall need to distinguish the logical relation of
identity of two classes in extension from the relation of numerical equality.
We may, then, express the fact that the class a has the same membership
as the class b, or all a's are all fe's, by a = b, and the fact that the number
of members of a is the same as the number of members of b, by a = b.
Also we must remember the distinction between the logical relations ex
pressed by a + b, ab, a \-b, and the corresponding arithmetical relations
expressed by a + b, a X b, and a - b. Peirce says: 159
"Let every expression for a class have a second meaning, which is its
meaning in a [numerical] equation. Namely, let it denote the proportion
of individuals of that class to be found among all the individuals examined
in the long run.
"Then we have
If a = b a = b
" Let ba denote the frequency of the 6's among the a's. Then considered
as a class, if a and b are events ba denotes the fact that if a happens b happens.
« X ba — ab
" It will be convenient to set down some obvious and fundamental proper
ties of the function ba.
a Xba = b X ab
<p(ba, Ca) = <p(b, C)a
(1 - b)a = 1 - ba
158 Proc. Amer. Acad., vn, 255 ff.
159 Ibid., pp. 255-56.
106 A Survey of Symbolic Logic
I - a
«& = 1 --- 7 - X 0(l_a)
The chief points of difference between this modified calculus of prob
abilities and the original calculus of Boole are as follows:
(1) Where Boole puts p, q, etc. for the "probability of a, of b, etc.",
in passing from the logical to the arithmetical interpretation of his equa
tions, Peirce simply changes the relations involved from logical relations to
the corresponding arithmetical relations, in accordance with the foregoing,
and lets the terms a, b, etc. stand for the frequency of the a's, o's, etc.
in the system under discussion.
(2) Boole has no symbol for the frequency of the a's amongst the 6's,
which Peirce represents by ab. As a result, Boole is led to treat the
probabilities of all unconditioned simple events as independent — a pro
cedure which involved him in many difficulties and some errors.
(3) Peirce has a complete set of four logical operations, and four
analogous operations of arithmetic. This greatly facilitates the passage
from the purely logical expression of relations of classes or events to the
arithmetical expression of their relative frequencies or probabilities.
Probably there is no one piece of work which would so immediately
reward an investigator in symbolic logic as would the development of this
calculus of probabilities in such shape as to make it simple and practicable.
Except for a monograph by Poretsky and the studies of H. MacColl,160
the subject has lain almost untouched since Peirce wrote the above in 1867.
Peirce's contribution to our subject is the most considerable of any up
to his time, with the doubtful exception of Boole's. His papers, however,
are brief to the point of obscurity: results are given summarily with little
or no explanation and only infrequent demonstrations. As a consequence,
the most valuable of them make tremendously tough reading, and they
have never received one-tenth the attention which their importance de
serves.161 If Peirce had been given to the pleasantly discursive style of
De Morgan, or the detailed and clearly accurate manner of Schroder, his
work on symbolic logic would fill several volumes.
160 Since the above was written, a paper by Couturat, posthumously published, gives
an unusually clear presentation of the fundamental laws of probability in terms of symbolic
logic. See Bibl.
161 Any who find our report of Peirce's work unduly difficult or obscure are earnestly
requested to consult the original papers.
The Development of Symbolic Logic 107
VIII. DEVELOPMENTS SINCE PEIRCE
Contributions to symbolic logic which have been made since the time
of Peirce need be mentioned only briefly. These are all accessible and in a
form sufficiently close to current notation to be readily intelligible. Also,
they have not been superseded, as have most of the papers so far discussed;
consequently they are worth studying quite apart from any relation to
later work. And finally, much of the content and method of the most
*
important of them is substantially the same with what will be set forth in
later chapters, or is such that its connection with what is there set forth
will be pointed out. But for the sake of continuity and perspective, a
summary account may be given of these recent developments.
We should first mention three important pieces of work contemporary
with Peirce's later treatises.162
Robert Grassmann had included in his encyclopedic Wissenschaftslehre
a book entitled Die Be griff slehre oder Logik,™3 containing (1) Lehre von den
Begriffen, (2) Lehre von den Urtheilen, and (3) Lehre von den Schlussen.
The Begriffslehre is the second book of Die Formerdehre oder Mathematik,
and as this would indicate, the development of logic is entirely mathematical.
An important character of Grassmann's procedure is the derivation of the
laws of classes, or Begriffe, as he insists upon calling them, from the laws
governing individuals. For example, the laws a + a = a and a- a = a,
where a is a class, are derived from the laws e + e = e, e-e = e, e\-e<i — 0,
where e, e\, e2 represent individuals. This method has much to commend
it, but it has one serious defect — the supposition that a class can be treated
as an aggregate of individuals and the laws of such aggregates proved
generally by mathematical induction. As Peirce has observed, this method
breaks down when the number of individuals may be infinite. Another
difference between Grassmann and others is the use throughout of the
language of intension. But the method and the laws are those of extension,
and in the later treatise, there are diagrammatic illustrations in which
"concepts" are represented by areas. Although somewhat incomplete, in
162 Alexander MacFarlane, Principles of the Algebra of Logic, 1879, gives a masterly
presentation of the Boolean algebra. There are some notable extensions of Boole's methods
and one or two emendations, but in general it is the calculus of Boole unchanged. Mac-
Farlane's paper "On a Calculus of Relationship" (Proc. Roy. Soc. Edin., x, 224-32) re
sembles somewhat, in its method, Peirce's treatment of "elementary relatives". But
the development of it seems never to have been continued.
163 There are two editions, 1872 and 1890. The later is much expanded, but the plan
and general character is the same.
108 A Survey of Symbolic Logic
other respects Grassmann's calculus is not notably different from others
which follow the Boolean tradition.
Hugh MacColl's first two papers on "The Calculus of Equivalent
Statements",164 and his first paper "On Symbolical Reasoning",165 printed
in 1878-80, present a calculus of propositions which has essentially the
properties of Peirce's, without II and 2 operators. In others words, it is
a calculus of propositions, like the Two-Valued Algebra of Logic as we know
it today. And the date of these papers indicates that their content was
arrived at independently of Peirce's studies which deal with this touic.
In fact, MacColl writes, in 1878, that he has not seen Boole.166
The calculus set forth in MacColl's book, Symbolic Logic and its
Application*,167 is of an entirely different character. Here the funda
mental symbols represent propositional functions rather than propositions;
and instead of the two traditional truth values, "true" and "false", we
have "true", "false", "certain", "impossible" and "variable" (not cer
tain and not impossible). These are indicated by the exponents r, i, e,
77, 8 respectively. The result is a highly complex system, the fundamental
ideas and procedures of which suggest somewhat the system of Strict
Implication to be set forth in Chapter V.
The calculus of Mrs. Ladd-Franklin, set forth in the paper "On the
Algebra of Logic" in the Johns Hopkins studies,168 differs from the other
systems based on Boole by the use of the copula v . Where a and b are
classes, a v b represents "a is-partly 6", or "Some a is 6", and its negative,
a v b, represents " a is-wholly-not-6 ", or " No a is b ". Thus a v 6 is equiva
lent to a b =h 0, and a v b to a b = 0. These two relations can, between
them, express any assertable relation in the algebra, a cb will be a v-6,
and a = b is represented by the pair, (a v-b)(-a v b). For propositions,
a v b denotes that a and b are consistent — a does not imply that b is false
and b does not imply that a is false. And a v6 symbolizes "a and b are
inconsistent" — if a is true, b is false; if 6 is true, a is false. The use of the
terms "consistent" and "inconsistent" in this connection is possibly mis
leading: any two true propositions or any two false propositions are con-
164 (1) Proc. London Math. Soc., ix, 9-20; (2) ibid., ix, 177-86.
™Mind, v (1880), 45-60.
166 Proc. London Math. Soc., ix, 178.
167 Longmans, 1906.
38 The same volume contains an interesting and somewhat complicated system by
O. H. Mitchell. Peirce acknowledged this paper as having shown us how to express uni
versal and particular propositions as n and S functions. B. I. Oilman's study of relative
number, also in that volume, belongs to the number of those papers which are important
in connecting symbolic logic with the theory of probabilities.
The Development of Symbolic Logic 109
sistent in this sense, and any two propositions one of which is true and
the other false are inconsistent. This is not quite the usual meaning of
"consistent" and " inconsistent "—it is related to what is usually meant by
these terms exactly as the " material implication a c b is related to what
is usually meant by "b can be inferred from a".
That a given class, x, is empty, or a given proposition, x, is false, x = 0,
may be expressed by x v oo, where co is "everything" — in most systems
represented by 1. That a class, y, has members, is symbolized by y v oo.
This last is of doubtful interpretation where y is a proposition, since Mrs.
Ladd-Franklin's system does not contain the assumption which is true
for propositions but not for classes, usually expressed, "If x =(= 0, then
x = 1, and if x =|= 1, then x = 0". x v oo may be abbreviated to xv,
a b v co to a b v , and T/VGO to yv , c dv co to c d v , etc., since it is always
understood that if one term of a relation v or v is missing, the missing
term is oo . This convention leads to a very pretty and convenient opera
tion: v or v may be moved past its terms in either direction. Thus,
(a v 6) = (a by) = ( v a 6)
and (xvy) = (xyv) = ( v x y)
But the forms (va6) and (vxy) are never used, being redundant both
logically and psychologically.
Mrs. Ladd-Franklin's system symbolizes the relations of the traditional
logic particularly well :
All a is b. a v -b, or a -6 v
No a is b. avb, or a b v
Some a is b. avb, or a b v
Some a is not 6. a v -6, or a -6 v
Thus v characterizes a universal, v a particular proposition. And any
pair of contradictories will differ from one another simply by the difference
between v and v . The syllogism, " If all a is b and all b is c, then all
a is c, " will be represented by
(a v -b) (b v -c) v (a v c)
where v, or v, within the parentheses is interpreted for classes, and v
between the parentheses takes the propositional interpretation. This ex
pression may also be read, "'All a is b and all b is c' is inconsistent with
the negative (contradictory) of 'Some a is not c'". It is equivalent to
(a v -b) (b v -c) (a v -c) v
110 A Survey of Symbolic Logic
"The three propositions, 'All a is 6', 'All b is c, ' and 'Some a is not c',
are inconsistent — they cannot all three be true". This expresses at once
three syllogisms:
(1) (a v-6)(6 v-c) v (a v-c)
"If all a is b and all 6 is c, then all a is c";
(2) (a v -b) (a v -c) v (b v -c)
"If all a is 6 and some a is not c, then some b is not c";
(3) (b v -c) (a v -c) v (a v -6)
" If all b is c and some a is not c, then some a is not b ".
Also, this method gives a perfectly general formula for the syllogism
(a v -b) (b v c) (a v c) v
where the order of the parentheses, and their position relative to the sign v
which stands outside the parentheses, may be altered at will. This single
rule covers all the modes and figures of the syllogism, except the illicit
particular conclusion drawn from universal premises. We shall revert to
this matter in Chapter III.169
The copulas v and v- have several advantages over their equivalents,
= 0 and =j= 0, or c and its negative: (1) v and v are symmetrical rela
tions whose terms can always be interchanged; (2) the operation, mentioned
above, of moving v and v with respect to their terms, accomplishes trans
formations which are less simply performed writh other modes of expressing
the copula; (3) for various reasons, it is psychologically simpler and more
natural to think of logical relations in terms of v and v than in terms
of = 0 and =|= 0. But v and v have one disadvantage as against = , 4= ,
and c , — they do not so readily suggest their mathematical analogues in
other algebras. For better or for worse, symbolic logicians have not
generally adopted v and v .
Of the major contributions since Peirce, the first is that of Ernst Schroder.
In his Operationskreis des Logikkalkuls (1877), Schroder pointed out that
the logical relations expressed in Boole's calculus by subtraction and divi
sion were all otherwise expressible, as Peirce had already noted. The
meaning of + given by Boole is abandoned in favor of that which it now
has, first introduced by Jevons. And the "law of duality", which con
nects theorems which involve the relation + , or + and 1, with corresponding
theorems in terms of the logical product x, or x and 0, is emphasized.
169 See below, pp. 188 ff.
The Development of Symbolic Logic 111
(This parallelism of formulae had been noted by Peirce, in his first paper,
but not emphasized or made use of.)
The resulting system is the algebra of logic as we know it today. This
system is perfected and elaborated in Vorlesungen uber die Algebra der
Logik (1890-95). Volume I of this work covers the algebra of classes;
Volume II the algebra of propositions; and Volume III is devoted to the
calculus of relations.
The algebra of classes, or as we shall call it, the Boole-Schroder algebra,
is the system developed in the next chapter.170 We have somewhat elabo
rated the theory of functions, but in all essential respects, we give the algebra
as it appears in Schroder. There are two differences of some importance
between Schroder's procedure and the one we have adopted. Schroder's
assumptions are in terms of the relation of subsumption, c , instead of the
relations of logical product and =; which appear in our postulates. And,
second, Schroder gives and discusses the various methods of his predecessors,
as well as those characteristically his own.
The calculus of propositions (Aussagenkalkid) is the extension of the
Boole-Schroder algebra to propositions by a method which differs little
from that adopted in Chapter IV, Section I, of this book.
The discussion of relations is based upon the work of Peirce. But
Peirce's methods are much more precisely formulated by Schroder, and
the scope of the calculus is much extended. We summarize the funda
mental propositions which Schroder gives for the sake of comparison both
with Peirce and with the procedure we shall adopt in Sections II and III
of Chapter IV.
1) A, B, C, D, E . . symbolize "elements" or individuals.171 These
are distinct from one another and from 0.
I1 symbolizes the universe of individuals or the universe of discourse of
the first order.
3) 2, j, k, /, m, n, p, q represent any one of the elements A, B, C, D, ...
of I1.
4) I1 = Sri
70 For an excellent summary by Schroder, see Abriss der Algebra der Logik ; ed. Dr.
Eugen Miillor, 1909-10. Parts i and n, covering Vols. i and n of Schroder's Vorlesungen,
have so far appeared.
171 The propositions here noted will be found in Vorksungen uber die Algebra der Logik,
in, 3-42. Many others, and much discussion of theory, have been omitted.
112 A Survey of Symbolic Logic
5) i : j represents any two elements, i and j, of I1 in a determined order.
6) (i = j) = (i : j = j : i), (i 4= j) - (» : j * j : 0
for every z and j.
7) i:j*0
Pairs of elements of I1 may be arranged in a "block":
A:A, A:B, A:C, A:D, ...
B:A, B:B, B:C, B:D, ...
C:A, C:B, C : C, C : D, ...
D :A, D:B, D:C, D:D,
These are the "individual binary relatives".
I2 = (A : A) + (A : B) + (A : C) + .
+ (B : A) + (B : B) + (B : C) + . .
+ (C : A) + (C : B) + (C : C) + . .
I2 represents the universe of binary relatives.
10) I2 = S,-S,- (i : j) = St-S,- (i :j) = Siy (i : j)
9) and 10) may be summarized in a simpler notation:
1 = Ziji :j = A :A+A :B + A :C+ ...
+ B :A + B :B + B : C+ .
11)
+ C :A + C :B + C : C+ ...
+ ........
12) i :j : h will symbolize an "individual ternary relative".
13) 1» = S^SyS,- (i : j : A) = 2«At:j:A
Various types of ternary relatives are
14) A : A : A, B : A : A, A : B : A, A : A : B, A : B : C
It is obvious that we may similarly define individual relatives of the
fourth, fifth, ... or any thinkable order.
The Development of Symbolic Logic 113
The general form of a binary relative, a, is
a = S»y an (i : j)
where ai; is a coefficient whose value is 1 for those (i : j) pairs in which i has
the relation a to j, and is otherwise 0.
1 == Ziii-.j
0 = the null class of individual binary relatives.
/= St-/(i =j)(i:j) = 2i(i:i) 172
JV = S,-y(i=t= j)(i:j)
(a b)ij = aij bij (a + 6)i?- = a», + &,-/
-a.-/ = (-a),-,- = -(a,y)
(a 1 &);y = 2A a,-* 6A/ (a t 6),v = n* (a* + fe*,-)
The general laws which govern propositional functions, or Aussagen-
schemata, such as (ab)iit 2haihbhi, Uh (aih + bhj), IIaai}, Sa a,-y, etc., are as
follows :
Au symbolizes any statement about u\ UUAU will have the value 1 in
case, and only in case, Au = 1 for every u; 2UAU will have the value 1 if
there is at least one u such that Au == 1. That is to say, UUAU means
"Au for every u" t and 2UAU means " Au for some u".
a) UuAucAvc2uAu, -[2uAu]c-Avc-[UuAu]
j8) KUAU = AVUVAU1 2UAU = Av + 2UAU
(The subscript u, in a and 0, represents any value of the variable u.)
7) -[nu,4u] = 2U -/iu, -[s«^ „] = nu -A*
d) If ^4M is independent of u, then UUAU = A, and Su.4tt = A.
e) Uu(AcBu) = (AcnuBu), Uu(AucB) = (2uAucB)
r) n«,r or nMn,(^Mc5i;) = (2u^lMcny5v)
77) 2u(AucB) = (nuAucB), 2u(AcBu) = (Ac2uBu)
0) S«f, or 2u2v(AucBv) = (HuAuc2,B9)
I nw(.-iw = i) = (nu.4« = i), nu(,iu = 0) = (zM,iM = o)
\2u(Au = 0) = (nu/lu = 0), SB(.4« == 1) = (Stt.4« = 1)
172 We write I_ where Schroder has 1'; N where he has 0'; (a | 6) for (a; 6); (a f 6)
for (a j 6); -a for a; <-a for a.
9
114 A Survey of Symbolic Logic
\ (UUAU c UUBU)
K) Hu(AucBu)c]
[ (S«.4tt c ZU
(The reader should note that UU(AU^BU) is "formal implication", — in
Principia Mathematica, (x).<px o\f/x.)
X) A 2UBU = SM A BU) A + UUBU = UU(A + Bu)
//) (2UAU)(2VBV) = Sttf „ Au BV) IIUAU + UVBV = n», V(AU + Bv)
0 yl nu£M = nu A Bu, A + 2UBU = ?U(A + Bu)
£) (UUAU)(UVBV) = UU,VAUBV = UUAUBU,
o) 2un^4M, v c Iiv Su Au, v
From these fundamental propositions, the whole theory of relations is
developed. Though Schroder carries this much further than Peirce, the
general outlines are those of Peirce's calculus. Perhaps the most inter
esting of the new items of Schroder's treatment are the use of "matrices"
in the form of the two-dimensional array of individual binary relatives,
and the application of the calculus of relatives to Dedekind's theory of
"chains ", as contained in Was sind und was sollen die Zahlen.
Notable contributions to the Boole-Schroder algebra were made by
Anton Poretsky in his three papers, Sept lois fondamentales de la theorie
des egalites logiques (1899), Quelques lois ulterieures de la theorie des egalites
logiques (1901), and Theorie des non-egalites logiques (1904). (With his
earlier works, published in Russian, 1881-87, we are not familiar.) Poret-
sky's Law of Forms, Law of Consequences, and Law of Causes will be
given in Chapter II. As Couturat notes, Schroder had been influenced
overmuch by the analogies of the algebra of logic to other algebras, and
these papers by Poretsky outline an entirely different procedure which,
though based on the same fundamental principles, is somewhat more
"natural" to logic. Poretsky 's method is the perfection of that type of
procedure adopted by Jevons and characteristic of the use of the Venn
diagrams.
The work of Frege, though intrinsically important, has its historical
interest largely through its influence upon Mr. Bertrand Russell. Although
the Begrifsschrift (1879) and the Grundlagen der Arithmetik (1884) both
The Development of Symbolic Logic 115
precede Schroder's Vorlesungen, Frege is hardly more than mentioned
there; and his influence upon Peano and other contributors to the Formu-
laire is surprisingly small when one considers how closely their task is re
lated to his. Frege is concerned explicitly with the logic of mathematics
but, in thorough German fashion, he pursues his analyses more and more
deeply until we have not only a development of arithmetic of unprecedented
rigor but a more or less complete treatise of the logico-metaphysical problems
concerning the nature of number, the objectivity of concepts, the relations
of concepts, symbols, and objects, and many other subtleties. In a sense,
his fundamental problem is the Kantian one of the nature of the judgments
involved in mathematical demonstration. Judgments are analytic, de
pending solely upon logical principles and definitions, or they are synthetic.
His thesis, that mathematics can be developed wholly by analytic judg
ments from premises which are purely logical, is likewise the thesis of
Russell's Principles of Mathematics. And Frege's Grundgesetee der Arith-
metik, like Principia Mathematica, undertakes to establish this thesis — for
arithmetic — by producing the required development.
Besides the precision of notation and analysis, Frege's work is important
as being the first in which the nature of rigorous demonstration is suf
ficiently understood. His proofs proceed almost exclusively by substitu
tion for variables of values of those variables, and the substitution of defined
equivalents. Frege's notation, it must be admitted is against him: it is
almost diagrammatic, occupying unnecessary space and carrying the eye
here and there in a way which militates against easy understanding. It is
probably this forbidding character of his medium, combined with the
unprecedented demands upon the reader's logical subtlety, which accounts
for the neglect which his writings so long suffered. But for this, the revival
of logistic proper might have taken place ten years earlier, and dated from
Frege's Grundlagen rather than Peano's Formulaire.
The publication, beginning in 1894, of Peano's Formulaire de Mathe-
matiques marks a new epoch in the history of symbolic logic. Heretofore,
the investigation had generally been carried on from an interest in exact
logic and its possibilities, until, as Schroder remarks, we had an elaborated
instrument and nothing for it to do. With Peano and his collaborators, the
situation is reversed : symbolic logic is investigated only as the instrument
of mathematical proof. As Peano puts it: 173
" The laws of logic contained in what follows have generally been found
173 Formulaire, i (1901), 9.
116 A Survey of Symbolic Logic
by formulating, in the form of rules, the deductions which one comes upon
in mathematical demonstrations."
The immediate result of this altered point of view is a new logic, no
less elaborate than the old — destined, in fact, to become much more elabo
rate — but with its elaboration determined not from abstract logical con
siderations or by any mathematical prettiness, but solely by the criterion
of application. De Morgan had said that algebraists and geometers live
in "a higher realm of syllogism": it seems to have required the mathe
matical intent to complete the rescue of logic from its traditional inanities.
The outstanding differences of the logic of Peano from that of Peirce
and Schroder are somewhat as follows: m
(1) Careful enunciation of definitions and postulates, and of possible
alternative postulates, marking an increased emphasis upon rigorous
deductive procedure in the development of the system.
(2) The prominence of a new relation, e, the relation of a member of a
class to the class.
(3) The prominence of the idea of a prepositional function and of
"formal implication" and " formal equivalence", as against "material
implication" and "material equivalence".
(4) Recognition of the importance of "existence" and of the properties
of classes, members of classes, and so on, with reference to their "existence".
(5) The properties of relations in general are not studied, and "relative
addition" does not appear at all, but various special relations, prominent
in mathematics, are treated of.
The disappearance of the idea of relation in general is a real loss, not a
gain.
(6) The increasing use of substitution (for a variable of some value in
its range) as the operation which gives proof.
We here recognize those characteristics of symbolic logic which have
since been increasingly emphasized.
The publication of Principia Mathematica would seem to have deter
mined the direction of further investigation to follow that general direction
indicated by the work of Frege and the Formulaire. The Principia is con
cerned with the same topics and from the same point of view. But we see
here a recognition of difficulties not suggested in the Formulaire, a deeper
and more lengthy analysis of concepts and a corresponding complexity of
procedure. There is also more attention to the details of a rigorous
method of proof.
174 All these belong also to the Logica Mathematica of C. Burali Forti (Milan, 1894).
The Development of Symbolic Logic 117
The method by which the mathematical logic of Principia Mathematica
is developed will be discussed, so far as we can discuss it, in the concluding
section of Chapter IV. We shall be especially concerned to point out the
connection, sometimes lost sight of, between it and the older logic of Peirce
and Schroder. And the use of this logic as an instrument of mathematical
analysis will be a topic in the concluding chapter.
CHAPTER II
THE CLASSIC, OR BOOLE-SCHRODER, ALGEBRA OF LOGIC
I. GENERAL CHARACTER OF THE ALGEBRA. THE POSTULATES AND
THEIR INTERPRETATION
The algebra of logic, in its generally accepted form, is hardly old enough
to warrant the epithet "classic". It was founded by Boole and given its
present form by Schroder, who incorporated into it certain emendations
which Jevons had proposed and certain additions — particularly the relation
"is contained in" or " implies" — which Peirce had made to Boole's system.
It is due to Schroder's sound judgment that the result is still an algebra,
simpler yet more powerful than Boole's calculus. Jevons, in simplifying
Boole's system, destroyed its mathematical form; Peirce, retaining the
mathematical form, complicated instead of simplifying the original calculus.
Since the publication of Schroder's Vorlesungen uber die Algebra der Logik
certain additions and improved methods have been offered, the most notable
of which are contained in the studies of Poretsky and in Whitehead's Uni
versal Algebra.1
But if the term "classic" is inappropriate at present, still we may
venture to use it by way of prophecy. As Whitehead has pointed out,
this system is a distinct species of the genus "algebra", differing from all
other algebras so far discovered by its non-numerical character. It is
certainly the simplest mathematical system with any wide range of useful
applications, and there are indications that it will serve as the parent stem
from which other calculuses of an important type will grow. Already sev
eral such have appeared. The term "classic" will also serve to distinguish
the Boole-Schroder Algebra from various other calculuses of logic. Some
of these, like the system of Mrs. Ladd-Franklin, differ through the use
of other relations than + , x , c , and = , and are otherwise equivalent —
1 For Poretsky 's studies, see Bibliography; also p. 114 above. See Whitehead's Uni
versal Algebra, Bk. n. Whitehead introduced a theory of "discriminants" and a treatment
of existential propositions by means of umbral letters. This last, though most ingenious
and interesting, seems to me rather too complicated for use; and I have not made use of
"discriminants ", preferring to accomplish similar results by a somewhat extended study of
the coefficients in functions.
118
The Classic, or Boole-Schroder, Algebra of Logic 119
that is to say, with a " dictionary" of equivalent expressions, any theorem
of these systems may be translated into a theorem of the Boole-Schroder
Algebra, and vice versa. Others are mathematically equivalent as far as
they go, but partial. And some, like the calculus of classes in Principia
Mathematica, are logically but not mathematically equivalent. And,
finally, there are systems such as that of Mr. MacColl's Symbolic Logic
which are neither mathematically nor logically equivalent.
Postulates for the classic algebra have been given by Huntington,
by Schroder (in the Abriss}, by Del Re, by Sheffer and by Bernstein.2 The
set here adopted represents a modification of Huntington's third set.*
It has been chosen not so much for economy of assumption as for " natural
ness" and obviousness.
Postulated:
A class K of elements a, b, c, etc., and a relation x such that:
1-1 If a and b are elements in K, then a x b is an element in K, uniquely
determined by a and b.
1 • 2 For any element a, a x a = a.
1 • 3 For any elements a and 6, a x b = b x a.
1 • 4 For any elements a, b, and c, a x (b x c) = (a x 6) x c.
1 • 5 There is a unique element, 0, in K such that a x 0 = 0 for every ele
ment a.
1 • 6 For every element a, there is an element, -a, such that
1-61 If x x-a = 0, then x xa = x,
and 1-62 If y x a = y and y x -a = y, then y = 0.
The element 1 and the relations + and c do not appear in the above.
These may be defined as follows :
1-7 1 = -0 Def.
1-8 o + 6 = -(-ax-6) Def.
1-9 a c b is equivalent to a x6 = a Def.
It remains to be proved that -a is uniquely determined by a, from
which it will follow that 1 is unique and that a + b is uniquely determined
by a and b.
2 See Bibl.
3 See "Sets of Independent Postulates for the Algebra of Logic", Trans. Amer. Math.
Soc., v (1904), 288-309. Our set is got by replacing + in Huntington's set by x , and
replacing the second half of G, which involves 1, by its analogue with 0. Thus 1 can be
defined, and postulates E and H omitted. Postulate J is not strictly necessary.
120 A Survey of Symbolic Logic
The sign of equality in the above has its usual mathematical meaning;
j. e., j = j is a relation such that if x = y and <p(x) is an unambiguous
function in the system, then <p(x) and <p(y) are equivalent expressions and
interchangeable. It follows from this that if $(x) is an ambiguous function
in the system, and x = y, every determined value of ifr(x), expressible in
terms of x, is similarly expressible in terms of y. Suppose, for example,
that -a, "negative of a", is an ambiguous function of a. Still we may write
-a to mean, not the function "negative of a" itself, but to mean some
(any) determined value of that function — any one of the negatives of a—
and if -a = b, then <p(-a) and <p(b) will be equivalent and interchangeable.
This principle is important in the early theorems which involve negatives.
We shall develop the algebra as an abstract mathematical system: the
terms, a, b, c, etc., may be any entities which have the postulated properties,
and x , + , and c may be any relations consistent with the postulates.
But for the reader's convenience, we give two possible applications: (1) to
the system of all, continuous and discontinuous, regions in a plane, the
null-region included, and (2) to the logic of classes.4
(1)
For the first interpretation, a x b will denote the region common to a
and b (their overlapping portion or portions), and a + b will denote that
which is either a or 6 or both, a c b will represent the proposition, "Region
a is completely contained in region b (with or without remainder)". 0 will
represent the null-region, contained in every region, and 1 the plane itself,
or the "sum" { + } of all the regions in the plane. For any region a, -a
will be the plane except a, all that is not-a. The postulates will then hold
as follows:
1 • 1 If a and b are regions in the plane, the region common to a and b,
a x b, is in the plane. If a and 6 do not overlap, then a x b is the null-
region, 0.
1 • 2 For any region a, the region common to a and a, a x a, is a itself.
1 • 3 The region common to a and b is the region common to b and a.
1 • 4 The region common to a and b x c is the region common to a x b
and c — is the region common to all three. •
1-5 The region common to any region a and the null-region, 0, is 0.
1 • 6 For every region a, there is its negative, -a, the region outside or
4 Both of these interpretations are more fully discussed in the next chapter.
The Classic, or Boole-Schroder, Algebra of Logic 121
not contained in a, and this region is such that
1-61 If -a and any region x have only the null-region in
common, then the region common to x and a is x itself, or x is contained in a;
and 1-62 If the region common to y and a is y, or y is contained
in a, and the region common to y and -a is y, or y is contained in -a, then y
must be the null-region which is contained in every region.
That the definitions, 1-7, 1-8, and 1-9, hold, will be evident.
(2)
For the second interpretation, a, b, c, etc., will be logical classes, taken
in extension — that is, a = b will mean that a and b are classes composed of
identically the same members, a x b will represent the class of those
things which are members of a and of b both; a + b, those things which
are either members of a or members of b or both, a c b will be the proposi
tion that all members of a are also members of b, or that a is contained in b
(with or without remainder). 0 is the null-class or class of no members;
and the convention is required that this class is contained in every class.
1 is the "universe of discourse" or the class wrhich contains every entity
in the system. For any class a, -a represents the negative of a, or the class
of all things which are not members of a. The postulates will hold as fol
lows :
1-1 If a and b are logical classes, taken in extension, the members com
mon to a and b constitute a logical class. In case a and b have no members
in common, this class is the null-class, 0.
1-2 The members common to a and a constitute the class a itself.
1 • 3 The members common to a and b are the same as those common to
b and a.
1 -4 The members common to a, b, and c, all three, are the same, whether
we first find the members common to b and c and then those common to a
and this class, or wrhether we first find the common members of a and b
and then those common to this class and c.
1 • 5 The members common to any class a and the null-class are none, or
the null-class.
1 • 6 For every class a, there is its negative, -a, constituted by all members
of the " universe of discourse" not contained in a, and such that:
1-61 If -a and any class x have no members in common,
122 A Survey of Symbolic Logic
then all members of x are common to x and a, or x is contained in a;
and 1-62 If all members of any class y are common to y and a,
and common also to y and -a, then y must be null.
1-7 The "universe of discourse", "everything", is the negative of the
null-class, "nothing".
1 • 8 That which is either a or 6 or both is identical with the negative of
that which is both not-a and not-6.
1-9 That "a is contained in b " is equivalent to "The class a is identical
with the common members of a and b ".
That the postulates are consistent is proved by these interpretations.
In the form given, they are not independent, but they may easily be made
so by certain alternations in the form of statement.5
The following abbreviations and conventions will be used in the state
ment and proof of theorems:
1. ax6 will generally be abbreviated to a b or a -b, ax(bxc) to a (be),
ax-(6x-c) to a-(fc-c) or a--(&-c), etc.
2. In proofs, we shall sometimes mark a lemma which has been established
as (1), or (2), etc., and thereafter in that proof refer to the lemma by this
number. Also, we shall sometimes write "Q.E.D." instead of repeating
the theorem to be proved.
3. The principles (postulates, definitions, or previous theorems) by which
any step in proof is taken will usually be noted by a reference in square
brackets, thus: If x = 0, then [1-5] a x = 0. Reference to principles
whose use is more or less obvious will gradually be omitted as we proceed.
Theorems will be numbered decimally, for greater convenience in the
insertion of theorems without alteration of other numbers. The non-
decimal part of the number will indicate some major division of theorems,
as 1- indicates a postulate or definition. Theorems which have this digit
and the one immediately following the decimal point in common will be
different forms of the same principle or otherwise closely related.
II. ELEMENTARY THEOREMS
2-1 If a = 6, then a c = b c and c a = c b.
This follows immediately from the meaning of = and 1 • 1.
v 2 • 2 a = b is equivalent to the pair, a c b and b c a.
If a = 6, then [1-2] ab = a and b a = b.
6 On this point, compare with Huntington's set.
The Classic, or Boole-Schroder, Algebra of Logic 123
And if a b = a and b a = b, then [1 -3] a = a b = b a = b.
But [1 -9] a b = a is equivalent to a c b and b a = b to 6 c a.
Equality is, then, a reciprocal inclusion relation.
' 2-3 aca.
a = a, hence [2-2] Q.E.D.
Every element is "contained in" itself.
'2-4 a -a = 0 = -a a.
[1-2] a a = a.
Hence [2-1, 1-4, 1 -3] a -a = (a a) -a = a (a -a) = (a -a) a.
Also [1-2] -a -a = -a. Hence a -a = a (-a -a) = (a -a) -a.
But [1-62] if (a -a) a = (a -a) -a = a -a, then a -a = 0.
And [1-3] -a a = a -a. Hence also, -a a = 0.
Thus the product of any element into its negative is 0, and 0 is the
modulus of the operation x .
/ 2-5 a -b = 0 is equivalent to a b = a and to a c b.
If a b = a, then [1-4-5,. 2-1-4] a -b = (a b) -6 = a (b -b)
And [1-61] if a -6 = 0, then ab = a (2)
By (1) and (2), a -b = 0 and ab = a are equivalent.
And [1 • 9] a b = a is a c 6.
We shall derive other equivalents of a c 6 later. The above is required
immediately. In this proof, we have written "1-4-5" and "2-1-4"
instead of "1-4, 1-5" and "2-1, 2-4". This kind of abbreviation in
references will be continued.
2-6 If acO, then a = 0.
If a cO, then [1-9] o-O = a. But [1-5] a-0 = 0.
2-7 If a c b, then a c c b c, and c a c c b.
If acb, then [1-9] a b = a and [2-1] (« b) c = a c (1)
But [1-2-3-4] (a b) c = (b a) c = b (ac) = (ac) b = [a (c c) b] = [(ac) c] b
= (ac)(cb) = (ac)(bc) (2)
Hence, by (1) and (2), if a c 6, then (a c)(b c) = ac and [1 -9] a c c b c.
And [1-3] c a = ac and c b = b c. Hence also c a c c b.
2-8 -(-a) = a.
[2-4] -(-a) --a - 0. Hence [2-5] -(-a) ca (1)
By (1), -[-(-«)] c-a. Hence [2-7] a- -[-(-a)] ca-a.
124 A Survey of Symbolic Logic
But [2-4] a -a = 0. Hence a --[-(-a)] cO.
Hence [2-G] a --[-(-a)] = 0 and [2-5] a c-(-a) (2)
[2-2] (1) and (2) are equivalent to -(-a) = a.
3-1 a c b is equivalent to -b c -a.
[2-5] a c 6 is equivalent to a -6 = 0.
And [2-8] a-b = -ba = -b-(-a).
And -6 -(-a) = 0 is equivalent to -b c-a.
The terms of any relation c may be transposed by negating both.
If region a is contained in region b, then the portion of the plane not in b
is contained in the portion of the plane not in a: if all a's are 6's, all non-6's
are non-a's. This theorem gives immediately, by 2-8, the two corollaries:
3-12 a c -b is equivalent to b c-a; and
3-13 -a c 6 is equivalent to -6 c a.
3-2 a = b is equivalent to -a = -6.
[2-2] a = b is equivalent to (a cb and bed).
[3-1] a cb is equivalent to -b c-a, and b ca to -a c-6.
Hence a = b is equivalent to (-a c-b and -6 c-a), which is equiva
lent to -a = -6.
The negatives of equals are equals. By 2-8, we have also
3-22 a = -b is equivalent to -a = b.
Postulate 1-6 does not require that the function "negative of" be
unambiguous. There might be more than one element in the system having
the properties postulated of -a. Hence in the preceding theorems, -a
must be read "any negative of a ", -(-6) must be regarded as any one of
the negatives of any given negative of 6, and so on. Thus what has been
proved of -a, etc., has been proved to hold for every element related to a
in the manner required by the postulate. But we can now demonstrate
that for every element a there is one and only one element having the
properties postulated of -a.
3-3 -a is uniquely determined by a.
By 1-6, there is at least one element -a for every element a.
Suppose there is more than one: let -ax and -a2 represent any two
such.
Then [2-8] -(-Gl) = a = -(-a2). Hence [3-2] -a, = -a2.
Since all functions in the algebra are expressible in terms of a, b, c, etc.,
the relation x , the negative, and 0, while 0 is unique and a x b is uniquely
The Classic, or Boole-Schroder, Algebra of Logic 125
determined by a and b, it follows from 3-3 that all functions in the algebra
are unambiguously determined when the elements involved are specified.
(This would not be true if the inverse operations of "subtraction" and
"division" were admitted.)
3-33 The element 1 is unique.
[1-5] 0 is unique, hence [3-3] -0 is unique, and [1-7] 1 = -0.
3-34 -1 = 0.
[1-7] 1 = -0. Hence [3-2] Q.E.D.
3-35 If a and b are elements in K, a + 6 is an element in K uniquely deter
mined by a and b.
The theorem follows from 3.3, 1-1, and 1 • 8.
3-37 If a = 6, then a + c = b + c and c + a = c + b.
The theorem follows from 3-35 and the meaning of =.
, 3-4 -(a + 6) = -a -b.
[1-8] a + b = -(-a -b).
Hence [3-3, 2-8] -(a + b) = -[-(-a-b)} = -a -b.
.3-41 -(a 6) = -a + -6.
[1-8, 2-8] -a + -6 = -[-(-a) --(-6)] = -(a 6).
3-4 and 3-41 together state De Morgan's Theorem: The negative of a
sum is the product of the negatives of the summands; and the negative of a
product is the sum of the negatives of its factors. The definition 1-8 is a
form of this theorem. Still other forms follow at once from 3-4 and 3-41,
by 2-8:
3-42 -(-a + -b) = ab.
3-43 -(a + -6) = -ab.
3-44 -(-a + b) = a-b.
3-45 -(a-b) = -a + b.
3-46 -(-ab) = a + -b.
From De Morgan's Theorem, together with the principle, 3-2, "The
negatives of equals are equals", the definition 1-7, 1 = -0, and theorem
3-34, -1 = 0, it follows that for every theorem in terms of x there is a
corresponding theorem in terms of + . If in any theorem, each element be
replaced by its negative, and x and + be interchanged, the result is a
valid theorem. The negative terms can, of course, be replaced by positive,
126 A Survey of Symbolic Logic
since we can suppose x = -a, y = -b, etc. Thus for every valid theorem
in the system there is another got by interchanging the negatives 0 and 1
and the symbols x and + . This principle is called the Law of Duality.
This law is to be illustrated immediately by deriving from the postulates
their correlates in terms of + . The correlate of 1 • 1 is 3 • 35, already proved.
4-2 a + a = a. ,
[1-2] -a-a = -a. Hence [1-8, 3-2, 2-8] a + a = -(-a -a) =
-(-a) = a.
4-3 a + b = b + a.
[1-3] -a -b = -b -a. Hence [3-2] -(-a -6) = -(-6 -a).
Hence [1-8] Q.E.D.
4-4 a + (b + c) = (a + b) + c.
\1 4.1 n ( A ^\ - ( h\
Hence [3-2] -[-a (-b -c)} = -[(-a -6) -c].
But [3-46, 1-8] -[-a(-fc-c)] = a + -(-6-c) = a+(6 + c).
And [3-45, 1-8] -[(-a-6)-c] = -(-a-b)+c = (a + b)+c.
4-5 a+1 = 1.
[1-5] -a-0 = 0. Hence [3-2] -(-a-0) = -0.
Hence [3-46] a + -0 = -0, and [1-7] a+1 = 1.
4-61 If -x + a = 1, then x a = x.
If -x + a = 1, then [3-2-34-44] x -a = -(-z + a) =-1=0.
And [2-5] x -a = 0 is equivalent to x a = x.
4-612 If -a; + a = 1, then x + a = a.
[4-61] If -a + x = 1, then a x = a, and [3-2] -a + -x = -a (1)
By (1) and 2-8, if -x + a = 1, x + a = a.
4-62 If y + a = y and y + -a = y, then y = 1.
If y + a = y, [3-2] -y -a = -(y + a) = -y.
And if y + -a = y, -y a = -(y + -a) = -y.
But [1-62] if -y a = -y and -y -a = -y, -y = 0 and y = -0 = 1.
4-8 a + -a = l=-a + a. (Correlate of 2-4)
[2-4] -a a = 0. Hence [3-2] a + -a = -(-a a) =-0 = 1.
Thus the modulus of the operation + is 1.
4-9 -a + 6 = 1, a + b = b, a -6 = 0, a b = a, and a c b are all equivalent.
[2 • 5] a -b — 0, a 6 = a, and a cb are equivalent.
[3-2] -a + 6 = 1 is equivalent to a -b = -(-a + b) =-1=0.
n -M,V)
The Classic, or Boole-Schroder, Algebra of Logic 127
[4-612] If -a + b = 1, a + b = b.
And if a + b = b, [3 • 37] -a + b = -a + (a + b) = (-a + a) + 6 = 1+6
= 1.
Hence a + b = b is equivalent to -a + b = 1.
We turn next to further principles which concern the relation c .
5-1 If a c 6 and b cc, then ace.
[1-9] a cb is equivalent to a b = a, and 6 cc to b c = b.
If a b = a and b c = b, a c = (a b) c = a (b c) = a b = a.
But a c = a is equivalent to a c c.
This law of the transitivity of the relation c is called the Principle of
the Syllogism. It is usually included in any set of postulates for the algebra
which are expressed in terms of the relation c .
v 5-2 a b ca and ab cb.
(a b) a = a (a b) = (a a) b = a b.
But (a b) a = a b is equivalent to ab ca.
Similarly, (a b) b = a (b b) = a b, and ab cb.
5-21 a c a + b and b ca + b.
[5 • 2] -a -b c -a and -a -b c -b.
Hence [3-12] ac-(-a-b) and 6c-(-a-6).
But -(-a -b) = (a + b).
Note that 5-2 and 5-21 are correlates by the Law of Duality. In
general, having now deduced the fundamental properties of both x and + ,
we shall give further theorems in such pairs.
A corollary of 5-21 is:
5-22 a b ca + b.
[5- 1-2- 21]
5-3 If a c b and c c d, then a c c b d.
[1-9] If a c b and c cd, then a b = a and c d = c.
Hence (a c) (b d) = (a b) (c d) = a c, and a c c b d.
5-31 If a c b and c c d, then a + c c b + d.
If a c 6 and c cd, [3-1] -b c -a and -d c -c.
Hence [5-3] -b-dc-a-c, and [3-1] -(-a -c) c-(-fc-d).
Hence [1-8] Q.E.D.
By the laws, aa = a and a + a = a, 5-3 and 5-31 give the corollaries:
5-32 If a c c and b cc, then ab cc.
128 A Survey of Symbolic Logic
5-33 If a c c and b c c, then a + 6 c c.
5-34 If a c 6 and ace, then a cb c.
5-35 If a c 6 and ace, then a c 6 + c.
5-37 If a c6, then a + c c6 + c. (Correlate of 2-7)
[2-3]cce. Hence [5-31] Q.E.D.
5-4 a + ab = a.
[5-21] aca + ab (1)
[2-3] a c a, and [5-2] ab ca. Hence [5 • 33] a + a 6 c a (2)
[2-2] If (1) and (2), then Q.E.D.
5-41 a (a + b) = a.
[5-4] -a + -a-b = -a. Hence [3-2] -(-a + -a-b) = -(-a) = a.
But [3-4] -(-a + -«-&) = a --(-a -6) =0(0 + 6).
5-4 and 5-41 are the two forms of the Law of Absorption. We have
next to prove the Distributive Law, which requires several lemmas.
5-5 a (b + c) = ab + ac.
Lemma 1: ab + acca(b + c).
[5-2] a6ca and acca. Hence [5-33] a b + a c ca (1)
[5-2] a b c b and a c c c. But [5-21] b c 6 + c and c c b + c.
Hence [5-1] a b c b + c and ac cb + c.
Hence [5 • 33] a b + a c c 6 + c (2)
[5-34] If (1) and (2), then a b + a c ca (b + c).
Lemma 2 : If p c q is false, then there is an element x, 4= 0, such that
x c p and x c -q.
p-q is such an element, for [5-2] p-qcp and p-qc-q- and
[4-9] if p-q = 0, then pcq, hence if p c q is false, then p -q 4= 0.
(This lemma is introduced in order to simplify the proof of Lemma 3.)
Lemma 3 : a (6 + c) c b + a c.
Suppose this false. Then, by lemma 2, there is an element x,
4= 0, such that
xca(b + c) (1)
and .TC-(6 + ac)
But [3-12] if xc-(b + ac), then b + acc-x (2)
[5-1] If (1), then since [5-2] o (6 + oc) co, z co (3)
and also, since a(6 + a c) c 6 + c, .r c 6 + c (4)
[5-1] If (2), then since [5-21] b cb + a c, b c -x and [3-12] xc-b (5)
Also [5 • 1] if (2), then since [5 • 21] a c c b + a c, a c c -x and [3-12]
zc-(oc) (6)
The Classic, or Boole-Schroder, Algebra of Logic 129
From (6) and (3), it follows that xcc must be false; for if .TCC
and (3) x c a, then [5 • 34] x c a c. But if x c a c and (6) x c -(a c),
then [1-62] x = 0, which contradicts the hypothesis x 4= 0.
But if x c c be false, then by lemma 2, there is an element y, 4= 0,
such that
ycx (7)
and ?/c-c, or [3-12] cc-y (8)
[5-1] If (7) and (5), then y c-b and [3-12] bc-y (9)
If (8) and (9), then [5-33] b + cc-y and [3-12] yc-(b + c) (10)
If (7) and (4), then [5-1] ycb + c (11)
[1-9] If (11), then y(b + c) =y, and if (10), y~(b + c) = y (12)
But if (12), then [1-62] y = 0, which contradicts the condition,
y *0.
Hence the supposition — that a(b + c)cb + ac be false — is a false
supposition, and the lemma is established.
Lemma 4 : a(b + c)cab + ac.
By lemma 3, a (b + c) c b + a c.
Hence [2-7] a [a (b + c)] c a (b + a c) .
But a [a (b + c)] = (a a)(b + c) = a (b + c).
And a (b + a c) = a (a c + b). Hence a (b + c) c a (a c + b).
But by lemma 3, a (a c + b) c a c + a b.
And a c + a b = a b + a c. Hence a (b + c) c a b + a c.
Proof of the theorem: [2-2] Lemma 1 and lemma 4 are together equiva
lent to a (b + c) = a b + a c.
This method of proving the Distributive Law is taken from Huntington,
"Sets of Independent Postulates for the Algebra of Logic ". The proof of
the long and difficult lemma 3 is due to Peirce, who worked it out for his
paper of 1880 but mislaid the sheets, and it was printed for the first time in
Huntington 's paper.6
5-51 (a + b)(c + d) = (a c + b c) + (a d + b d).
[5 • 5] (a + b)(c + d) = (a + b) c + (a + b) d = (a c + b c) + (a d + b d).
5-52 a + b c = (a + b)(a + c). (Correlate of 5 • 5)
[5-51] (a + b) (a + c) = (a a + b a) + (a c + b c)
= [(a + a b) + a c] + b c.
But [5-4] (a + a b) + a c = a + a c = a. Hence Q.E.D.
Further theorems which are often useful in working the algebra and
which follow readily from the preceding are as follows:
6 See "Sets of Independent Postulates, etc.", loc. rit., p. 300, footnote.
10
130 A Survey of Symbolic Logic
5-6 a-1 = a = 1 -a.
[1-5] a-0 = 0. Hence a--l = 0.
But [1-01] if a--l = 0, then a-1 = a.
5-61 acl.
[1-9] Since a-1 = a, acl.
-a--0 = -a-1 = -a. Hence [3 • 2] a + 0 = -(-a--0) = -(-a) = a.
5-63 Oca.
0-a = a-0 = 0. Hence [1-9] Q.E.D.
5-64 1 ca is equivalent to a = 1.
[2-2] a = 1 is equivalent to the pair, acl and 1 ca.
But [5-61] a c 1 holds always. Hence Q.E.D.
5-65 a c 0 is equivalent to a = 0.
[2 • 2] a = 0 is equivalent to the pair, a c 0 and 0 c a.
But [5-63] 0 ca holds always. Hence Q.E.D.
5-7 If a + 6 = a; and a = 0, then 6 = x.
If a = 0, a + 6 = 0 + 6 = 6.
5-71 If a 6 = x and a = 1, then 6 = x.
If a = 1, a 6 = 1-6 = 6.
5-72 a + 6 = 0 is equivalent to the two equations, a = 0 and 6 = 0.
If a = 0 and 6 = 0, then a + 6 = 0 + 0 = 0.
And if a + 6 = 0, -a -6 = -(a + 6) = -0 = 1.
But if -a -6 = 1, a = a-1 = a(-a -6) = (a -a) -6 = 0--6 = 0.
And [5-7] if a + 6 = 0 and a = 0, then 6 = 0.
5-73 a 6 = 1 is equivalent to the two equations, a = 1 and 6 = 1.
If a == 1 and 6 = 1, then a 6 = 1 - 1 = 1.
And if a 6 = 1, -a + -6 = -(a 6) =-1=0. Hence [5-72] -a = 0
and -6 = 0.
But [3-2] if -a = 0, a == 1, and if -6 = 0, 6 = 1.
5-7 and 5-72 are important theorems of the algebra. 5-7, "Any null
term of a sum may be dropped", would hold in almost any system; but
5-72, "If a sum is null, each of its summands is null", is a special law
characteristic of this algebra. It is due to the fact that the system con
tains no inverses with respect to + and 0. a and -a are inverses with
The Classic, or Boole-Schroder, Algebra of Logic 131
respect to x and 0 and with respect to + and 1. 5-71 and 5-73, the
correlates of 5-7 and 5-72, are less useful.
5-8 a (b + -b) = a b + a -b = a.
[5-5] a (b + -6) = a b + a -b.
And [4-8] b + -b = 1. Hence a (b + -b) = a-l = a.
5-85 a + b = a + -a b.
[5-8] b = ab + -ab.
Hence a + b = a + (a b + -a b) = (a + a b) + a flb.
But [5-4] a + a b = a. Hence Q.E.D.
It will be convenient to have certain principles, already proved for two
terms or three, in the more general form which they can be given by the
use of mathematical induction. Where the method of such extension is
obvious, proof will be omitted or indicated only. Since both x and +
are associative, we can dispense with parentheses by the definitions:
5-901 a + b + c=(a + b)+c Def.
5-902 abc = (ab) c Def.
5-91 a = a (b + -b) (c + -c) (d + -d) . . .
[5-8]
5-92 1 - (a + -a)(6 + -6)(c + -c)...
[4-8]
5-93 a = a + ab + ac + ad+...
[5-4]
5 • 93 1 a = a (a + b) (a + c) (a + d) . . .
[5-41]
5-94 a (b + c + d + ...) = ab + ac + ad+ ...
[5-5]
5-941 a + bcd.,.. --= (a + b)(a + c)(a + d) . . .
[5-52]
5-95 -(a + b + c + . . . ) = -a -b -c ...
If the theorem hold for n terms, so that
-(ai + a2 + . . . + an) = -a-i -az . . . -an
then it will hold for n + 1 terms, for by 3-4,
And [3-4] the theorem holds for two terms. Hence it holds for any
number of terms.
132 A Survey of Symbolic Logic
5-951 -(a bed...) = -a + -b + -c + -d+ . . .
Similar proof, using 3-41.
5-96 1 = a + b + c + . . . +-a-b -c . . .
[4-8, 5-951]
5-97 a + b + c + . . . = 0 is equivalent to the set, a = 0, 6 = 0, c = 0, . . .
[5-72]
5-971 abed... --= 1 is equivalent to the set, a == 1, b = 1, c = I, . . .
[5-73]
5-98 a-b c d . . . =ab-ac-ad...
[1-2] a a a a . . . = a.
5-981 a + (b + c + d + . . . ) = (a + b) + (a + c) + (a + d) + . . .
[4-2] a + a + a + . . . = a.
The extension of De Morgan's Theorem by 5-95 and 5-951 is especially
important. 5-91, 5-92, and 5-93 are different forms of the principle by
which any function may be expanded into a sum and any elements not
originally involved in the function introduced into it. Thus any expression
whatever may be regarded as a function of any given elements, even though
they do not appear in the expression,— a peculiarity of the algebra. 5-92,
the expression of the universe of discourse in any desired terms, or expansion
of 1, is the basis of many important procedures.
The theorems 5-91-5-981 are valid only if the number of elements
involved be finite, since proof depends upon the principle of mathematical
induction.
III. GENERAL PROPERTIES OF FUNCTIONS
We may use/Or), $(x, y), etc., to denote any expression which involves
only members of the class K and the relations x and + . The further
requirement that the expression represented by /(.r) should involve x or
its negative, -x, that $0, y) should involve x or -.r and y or -y, is unnecessary,
for if .r and -x do not appear in a given expression, there is an equivalent
expression in which they do appear. By 5-91,
a = a (x + -.r) = a x + a -x = (a x + a -x) (y + -?/)
= axy + ax-y + a-xy + a -x -y, etc.
a x + a -x may be called the expansion, or development, of a with reference
to x. And any or all terms of a function may be expanded with reference
to x, the result expanded with reference to y, and so on for any elements
and any number of elements. Hence any expression involving only ele-
The Classic, or Boole-Schroder, Algebra of Logic 133
ments in K and the relations x and + may be treated as a function of
any elements whatever.
If we speak of any a such that x = a as the "value of x ", then a value
of x being given, the value of any function of x is determined, in this algebra
as in any other. But functions of .r in this system are of two types: (1)
those whose value remains constant, however the value of x may vary, and
(2) those such that any value of the function being assigned, the value of x
is thereby determined, within limits or completely. Any function which
is symmetrical with respect to x and -x will belong to the first of these
classes; in general, a function which is not completely symmetrical with
respect to x and -x will belong to the second. But it must be remembered,
in this connection, that a symmetrical function may not look symmetrical
unless it be completely expanded with reference to each of the elements
involved. For example,
a + -a b + -b
is symmetrical with respect to a and -a and with respect to b and -b. Ex
panding the first and last terms, we have
a (b + -b) + -« b + (a + -a) -b = a b + a -b + -a b + -a -b = 1
whatever the value of a or of b. Any function in which an element, .r,
does not appear, but into which it is introduced by expanding, will be
symmetrical with respect to x and -x.
The decision what elements a given expression shall be considered a
function of is, in this algebra, quite arbitrary except so far as it is deter
mined by the form of result desired. The distinction between coefficients
and "variables" or "unknowns" is not fundamental. In fact, we shall
frequently find it convenient to treat a given expression first as a function-
say — of x and y, then as a function of z, or of x alone. In general, coef
ficients will be designated by capital letters.
The Normal Form of a Function. — Any function of one variable, /(.r),
can be given the form
A x + B -x
where A and B are independent of x. This is the normal form of functions
of one variable.
6 • 1 Any function of one variable, /(.r), is such that, for some A and some B
which are independent of .r,
/(.r) = A x + B -x
134 A Survey of Symbolic Logic
Any expression which involves only elements in the class K and
the relations x and + will consist either of a single term — a single
element, or elements related by x — or of a sum of such terms. Only
four kinds of such terms are possible: (1) those which involve x,
(2) those which involve -x, (3) those which involve both, and (4)
those which involve neither.7
Since the Distributive Law, 5 • 5, allows us to collect the coefficients
of x, of -x and of (x -x), the most general form of such an expres
sion is
px + q -x + r (x -x) + s
where p, q, r, and s are independent of x and -x.
But [2-4] r(x-x) = r-0 - 0.
And [5-9] s = s x + s -x.
Hence px + q -x + r (x -x) + s = (p + s) x + (q + s) -x.
Therefore, A = p + s, B = q + s, gives the required reduction.
The normal form of a function of n + 1 variables,
C2, • • • Xn, £„+!
may be defined as the expansion by the Distributive Law of
f(Xi, X2, ... Xn) -Xn+i +/ '(.Tl, .T2, . . . Xn) '-Xn+i
where/ and/ ' are each some function of the n variables, Xi, x*>, ... xn, and
in the normal form. This is a "step by step" definition; the normal form
of a function of two variables is defined in terms of the normal form of
functions of one variable; the normal form of a function of three variables
in terms of the normal form for two, and so on.8 Thus the normal form
of a function of two variables, $(x, y), will be found by expanding
(A x + B -x) y+(Cx + D -x) -y
It will be, Axy + B -x y + C x -y + D -x -y
The normal form of a function of three variables, ty(x, y, z), will be
Axyz + B-xyz + Cx-yz + D-x-yz + Exy-z + F-xy-z
+ G x -y -z + II -x -y -2
And so on. Any function in the normal form will be fully developed with
7 By a term which "involves" x is meant a term which either is x or has x "as a
factor". But "factor" seems inappropriate in an algebra in which h x is always contained
in x, h x ex.
8 This definition alters somewhat the usual order of terms in the normal form of func
tions. But it enables us to apply mathematical induction and thus prove theorems of a
generality not otherwise to be attained.
The Classic, or Boole-Schroder, Algebra of Logic 135
reference to each of the variables involved — that is, each variable, or its
negative, will appear in every term.
6-11 Any function may be given the normal form.
(a) By (> • 1 , any function of one variable may be given the
normal form.
(6) If functions of n variables can be given the normal form,
then functions of n + 1 variables can be given the normal form, for,
Let $(£1, Xz, ... xn, Xn+i) be any function of n + 1 variables.
By definition, its normal form will be equivalent to
f(Xi, Xz, ... Xn) 'Xn+l +f '(Xi, XZ, ... Xn)'-Xn+l
wLere / and / ' are functions of x\, x2, ... xn and in the normal
form.
By the definition of a function, $(#1, x2) . . . xn, xn+i) may be re
garded as a function of xn+i. Hence, by 6-1, for some A and some
B which are independent of xn^ i
$(Xi, Xz, . . . Xn, Xn+l) = A Xn+i + B -Xn+l
Also, by the definition of a function, for some/ and some/7
A = f(xi, x2, ... xn)
and B = / r(xi, Xz, . . . xn)
Hence, for some / and / ' which are independent of xn+\
&(Xi, X2, . . . Xn, Xn+l) — f(Xi, -To, . . . Xn)'Xn+l
+ /'(.Ti, XZt • • • Xn)—Xn+l
Therefore, if the functions of n variables, / and / ', can be given the
normal form, then $(0*1, Xz, . . . xn> xn+\) can be given the normal form.
(c) Since functions of one variable can be given the normal form,
and since if functions of n variables can be given the normal form,
functions of n + 1 variables can be given the normal form, therefore
functions of any number of variables can be given the normal form.
The second step, (6), in the above proof may seem arbitrary. That it
is valid, is due to the nature of functions in this algebra.
6-12 For a function of n variables, &(xi, x2, . . - xn), the normal form will
be a sum of 2n terms, representing all the combinations of xit positive or
negative, with xz, positive or negative, with . . . with .rn, positive or
negative, each term having its coefficient.
136 A Survey of Symbolic Logic
(a) A normal form function of one variable has two terms, and
by definition of the normal form of functions of n + 1 variables, if
functions of k variables have 2* terms, a function of k + 1 variables
will have 2k + 2k, or 2A+1, terms.
(b) A normal form function of one variable has the further
character described in the theorem; and if normal form functions
of k variables have this character, then functions of k -f- 1 variables
will have it, since, by definition, the normal form of a function of
k + 1 variables will -consist of the combinations of the (k + l)st
variable, positive or negative, with each of the combinations repre
sented in functions of k variables.
Since any coefficient may be 0, the normal form of a function may con
tain terms which are null. Where no coefficient for a term appears, the
coefficient is, of course, 1. The order of terms in the normal form of a
function will vary as the order of the variables in the argument of the
function is varied. For example, the normal form of 3>(x, y) is, by defini
tion,
A x y + B -x y + C x -y + D -x -y
and the normal form of V(y, x) is
P yx + Q-yx + Ry-x + S-y -x
Except for the coefficients, these differ only in the order of the terms and
order of the elements in the terms. And since + and x are both associa
tive and commutative, such a difference is not material.
6-15 Any two functions of the same variables can differ materially only
in the coefficients of the terms.
The theorem follows immediately from 6-12.
In consequence of 6-15, we can, without loss of generality, assume
that, for any two normal form functions of the same variables with which
we may be concerned, the order of terms and the order of variables in the
arguments of the functions is the same. And also, in any function of
n + 1 variables, 3>(Xl, x2, . . . xn, .rn+1), which is equated to
/Ol, .1-2, . . . Xn)'Xn+1 +f '(X,, .To, . . . Xn)'-Xn+l
xn+l may be any chosen one of the n + 1 variables. The convention that
it is always the last is consistent with complete generality of the proofs.
6-17 The product of any two terms of a function in the normal form is
null.
The Classic, or Boole-Schroder, Algebra of Logic 137
By 6-12, for any two terms of a function in the normal form,
there will be some variable, xn, such that xn is positive in one of
them and negative in the other; since otherwise the two terms
would represent the same combination of x\t positive or negative,
with 0*2, positive or negative, etc. Consequently, the product of
any two terms will involve a factor of the form xn -orn, and will
therefore be null.
Unless otherwise specified, it will be presumed hereafter that any func
tion mentioned is in the normal form.
The Coefficients in a Function. — The coefficients in any function can be
expressed in terms of the function itself.
6-21 If f(x) = A y. + 11 -.r, then /(I) = A.
For /(I) = A-1 + U--1 = A + B-0 = A.
6-22 If /(.r) = A x + B -.r, then /(O) = B.
For/(0) = A-O + B-Q = 0 + 7M == B.
6-23 /(.r) =/(!)• a; +/(0)~z.
The theorem follows immediately from 6-1, 0-21 and 0-22.
These laws, first stated by Boole, are very useful in reducing compli
cated expressions to normal form. For example, if
^(.r) = a c (d x + -d -x) + (c + .r) d
reduction by any other method would be tedious. But we have
*(1) = ac (VM+-d-0) + (c+l) d = acd+cd+d = d
and *(0) = a c (d-Q + -rf-1) + (c + 0) d = a c -d + c d
Hence the normal form of ty(x) is given by
\F(.r) = d x + (a c -d + c d) -x
Laws analogous to 6-23, also stated by Boole, may be given for functions
of more than one variable. For example,
/(.r, y) = /(I, l)-x y +/(0, 1) -a y +/(!, 0) -.r -y +/((), 0) •-.*• -//
and $(.r, y, z) = $(1, 1, l)-.r ?/ z + $(0, 1, 1) •-.)* y z + <I>(1, 0, l)-x-ijz
+ $(0,0, l)--.r-z/2+ $(1, l,0)-.r z/-s + $(0, 1,0) --.r y-z
+ $(1, 0, 0) • x -y -z + $(0, 0, 0) • -.r -// -z
We can prove that this method of determining the coefficients extends to
functions of anv number of variables.
138 A Surrey of Symbolic Logic
6-24 If C •{ f be any term of ^(.TI, xz, x3, ... #„)> then
I -.TI -.i-2 -.r3 . . . -xn J
^ -j [- will be the coefficient, C.
I 0, 0, 0, ... 0 J
(a) By 6-23, the theorem holds for functions of one variable.
(6) If the theorem hold for functions of A* variables, it will hold
for functions of k + 1 variables, for,
By 6-11, any function of k + 1 variables, $(#1, Xz, ... .T*, .
is such that, for some / and some / ',
—*
0, 0, ... 0 j
Therefore, if every term of / be of the form
{1,1, ... ll f .n xz... xk\
J I 0, 0, ... 0 j I -x, -x2 . . . -xk }
then every term of $ in which xk+i is positive will be of the form
ffi,i,...ii r x, x^.. xk\
] I 0, 0, ... 0 J I -.n -x2 . . . -xk J
and the coefficient of any such term will be / \ ' ' r , which,
(^ U, U, ... U J
And similarly, if every term of / ' be of the form
.,f 1,1, ... 11
• I 0, 0, ... 0 j -.rj -,r, . . . -xt j
then every term of * in which x/,+i is negative will be of the form
fl, 1, ... l| f .Tl x,... .rtl
Mo.o,. .ol'i-.,,-.,,. .-,,|'-
The Classic, or Boole-Schroder, Algebra of Logic 139
and the coefficient of any such term will be / ' f , which,
Hence every term of <£ will be of the form
r 1,1,. ..MI r « x2... „ .
I 0, 0, ... 0, 0 j I -.?! -.r2 . . . -xk -xk+l
(c) Since the theorem holds for functions of one variable, and
since if it hold for functions of k variables, it will hold for functions
of k + 1 variables, therefore it holds for functions of any number of
variables.
For functions of one variable, further laws of the same type as 6-23—
but less useful — have been given by Peirce and Schroder.
If f(x) = Ax + B-x:
6-25 /(I) = f(A + B) = f(-A + -B).
6-26 /(()) =f(A-B) =f(-A-B).
6-27 j(A) = A + B =/(-J8) = f(A-B) =f(A+-B)
6-28 f(B) = A-B = /(-4) = f(-A-B) = f(-A + B)
The proofs of these involve no difficulties and may be omitted.
In theorems to be given later, it will be convenient to denote the coef
ficients in functions of the form $(.?!, x2, . . . xn) by A\, Az, A3, . . . A^,
or by Ci, C2, C3, . . ., etc. This notation is perfectly definite, since the
order of terms in the normal form of a function is fixed. If the argument
of any function be (a*i, .r2, . . . xn), then any one of the variables, xk, will
be positive in the term of which Cm is the coefficient in case
p-2k~l < m ^ (p + l)-2k~l
where p = any even integer (including 0). Otherwise .r^ will be negative
in the term. Thus it may be determined, for each of the variables in the
function, whether it is positive or negative in the term of which Cm is the
coefficient, and the term is thus completely specified. We make no use
of this law, except that it validates the proposed notation.
Occasionally it will be convenient to distinguish the coefficients of those
terms in a function in which some one of the variables, say xk, is positive
from the coefficients of terms in which xk is negative. We shall do this
140 A Survey of Symbolic Logic
by using different letters, as PI, P2, PS, . . . , for coefficients of terms in
which Xk is positive, and Qb Q2, Q3, • • • for coefficients of terms in which Xk
is negative. This notation is perfectly definite, since the number of terms,
for a function of n variables, is always 2n, the number of those in which Xk
is positive is always equal to the number of those in which it is negative,
and the distribution of the terms in which xk is positive, or is negative, is
determined by the law given above.
The sum of the coefficients, Ai + Ao + A3+ . . ., will frequently be indi
cated by ^A or ^Ah', the product, Ai-A2'A3'... by JJ/l or YL^h-
h h
Since the number of coefficients involved will always be fixed by the func
tion which is in question, it wrill be unnecessary to indicate numerically the
range of the operators ^ and IJ .
The Limits of a Function. — The lower limit of any function is the prod
uct of the coefficients in the function, and the upper limit is the sum of
the coefficients.
6-3 A B cAx + B-xcA + B.
(A B)(A x + B -x) = A B x + AB -x = A B
Hence [1-9] ABcAx + B-x.
And (A x + B -x)(A + B) = Ax + AB -x + ABx + B -x
= (A B + A) x + (AB + B) -x.
But [5-4] A B + A == A, and AB + B = B.
Hence (.1 x + B -x) (A + B) = A x + B -x, and [1-9] A x + B -x
cA + B.
6-31 f(B)cf(x)cf(A).
[(3-3 and 6-26, 6-27]
6-32 If the coefficients in any function, F(xlf x2, . . . ;i-»), be Clt C2, C3,
then
(a) By 6 • 3, the theorem holds for functions of one variable.
(b) Let $(xi, xz, . . . xk, xk+i) be any function of k + 1 variables.
By (Ml, for some / and some / ',
$(.ri, *2, . . . xk, Xk+i) = f(xlt xz, . . . xk) -iCfc+i
+ /'(.TI, x2, . . . xk)'-xk+i (1)
Since this last expression may be regarded as a function of Xk+i in
which the coefficients are the functions / and / ', [6-3]
i, x2, . . . xk) x/ '(#,, x2, . . . xh) c $(.ri, x2, . . . xk, xk+i)
The Classic, or Boole-Schroder, Algebra of Logic 141
Let Ai{$}, Ai[3>\, A3{3>}, etc., be here the coefficients in <J>; Ai{f\,
^2 {/I, A*\f}, etc., the coefficients in/; and A,{ff\, J2{/'[, A3{f'},
etc., the coefficients in / '.
If rU{/} c/and IU{/'} c/', then [0-3]
and, by (1), EU {/J x JL'1 !/ ' } c <£.
But since (1) holds, any coefficient in $ will be either a coefficient
in / or a coefficient in / ', and hence
lUf/! xIL-i{/'! ==IU!*i
Hence if the theorem hold for functions of k variables, so that
II A {/} c/Cd, *2, . . . xk) and EU {/ ' ) c/ '(xlt x2, . . . xk),
then II^l^! c$(a-i, xz, ... xk, xk+}).
Similarly, since (1) holds, [0-23] $c/+/'.
Hence if /c £.•!{/} and / ' c Z^f/'l, then [5-31]
*<=Z^{/1 + Z^{/')
But since any coefficient in $ is either a coefficient in / or a coef
ficient in /', z^f/i + z^m == z^j.
Hence $ c ^A{$}.
Thus if the theorem hold for functions of k variables, it will
hold for functions of k + 1 variables.
(c) Since the theorem holds for functions of one variable, and
since if it hold for functions of k variables, it will hold for functions
of k -f 1 variables, therefore it holds generally.
As we shall see, these theorems concerning the limits of functions are
the basis of the method by which eliminations are made.
Functions of Functions. — Since all functions of the same variables may
be given the same normal form, the operations of the algebra may frequently
be performed simply by operating upon the coefficients.
6-4 If f(x) = A x + B -x, then 4/0)] - -.-1 .r + -B -x.
[3-4] -(A x + B -x) = -(A x)—(B -.r)
= (-A + -x) (-B + .r) = -.1 -B + -.1 .T + -B -x
= (-A -B + -A) x + (-A -B + -B) -x
But [5-4] -.1 -B + -.1 == -A and -.1 -B + -B = -B.
Hence -(,1 x + B -x) = -A x + -B -.r.
142 A Survey of Symbolic Logic
6-41 The negative of any function, in the normal form, is found by re
placing each of the coefficients in the function by its negative.
(a) By 6-4, the theorem is true for functions of one variable.
(b) If the theorem hold for functions of k variables, then it will
hold for functions of k + 1 variables.
Let F(XI, xz, • • . Xk, Xk+i) be any function of k + 1 variables.
Then by 6-11 and 3 • 2, for some / and some / ',
+f'(xi, x2, . . . xk)--xk+l]
But f(xi, Xz, . . . xk)-xk+i +f 'Oi> ^, • • • Zk)'-Xk+i may be regarded
as a function of a^+i.
Hence, by 6-4,
, a-o, . . . xk) -xk+l +/ '(xi, xz, . . . xk)'-xk+i]
Hence if the theorem be true for functions of k variables, so that
the negative of / is found by replacing each of the coefficients in / by
its negative and the negative of / ' is found by replacing each of the
coefficients in / ' by its negative, then the negative of F will be
found by replacing each of the coefficients in F by its negative, for,
as has just been shown, any term of
-xl} xz, . . . xkj
in which xk+i is positive is such that its coefficient is a coefficient in
-[/(>!, X2, ... Xk)]
and any term of
-[F(xlt xz, . . . xk,
in which xk+\ is negative is such that its coefficient is a coefficient in
-[/'(*„ X* ... .T,;)]
(c) Since (a) and (b) hold, therefore the theorem holds generally.
Since a difference in the order of terms is not material, 6-41 holds not
only for functions in the normal form but for any function which is com
pletely expanded so that every element involved appears, either positive
or negative, in each of the terms. It should be remembered that if any
term of an expanded function is missing, its coefficient is 0, and in the
negative of the function that term will appear with the coefficient 1.
The Classic, or Boole-Schroder, Algebra of Logic 143
6-42 The sum of any two functions of the same variables, $(0:1, T2, . . . xn)
and V(xi, #2, . . . xn), is another function of these same variables,
F(Xi, To, ... Xn),
such that the coefficient of any term in F is the sum of the coefficients of
the corresponding terms in $ and ^.
By 6-15, $(0:1,0:2, ... xn) and V(xi, x2, ... xn) cannot differ
except in the coefficients of the terms.
Let AI, At, A3, etc., be the coefficients in $; BI, B2, B3, etc., the
coefficients of the corresponding terms in ^. For any two such cor-
r J TI TO . . . Tn I f TI x* . xn]
responding terms, Ak 1 r and Bk l
L -0*1 -.1'2 . . . ~.l'n J I -.Ti -.r2 . . . -Xn }
f f *1 *2 ... .!•„ 1 f T! 0:2 ... Tn 1
Ah ] r + nk -j f-
I -Ti -1'2 • • • -Xn } t -Xi -X2 . . . 'Xn }
Ti T2 . . . Xn
-TI -T2 . . -xn
And since addition is associative and commutative, the sum of the
two functions is equivalent to the sum of the sums of such corre
sponding terms, pair by pair.
6-43 The product of two functions of the same variables, $(£1, o:2, ... xn)
and ^f(xi, 0*2, ... xn), is another function of these same variables,
F(Xi, .T2, • • - Xn),
such that the coefficient of any term in F is the product of the coeffi
cients of the corresponding terms in $ and ^.
T J Xi X2 . . . Xn } f Ti T2 ... Xn } .
Let Ak "j r and Bk'] r be any two
L -TI -T2 . . . -xn J I -T! -T2 . . . -xn }
corresponding terms in $ and ^.
, [ TI T2 . . . O'n 1 J Tl 0*2 ... Xn]
Ak\ fx5&H
( -Ti -T^ . . . -Xn J I -Ti -T2 . . . -Xn J
-(4*Jfc>{
t -Ti -To . . . -Tn J
By 6-15, $ and ^ do not differ except in the coefficients, and by
6-17, whatever the coefficients in the normal form of a function, the
product of any two terms is null. Hence all the cross-products of
terms in $ and ^ will be null, and the product of the functions will
144 A Survey of Symbolic Logic
be equivalent to the sum of the products of their corresponding terms,
pair by pair.
Since in this algebra two functions in which the variables are not the
same may be so expanded as to become functions of the same variables,
these theorems concerning functions of functions are very useful.
TV. FUNDAMENTAL LAWS OF THE THEORY OF EQUATIONS
We have now to consider the methods by which any given element
may be eliminated from an equation, and the methods by which the value
of an " unknown " may be derived from a given equation or equations. The
most convenient form of equation for eliminations and solutions is the
equation with one member 0.
Equivalent Equations of Different Forms. — If an equation be not in the
form in which one member is 0, it may be given that form by multiplying
each side into the negative of the other and adding these two products.
7-1 a = b is equivalent to a -b + -a b = 0.
[2-2] a = 6 is equivalent to the pair, a c b and b c a.
[4-9] a c b is equivalent to a -b = 0, and b c a to -a b = 0.
And [5 • 72] a -b = 0 and -a b = 0 are together equivalent to a -b
+ -a b = 0.
The transformation of an equation with one member 1 is obvious:
7-12 a = 1 is equivalent to -a = 0.
[3-2]
By 6-41, any equation of the form f(xi, ,r2, . . . xn) = 1 is reduced to the
form in which one member is 0 simply by replacing each of the coefficients
in / by its negative.
Of especial interest is the transformation of equations in which both
members are functions of the same variables.
7-13 If <J>(a,-i, .T2, . . . Xn) and ty(x1} x2, ... xn) be any two functions of the
same variables, then
is equivalent to F(XI, x2, . . . xn) = 0, where F is a function such that
if A i, A2, A3, etc., be the coefficients in <£, and B1} B2, Bs, etc., be the coef
ficients of the corresponding terms in ty, then the coefficients of the corre
sponding terms in F will be (Ai -Bi + -Al #0, (A2 -B2 + -A2 B2), (A3 -J53
+ -.43#3), etc.
The Classic, or Boole-Schroder, Algebra of Logic 145
By 7- 1, $ = ^ is equivalent to ($ x-\F) + (-<!> x ^) = 0.
By 6-41, -<£ and -^ are functions of the same variables as <£ and ^.
Hence, by 6-43, $ x-^ and -$ x ^ will each be functions of these
same variables, and by (i-42, (3> x-^) + (-$ x ty) will also be a
function of these same variables.
Hence 3>, ^, -<£, -^, <J> x-^, -$ x ^, and ($ x-^) + (-<£ x ^) are all
functions of the same variables and, by 6-15, will not differ except
in the coefficients of the terms.
If Ate be any coefficient in <£, and Bk the corresponding coefficient
in ^, then by 6-41, the corresponding coefficient in -$ will be -Ak
and the corresponding coefficient in -^ will be -Bk.
Hence, by 6-43, the corresponding coefficient in $x-^ will be
Ak -Bk, and the corresponding coefficient in -3> x fy will be -AkBk.
Hence, by 0-42, the corresponding coefficient in (<£ x-^) + (-<£ x ^)
will be Ak -Bk + -AkBk.
Thus ($ x-^) + (-$ x >^) is the function F, as described above, and
the theorem holds.
By 7- 1, for every equation in the algebra there is an equivalent equation
in the form in which one member is 0, and by 7 • 13 the reduction can usually
be made by inspection.
One of the most important additions to the general methods of the
algebra which has become current since the publication of Schroder's work
is Poretsky's Law of Forms.9 By this law, given any equation, an equiva
lent equation of which one member may be chosen at will can be derived.
7-15 a = 0 is equivalent to t = a -t + -a t.
If a - 0, a-t + -at = Q—t+l-t = t.
And if t = a -t + -a t, then [7-1]
(a -t + -a t) -t + (a t + -a -t) t = 0 = a -t + a t = a
Since t may here be any function in the algebra, this proves that every
equation has an unlimited number of equivalents. The more general form
of the law is :
7-16 a • = b is equivalent to t = (a b + -a -6) t + (a -b + -a b) -t.
[7 • 1] a = b is equivalent to a -b + -a b = 0.
And [6-4] -(a -b + -ab) = ab + -a -b.
Hence [7-15] Q.E.D.
The number of equations equivalent to a given equation and expressible
9 See Sept lois fondamentales de la theorie des egaliies logiques, Chap. i.
11
146 A Survey of Symbolic Logic
in terms of n elements will be half the number of distinct functions which
can be formed from n elements and their negatives, that is, 2'~n/2.
The sixteen distinct functions expressible in terms of two elements,
a and b, are:
a, -a, b, -b, 0 (i. e., a -a, b -b, etc.), 1 (i. e., a + -a, b + -b, etc.), a b,
a -^ _a if _rt -^ a + 6, a + -6, -a + 6, -a + -b, ab + -a -b, and a -b + -a b.
In terms of these, the eight equivalent forms of the equation a = b are:
a = b ; -«=-&; 0 = a -6 + -a & ; 1 = a 6 + -a -6 ; ab = a + b; a -b
= -ab; -a -b = -a + -b: and a + -6 = -a + b.
Each of the sixteen functions here appears on one or the other side of an
equation, and none appears twice.
For any equation, there is such a set of equivalents in terms of the
elements which appear in the given equation. And every such set has
what may be called its "zero member" (in the above, 0 = a -b + -a b)
and its "whole member" (in the above, 1 = a & + -«-&). If we observe
the form of 7-10, we shall note that the functions in the "zero member"
and "whole member" are the functions in terms of which the arbitrarily
chosen t is determined. Any t — the t which contains the function { = 0}
and is contained in the function { = 1 } . The validity of the law depends
simply upon the fact that, for any t, Octfcl, i. e., t = 1-/ + 0--/. It is
rather surprising that a principle so simple can yield a law so powerful.
Solution of Equations in One Unknown. — Every equation which is pos
sible according to the laws of the system has a solution for each of the un
knowns involved. This is a peculiarity of the algebra. We turn first to
equations in one unknown. Every equation in x, if it be possible in the
algebra, has a solution in terms of the relation c .
7-2 A x + B -x = 0 is equivalent to B c x c -A.
[5-72] A x + B -x = 0 is equivalent to the pair, Ax = 0 and
B -x = 0.
[4-9] B -x = 0 is equivalent to B ex.
And A x = 0 is equivalent to x -(-A) = 0, hence to x c-A.
7-21 A solution in the form // c x c A" is indeterminate whenever the equa
tion which gives the solution is symmetrical with respect to x and -.T.
First, if the equation be of the form A x + A -x = 0.
The solution then is, A ex c-A.
But if A x + A -x = 0, then A --= A (x + -x) = A x + A -x = 0, and
-A = 1.
The Classic, or Boole-Schroder, Algebra of Logic 147
Hence the solution is equivalent to Oc.rcl, which [5-61-63] is
satisfied by every value of x.
In general, any equation symmetrical with respect to x and -x
which gives the solution, // c x c K, will give also // c -x c K.
But if H c x and // c -x, then [4 • 9] // x = H and // -x = H.
Hence [1-62] II = 0.
And if xcK and -xcK, then [5-33] x + -xcK, and [4-8, 5-63]
K = I.
Hence H ex c K will be equivalent to 0 c a: c 1.
It follows directly from 7-21 that if neither x nor -x appear in an equa
tion, then although they may be introduced by expansion of the functions
involved, the equation remains indeterminate with respect to x.
7-22 An equation of the form A x + B -x = 0 determines x uniquely when
ever A = -B, B = -A.
[3-22] A = -B and -.1 = B are equivalent; hence either of
these conditions is equivalent to both.
[7 • 21 A x + B -x = 0 is equivalent to B c x c -A.
Hence if B = -A, it is equivalent to B ex cB and to -.1 c.r c-.4,
and hence [2-2] to x = B = -A.
In general, an equation of the form A x + B -x = 0 determines x be
tween the limits B and -A. Obviously, the solution is unique if, and only
if, these limits coincide; and the solution is wholly indeterminate only
when they are respectively 0 and 1, the limiting values of variables generally.
7-221 The condition that an equation of the form A x + B -x = 0 be pos
sible in the algebra, and hence that its solution be possible, is A B = 0.
By 6-3, A B cAx + B-x. Hence [5-65] if A x + B -x = 0, then
A B = 0.
Hence if A B =(= 0, then A x + B -x = 0 must be false for all values
of x.
And A x + B -x = 0 and the solution B ex c-A are equivalent.
A B = 0 is called the " equation of condition" of A x + B -x = 0: it is
a necessary, not a sufficient condition. To call it the condition that A x
+ B -x = 0 have a solution seems inappropriate: the solution Bc.xc.-A
is equivalent to A x + B -x = 0, whether A x + B -x = 0 be true, false, or
impossible. The sense in which A B = 0 conditions other forms of the
solution of A x + B -x = 0 will be made clear in what follows.
The equation of condition is frequently useful in simplifying the solution.
148 A Survey of Symbolic Logic
(In this connection, it should be borne in mind that A B = 0 follows from
A x + B -x = 0.) For example, if
a b x + (a + 6) -x = 0
then (a + b) c.r c-(a b). But the equation of condition is
a b (a + b) = ab = 0, or, -(a b) = 1
Hence the second half of the solution is indeterminate, and the complete
solution may be written
a + b ex
However, this simplified form of the solution is equivalent to the original
equation only on the assumption that the equation of condition is satisfied
and a b = 0.
Again suppose ax + b -x + c = 0
Expanding c with reference to x, and collecting coefficients, we have
(a + c) x + (b + c) -x = 0
and the equation of condition is
(a + c)(b + c) = a b + a c + b c + c = ab + c = 0
The solution is b + c c x c -a -c
But, by 5-72, the equation of condition gives c = 0, and hence -c = 1.
Hence the complete solution may be written
b c x c -a
But here again, the solution b ex c-a is equivalent to the original equation
only on the assumption, contained in the equation of condition, that c = 0.
This example may also serve to illustrate the fact that in any equation
one member of which is 0, any terms which do not involve x or -x may be
dropped without affecting the solution for x. If a x + b-x + c = 0, then
by 5-72, a x + b -x = 0, and any addition to the solution by retaining c will
be indeterminate. All terms which involve neither the unknown nor its
negative belong to the "symmetrical constituent" of the equation — to be
explained shortly.
Poretsky's Law of Forms gives immediately a determination of x which
is equivalent to the given equation, whether that equation involve x or not.
7 • 23 A x + B -x = 0 is equivalent to x = -A x + B -x.
[7-15] A x + B -x = 0 is equivalent to
x = (A x + B -x) -x + (-A x + -B -r) r - B -x + -Ax
The Classic, or Boole-Schroder, Algebra of Logic 149
This form of solution is also the one given by the method of Jevons.10
Although it is mathematically objectionable that the expression which
gives the value of x should involve x and -.r, this is in reality a useful and
logically simple form of the solution. It follows from 7-2 and 7-23 that
x = -A x + B -x is equivalent to B c x c -.1.
Many writers on the subject have preferred the form of solution in
which the value of the unknown is given in terms of the coefficients and an
undetermined (arbitrary) parameter. This is the most "mathematical"
form.
7-24 If A E = 0, as the equation A x + B -x = 0 requires, then A x
+ B -x = 0 is satisfied by x = B -u + -A u, or x = B + u -A, where u is
arbitrary. And this solution is complete because, for any x such that
A x + B -x = 0 there is some value of u such that x = B -u + -A u = B
+ u -A .
(a) By 6-4, if x — B -u + -A u, then -x = -B -u + A u.
Hence if x = B -u + -A ?/, then
A x + B -x = A (B -u + -B u) + B (-B -u + A u)
= A B -?/ + A B u = A B
Hence if A B = 0 and .r = B -u + -A u, then whatever the value
of u, A x + B -x = 0.
(b) Suppose x known and such that A x + B -x = 0.
Then if x = B -u + -A u, we have, by 7-1,
(B -u + -A if) -x + (-B -u + A n) x
= (A x + -A -x) u + (B -x + -B x) -u = 0
The condition that this equation hold for some value of u is, by 7 • 221,
(A x + -A -x) (B -x + -B x) = A -B x + -.1 B -x = 0
This condition is satisfied if A x + B -x = 0, for then
A (B + -B) x + (A + -A) B -.r = AB + A -B x + -A B -x = 0
and by 5-72, A -B x + -A B -x = 0.
(c) If A B = 0, then B -u + -A u = B + u-A, for:
If A B = 0, then A B u = 0.
Hence B -u + -A u = B -u + -A (B + -B) u + A B u
= B -u + (A + -A) B u + -.1 -B u = B (-u + u) + -A -B u
= B + -A -B u.
But [5-85] B + -A -Bu = B+n -A.
10 See above, p. 77.
150 A Survey of Symbolic Logic
Only the simpler form of this solution, x = B + u -A, will be used hereafter.
The above solution can also be verified by substituting the value given
for x in the original equation. We then have
A (B -u + -A ?/) + B (-B -u + Au) = A B-u + ABu = A B
And if A B = 0, the solution is verified for every value of u.
That the solution, x = B-u + -Au = B + u-A, means the same as
B ex c-A, will be clear if we reflect that the significance of the arbitrary
parameter, u, is to determine the limits of the expression.
If u = 0, B -u + -A u = B + u -A = B.
If u = 1, B-u + -Au = -A and B + u-A = B + -A. But when
AB = 0, B + -A = -A B + -A = -A.
Hence x = B -u + -A u = B + u-A simply expresses the fact, otherwise
stated by B ex c-A, that the limits of x are B and -A.
The equation of condition and the solution for equations of the form
C x + D -x = 1, and of the form A x + B -x = C x + D -x, follow readily
from the above.
7 • 25 The equation of condition that C x + D -x = 1 is C + D = 1, and the
solution of C x + D -x = 1 is -D c x c C.
(a) By 6-3, Cx + D-xcC + D.
Hence if there be any value of x for which C x + D -x = 1, then
necessarily C + D = 1.
(6) If Cx + D-x == 1, then [6-4] -Cx + -D-x = 0, and [7-2]
-DcxcC.
7-26 If C + D = 1, then the equation Cx + D-x = 1 is satisfied by
x = -D + uC, where u is arbitrary.
Since [6-4] Cx + D-x = 1 is equivalent to -Cx + -D-x = 0,
and C + D = 1 is equivalent to -C -D = 0, the theorem follows
from 7-24.
7-27 If A x + B -x = C x + D -x, the equation of condition is
(A -C + -A C)(B -D + -BD) =0
and the solution is B -D + -B D c x c A C + -A -C, or
x = B -D +-B D + u (A C + -A -C), where u is arbitrary.
By 7 • 13, A x + B -x = C x + D -x is equivalent to
(A-C + -AC)x + (B-D + -Bd-x = 0.
The Classic, or Boole-Schroder, Algebra of Logic 151
Hence, by 7-221, the equation of condition is as given above.
And by 7-2 and 7-24, the solution is
B -D + -B D c x c - (A -C + -A C) , or
x = B -D +-B D + u--(A -C + -A C), where u is arbitrary.
And [6-4] -(A -C + -AC) = A C + -A -C.
The subject of simultaneous equations is very simple, although the
clearest notation we have been able to devise is somewhat cumbersome.
7-3 The condition that n equations in one unknown, Alx + B1-x = 0,
A*x + B2 -x = 0, . . . Anx + Bn -x — 0, may be regarded as simultaneous, is
the condition that
£ (Ah Bk) = 0
fi, k
And the solution which they give, on that condition, is
^Bkcxc H-Ak
k k
or x = ^Bk + u> II -^*» where u is arbitrary.
k k
By 6-42 and 5-72, Alx + 51 -x = 0, A*x + B2 -x = 0,
Anx + Bn -x = 0, are together equivalent to
(A1 + A2 + . . . + An)x + (B1 + B* + . . . + B") -x = 0
or , z +
k k
-* = 0
By 7 • 23, the equation of condition here is
A* x £ Bk = 0
A: A'
But ZM*xZ£fc = (.I1 + , I2 +...+.4«
A- k
= A1 Bl + A1 B- + . . . + A1 Bn + A2 B1 + A* B2 + . . + A2 B
+ A*Bl + A* B2 + . . . + A* B- + . . . + A" B1 + . . . + An B
= Z(^£*).
h,k
And by 7-2 and 7-24, the solution here is
or *= «-
A- A-
And by 5-95, -{£^M == II ~Ak.
k k
It may be noted that from the solution in this equation, ?i2 partial solu
152 A Survey of Symbolic Logic
tions of the form Bh c.r c-A> can be derived, for
Wc^B* and H -A* c-A>'.
k k
Similarly, 22" — 1 partial solutions can be derived by taking selections of
members of ^Z Hk and II -Ak.
Symmetrical and Unsymmetrical Constituents of Equations. — Some of
the most important properties of equations of the form A x + B -x = 0 are
made clear by dividing the equation into two constituents — the most
comprehensive constituent which is symmetrical with respect to .r and -x,
and a completely unsymmetrical constituent. For brevity, these may
be called simply the "symmetrical constituent" and the ''unsymmetrical
constituent ". In order to get the symmetrical constituent complete, it
is necessary to expand each term with reference to every element in the
function, coefficients included. Thus in .1 x + B -x = 0 it is necessary to
expand the first term with respect to B, and the second with respect to A.
A (B + -B) x + (A + -A) B -x = A Bx + AB -x + A -B x + -A B -x = 0
By 5 • 72, this is equivalent to the two equations,
.1 B (x + -.r) = .1 B = 0 and .1 -B x + -.4 B -x = 0
The first of these is the symmetrical constituent; the second is the unsym
metrical constituent. The symmetrical constituent will always be the equa
tion of condition, while the unsymmetrical constituent will give the solution.
But the form of the solution will most frequently be simplified by con
sidering the symmetrical constituent also. The unsymmetrical constituent
will always be such that its equation of condition is satisfied a priori. Thus
the equation of condition of
A -B x + -A B -x = 0
is (.1 -B)(-A B) = 0, which is an identity.
By this method of considering symmetrical and unsymmetrical con
stituents, equations which are indeterminate reveal that fact by having
no unsymmetrical constituent for the solution. Also, the method enables
us to treat even complicated equations by inspection. Remembering that
any term in which neither x nor -x appears belongs to the symmetrical
constituent, as does also the product of the coefficients of x and -.1-, the
separation can be made directly. For example,
(c + x) d + -c -d + (-a + -x) b = 0
The Classic, or Boole-Schroder, Algebra of Logic 153
will have as its equation of condition
c d + -c -d + -a b + b d = 0
and the solution will be
b c .r c -d
Also, as we shall see shortly, the symmetrical constituent is always the
complete resultant of the elimination of x.
The method does not readily apply to equations which do not have
one member 0. But these can always be reduced to that form. Plow it
extends to equations in more than one unknown will be clear from the
treatment of such equations.
Eliminations. — The problem of elimination is the problem, what equa
tions not involving x or -x can be derived from a given equation, or equa
tions, which do involve x and -.r. In most algebras, one term can, under
favorable circumstances, be eliminated from two equations, two terms
from three, n terms from n + 1 equations. But in this algebra any number
of terms (and their negatives) can be eliminated from a single equation;
and the terms to be eliminated may be chosen at will. The principles
whereby such eliminations are performed have already been provided in
theorems concerning the equation of condition.
7-4 A B = 0 contains all the equations not involving .r or -.r which can
be derived from A x + B -x = 0.
By 7-24, the complete solution of A x + B -.r = 0 is
x = B -?/ + -A u
Substituting this value of x in the equation, we have
.1 (B -u + -A 11) + B (-B -u + A u) = A B -u + A B u = A B = 0
Hence A B = 0 is the complete resultant of the elimination of .r.
It is at once clear that the resultant of the elimination of x coincides
with the equation of condition for solution and with the symmetrical con
stituent of the equation.
7-41 If n elements, x\, .r2, .r3, . . . xn, be eliminated from any equation,
F(XI, #2, ,T3, . . . xn) = 0, the complete resultant is the equation to 0 of the
product of the coefficients in F(xit x2, .r3, . . . xn).
(a) By 6-1 and 7-4, the theorem is true for the elimination of
one element, .r, from any equation, /(.r) = 0.
(b) If the theorem hold for the elimination of k elements, x\, X2,
154 A Survey of Symbolic Logic
. . . xk, from any equation, $(.TI, xZt ... xk) = 0, then it will hold
for the elimination of k + 1 elements, .TI, .TL>, . . . xk, Xk+i, from any
equation, V(xi, xz, . . . xki xk+i) = 0, for:
By (Ml, V(xi, .1-2, ... xk, xk+i) = /(.ri, xz, . . . xk)-xk+i
+f'(xi, xz, . . . xk)--Xk+i-
And the coefficients in ¥ will be the coefficients in / and / '. By
7-4, the complete resultant of eliminating xk+i from
/(.Ti, .To, . . . Xk)'Xk+i + / '(Xi, XZ, - . . Xk)~Xk+i = 0
is f(xi, .T2, . . . xk) x/ 'Oi, .r2, . • • a'*) = 0
And by 6-43, /(.TI, .T2, . . . xk) X/'(.TI, or2, . . . a**) = 0 is equivalent
to <£(.TI, 2*2., ... a;*;) = 0, where $ is a function such that if the
coefficients in/ be PI, P2, P3, etc., and the corresponding coefficients
in/' be Qi, Qz, Qs, etc., then the corresponding coefficients in <£ will
be PiQi, P2§2, PsQa, etc. Hence if the theorem hold for the elimina
tion of k elements, .TI, xZ) ... xk, from $(.TI, xz, . . . xk) = 0, this
elimination will give
(P.Qi)(ft<2.)(PiC.). . • •• (f iftft- • .ddQ.. • •) = 0,
where PiP^Pz- • -QiQzQs- • - is the product of the coefficients in 3>,
or in / and / ' — i. e., the product of the coefficients in SF.
Hence if the theorem hold for the elimination of k elements, .TI, xz,
. . . xk, from <£(.TI, xz, . . . xk) = 0, it will hold for the elimination of
k + 1 elements, Xi, xz, ... xk, xk+i, from ^(.TI, xz, . . . xk, xk+i) = 0,
provided xk+i be the first eliminated.
But since the order of terms in a function is immaterial, and for
any order of elements in the argument of a function, there is a
normal form of the function, xk+i in the above may be any of the
k + 1 elements in \F, and the order of elimination is immaterial.
(c) Since (a) and (b) hold, therefore the theorem holds for the
elimination of any number of elements from the equation to 0 of
any function of these elements.
By this theorem, it is possible to eliminate simultaneously any number of
elements from any equation, by the following procedure: (1) Reduce the
equation to the form in which one member is 0, unless it already have that
form; (2) Develop the other member of the equation as a normal-form
function of the elements to be eliminated; (3) Equate to 0 the product of
the coefficients in this function. This will be the complete elimination
resultant.
The Classic, or Boole-Schroder, Algebra of Logic 155
Occasionally it is convenient to have the elimination resultant in the
form of an equation with one member 1, especially if the equation which
gives the resultant have that form.
7-42 The complete resultant of eliminating n elements, x\t x2, ... xn,
from any equation, F(XI, x2, ... xn) == 1, is the equation to 1 of the sum
of the coefficients in F(XI, x2, ... xn).
Let AI, A2, A3, etc., be the coefficients in F(XI, x2, . . . xn).
F(XI, xz, ... xn) = 1 is equivalent to -[F(xi, Xz, ... xn)] = 0. And
by 6-41, -[F(xi, x2, ... xn)] is a function, $(xi, x2, ... xn), such
that if any coefficient in F be AU, the corresponding coefficient in $
will be -A/c.
Hence, by 7-41, the complete resultant of eliminating x\9 x2, ... xn,
from F(XI, Xz, ... xn) = 1 is
U-A =0, or -{ 11-^1 1 -= 1
But [5-95] -{ 11-4} =-- Z A. Hence Q.E.D.
For purposes of application of the algebra to ordinary reasoning, elimina
tion is a process more important than solution, since most processes of
reasoning take place through the elimination of "middle" terms. For
example :
If all b is x, b c x, b -x = 0
and no a is x, a x = 0,
then ax + b -x = 0. Whence, by elimination, a b = 0, or no a is b.
Solution of Equations in more than one Unknown. — The complete solu
tion of any equation in more than one unknown may be accomplished by
eliminating all the unknowns except one and solving for that one, repeating
the process for each of the unknowns. Such solution will be complete
because the elimination, in each case, will give the complete resultant which
is independent of the unknowns eliminated, and each solution will be a
solution for one unknown, and complete, by previous theorems. How
ever, general formulae of the solution of any equation in n unknowns, for
each of the unknowns, can be proved.
7 • 5 The equation of condition of any equation in n unknowns is identical
with the resultant of the elimination of all the unknowns; and this resultant
is the condition of the solution with respect to each of the unknowns sepa
rately.
(a) If the equation in n unknowns be of the form
z, . . xn) = 0:
156 A Survey of Symbolic Logic
Let the coefficients in F(XI, x2, ... xn) be AI, A2, A3, etc. Then,
by 6 -32,
II A c F(xi, x2, ... xn)
and [5-65] JJ A = 0 is a condition of the possibility of
F(xlt 0-2, . . . z») - 0
And [7-41] U A = 0 is the resultant of the elimination of Xi, x2,
. . . xn, from F(x\, 0*2, . . . xn) = 0.
(6) If the equation in n unknowns have some other form than
F(XI, x2) . . . Xn) = 0, then by 7-1, it has an equivalent which is
of that form, and its equation of condition and its elimination
resultant are the equivalents of the equation of condition and
elimination resultant of its equivalent which has the form
F(xlf X2, ... Xn) =0
(c) The result of the elimination of all the unknowns is the
equation of condition with respect to any one of them, say xk,
because :
(1) The equation to be solved for Xk will be the result of eliminat
ing all the unknowns but Xk from the original equation; and
(2) The condition that this equation, in which Xk is the only
unknown, have a solution for Xk is, by (a) and (6), the same as the
result of eliminating Xk from it.
Hence the equation of condition with respect to Xk is the same as
the result of eliminating, from the original equation, first all the
other unknowns and then x^
And by 7-41 and (6), the result of eliminating the unknowns is
independent of the order in which they are eliminated.
Since this theorem holds, it will be unnecessary to investigate separately
the equation of condition for the various forms of equations; they are
already given in the theorems concerning elimination.
7-51 Any equation in n unknowns, of the form F(XI, x2, . . . xn) = 0,
provided its equation of condition be satisfied, gives a solution for each
of the unknowns as follows: Let xk be any one of the unknowns; let P], P2,
Pa, etc., be the coefficients of those terms in F(XI, x2, . . . xn) in which Xk
is positive, and Qi, Q2, Q3, etc., the coefficients of those terms in which Xk
is negative. The solution then is
II Q ex* c 2 -P, or Xk = II Q + u- 2 -P, where u is arbitrary.
The Classic, or Boole-Schroder, Algebra of Logic 157
(a) By (Ml, for some / and some / ', F(x\, .TO, . . . x,n) = 0 is
equivalent to /(.ri, .r2, . . . .rn_i)-.rn +/'(.ri, .T2, . . . .rn_i)--.rn --= 0.
Let the coefficients in / be PI, P2, P3, etc., in /' be ^i, Qz, Qs, etc.
Then PI, P2, P3, etc., will be the coefficients of those terms in F
in which Xk is positive, Qi, Q*, Q3, the coefficients of terms in F in
which Xk is negative.
If /(.TI, .T2, • • • .Tn_])-.rn be regarded as a function of the variables,
.TI, .TO, . . . £n_i, its coefficients will be Pi.rn, P2.rn, P3.Tn, etc.
And if /'Oi, .r2, . . . rn-i)'-xn be regarded as a function of .TI, o*2,
. . . xn-i, its coefficients will be Q1 -xn, Q2 -xn, (?3 -xn, etc.
Hence, by 6-42,
/(.?!, X'2, • • • Xn-l) 'Xn + / '(£1, .T2, . . . Xn-i)'-Xn = 0
is equivalent to ^(.TI, .TO, . . . xn-i) = 0, where ^ is a function in
which the coefficients are (PiXn + Qi-xn), (P2 xn + Q2 -.rn), (P3^n
+ §3-^n), etc.
And ^(iCi, T2, ... .rn_i) = 0 is equivalent to F(XI, .r2, . . xn) = 0.
By 7-41, the complete resultant of the elimination of 0*1, .r2, . . .rn_i
from ty(xi, x2, . . . xn-i) = 0 will be the equation to 0 of the product
of its coefficients,—
But any expression of the form Pnrn + Qr -xn is a normal form func
tion of xn. Hence, by 6 • 43,
By 7-2 and 7-24, the solution of H P^n+ Qr -^» = 0 is
And [5-951] -{II P] ---- T.-P-
(b) Since the order of terms in a function is immaterial, and
for any order of the variables in the argument of a function there is a
normal form of the function, xn in the above may be any one of the
variables in F(x^ .T2, . . . .Tn), and /(.TI, .i'2, . . . .rn_t) and /'(.n, .r2,
. . . .Tn_0 each some function of the remaining n — 1 variables.
Therefore, the theorem holds for any one of the variables, xk.
That a single equation gives a solution for any number of unknowns
is another peculiarity of the algebra, due to the fact that from a single
equation any number of unknowns may be eliminated.
158 A Survey of Symbolic Logic
As an example of the last theorem, we give the solution of the exemplar
equation in two unknowns, first directly from the theorem, then by elimina
tion and solution for each unknown separately.
(1) A x y + B -x y + C x -y + D -x -y = 0 has the equation of condition,
ABCD = 0
Provided this be satisfied, the solutions for x and y are
BDcxc-A+-C, or x = B D + u (-A + -C)
CDcyc-A+-B, or y = C D + u (-A+-B)
(2) A x y + B -x y + C x -y + D -x -y = 0 is equivalent to
(a) (Ax + B -x) y+(Cx + D -x) -y = 0
and to (b) (A y + C -y) x + (B y + D -y) -.r = 0
Eliminating y from (a), we have
(Ax + B-x)(Cx + D-x) = ACx + BD-x = 0
The equation of condition with respect to x is, then,
(AC)(BD) = ABCD = 0
And the solution for x is
BDcxc-(A C), or x = BD + u--(A C). And -(A C) = -A + -C
Eliminating x from (6), we have
(Ay+C-y)(By + D-y) = A B y + C D -y = 0
The equation of condition with respect to y is, then, A BCD = 0. And
the solution for y is
CDcyc-(AB), or y = CD + v-(AB). And -(A B) = -A+-B
Another method of solution for equations in two unknowns, x and y,
would be to solve for y and for -y in terms of the coefficients, with x and u
as undetermined parameters, then eliminate y by substituting this value
of it in the original equation, and solve for x. By a similar substitution,
x may then be eliminated and the resulting equation solved for y. This
method may inspire more confidence on the part of those unfamiliar with
this algebra, since it is a general algebraic method, except that in other
algebras more than one equation is required.
The solution of A x y + B -x y + C x -y + D -x -y = 0 for y is
y = (C x + D -x) + u--(A x + B -x) = (C + u -A) x +(D + u -B) -x
The Classic, or Boole-Schroder, Algebra of Logic 159
The solution for -y is
-y = (A x + B -x) + v-(C x + D -x) = (A + v -C) x + (B + v -D) -x
Substituting these values for y and -y in the original equation,
(A x + B -x)[(C + u -A) x + (D + n -B) -x]
+ (Cx + D -x)[(A + v -C) x + (B + v -D) -x\
= A (C + u -A) x + B (D + u -B) -x + C (A + v -C) x + 1) (B + v -D) -x
= AC x + 7? D -x = 0.
Hence B D ex c-A + -C.
Theoretically, this method can be extended to equations in any number of
unknowns: practically, it is too cumbersome and tedious to be used at all.
7 • 52 Any equation in n unknowns, of the form
F(XI, x2, ... xn) = f(xi, x2, ... xn)
gives a solution for each of the unknowns as follows: Let xk be any one
of the unknowns; let PI, P2, P3, • • • Qi, Qi, Q.a, ... be the coefficients in F,
and J/i, If 2, 3/3, . . . AI, AT2, AT3, ... the coefficients of the corresponding
terms in /, so that Pr and 1/V are coefficients of terms in which xk is positive,
and Qr and Nr are coefficients of terms in which Xk is negative. The solu
tion for Xk then is
II (Qr 'Nr + -Q, Nr) C Xk C £ (Pr Mr + ~P, - J/r)
or
Xk = II (Qr -Nr + -Qr Nr) + U ' £ (Pr ^
r r
By 7-13, 7'X.i'i, ;i'2, ... .rn) = /Oi, £2, ... .rn) is equivalent to
i, #2, • • • -I'n) = 0, where $ is a function such that if Ar and l?r
be coefficients of any two corresponding terms in F and /, then the
coefficient of the corresponding term in $ will be Ar -Br + -Ar B,.
Hence, by 7-51, the solution will be
II (Qr "Nr + -Qr N r) C Xk C £ -(Pr -Mr + -Pr 3/r)
Or
= II (Qr -Nr + -Qr N r) + H • £ -(Pr - J/r + -P, J/r)
And [0-4] -(Pr-J/r + -Pr J/r) = (Pr J/r + -Pr-J/r).
7-53 The condition that m equations in n unknowns, each of the form
F(XI, x», . . . xn) = 0, may be regarded as simultaneous, is as follows:
Let the coefficients of the terms in F1, in the equation Fl(xlf .r2, . . xn) = 0,
be Pi1, Pa1, P31, . Qi1, QJ, Qs1, • • • 5 let the coefficients of the corre-
160 A Survey of Symbolic Logic
spending terms in F2, in the equation F2(.ri, ,r2, . . . .rn) = 0, be Pi2, P22,
Pa2, • • • Q\~, Qz~, Qs2, • . . ; the coefficients of the corresponding terms in
Fm, in the equation Fm(xlt xz, . - . xn) = 0, be PIW, P2m, P3W, . . . Qim, Qzm,
zm, .... The condition then is
;• A r h
Or if Cr* be any coefficient, whether P or Q, in P\ the condition is
And the solution which n such equations give, on this condition, for any
one of the unknowns, Xk, is as follows: Let PI^, P/, P37i, ... be the coef
ficients of those terms, in any one of the equations Fh = 0, in which x-k is
positive, and let Qih, Qih, Q3h, ... be the coefficients of those terms, in
Fh = 0, in which Xk is negative. The solution then is
nizvic^czn-w]
r h r ft
or ** = ![ tZ Qr''] + w-Z III -Pr"]
r h r h
By 6-42, m equations in n unknowns, each of the form F(x\, xz,
. . . xn) = 0, are together equivalent to the single equation &(x-i, xz,
. . . xn) — 0, where each of the coefficients in <£ is the sum of the
corresponding coefficients in F1, F2, F3, . . . Fm. That is, if Prl, P,2,
. . . Prm be the coefficients of corresponding terms in P1, F2, . . . Fm,
then the coefficient of the corresponding term in <£ will be
P,1 + Pr2 + . . . + P, .-, or 2 Prh
h
and if Qr1, Qr2, . . . Qrm be the coefficients of corresponding terms in
F1, F2, . . . Fm, then the coefficient of the corresponding term in <£
will be
h
The equation of condition for <£> = 0, and hence the condition that
F1 = 0, F2 = 0, ... Fm = 0 may be regarded as simultaneous, is
the equation to 0 of the product of the coefficients in $; that is,
h h h h h h
or nizjvixiiizcr'i-o
r h r h
And by 7-51, the solution of $(#1, -T2, • • • #n) = 0 for Xk is
II I S Qrh] c TA c V
The Classic, or Boole-Schroder, Algebra of Logic 161
or % =
And by 5-95, -[£ P«\ = -P/.
h h
7 • 54 The condition that m equations in n unknowns, each of the form
F(XI, x2, ... xn) = f(xi, x2, ... xn)
may be regarded as simultaneous, is as follows: Let the coefficients in Fl,
in the equation F1 =fl, be Px1, P,1, Pj1, . . . QJ, Q1, Q.1, . . . , and let the
coefficients of the corresponding terms in / l, in the equation F1 = fl, be
Mi1, MJ, Ms1, . . . Ni1, As1, AY, . . . ; let the coefficients of the corresponding
terms in F2, in the equation F2 =f2, be P^, P22, P32, . . . QS, Q22, Q32,
and let the coefficients of the corresponding terms in / 2 be M i2, M22, Ms2,
. . . Ni2, A22, A32, . . . ; let the coefficients of the corresponding terms in Fm,
in the equation Fm = f m, be Pim, P2m, P3m, . . . Qim, Qzm, $3W, . . ., and
let the coefficients of the corresponding terms in / m be Mim, Mzm, Msm,
. . . ATim, A2m, AT3m, .... The condition then is
II I H (Prh -MS + ~Prh MS)] X JJ I E (Qr* -N,» + -(?r* A/)] = 0
r h r h
or if Arh represent any coefficient in Fh, whether P or Q, and Brh represent
the corresponding coefficient in / h, whether M or A, the condition is
II IZ (Arh -BS + -Arh Brh)} = 0
And the solution which m such equations give, on this condition, for any
one of the unknowns, .TA-, is as follows: Let Prh and Mrh be the coefficients
of those terms, in any one of the equations Fh = f h, in which a^ is positive,
and let Qrh and Nrh be the coefficients of the terms, in Fh = fh, in which xk
is negative. The solution then is
or a* - II [ Z (Qr» -A7/ + -9," AV)] + « • E [II
By 7-13, ^(.TI, #2, • • . arn) = /A(-TI» a?2, ... »n) is equivalent to
i, .r2, . . . xn) = 0, where ^ is a function such that if Qrh and Nrh
be coefficients of corresponding terms in Fh and / h, the coefficient
of the corresponding term in * will be Qrh -Nrh + -Qr* AV, and if
Prh and Jf rA be coefficients of corresponding terms in Fh and / h, the
coefficient of the corresponding term in ^ will be Prh -Mrh + -Prh Mrh.
And -(Prh -MS + -Prh Mrh) = Prh Mrh + -P/ -Mr\
Hence the theorem follows from 7 • 53.
12
162 A Survey of Symbolic Logic
F(XI, .1*2, . . . xn) = f(xi, xz, . . . a-n) is a perfectly general equation, since
F and / may be any expressions in the algebra, developed as functions of
the variables in question. 7-54 gives, then, the condition and the solution
of any number of simultaneous equations, in any number of unknowns, for
each of the unknowns. This algebra particularly lends itself to generaliza
tion, and this is its most general theorem. It is the most general theorem
concerning solutions in the whole of mathematics.
Boole's General Problem. — Boole proposed the following as the general
problem of the algebra of logic.11
Given any equation connecting the symbols x, y, ... w, z, .... Re
quired to determine the logical expression of any class expressed in any
way by the symbols x, y, ... in terms of the remaining symbols u\ z, ....
We may express this: Given t = f(x, y, . . .) and $>(.r, y, . . .) = ty(w, z,
. . . ) ; to determine t in terms of w, z, .... This is perfectly general, since
if x, y, . . . and w, z, . . . are connected by any number of equations, there
is, by 7-1 and 5-72, a single equation equivalent to them all. The rule
for solution may be stated: Reduce both t = f(x, y, . . .) and 3>(.r, y, . . .)
= V(w, 2, . . .) to the form of equations with one member 0, combine them
by addition into a single equation, eliminate x, y, . . . , and solve for t. By
7-1, the form of equation with one member 0 is equivalent to the other
form. And by 5-72, the sum of two equations with one member 0 is
equivalent to the equations added. Hence the single equation resulting
from the process prescribed by our rule will contain all the data. The
result of eliminating will be the complete resultant which is independent
of these, and the solution for t will thus be the most complete determination
of t in terms of w, z, . . . afforded by the data.
Consequences of Equations in General. — A word of caution with refer
ence to the manipulation of equations in this algebra may not be out of
place. As compared with other algebras, the algebra of logic gives more
room for choice in this matter. Further, in the most useful applications
of the algebra, there are frequently problems of procedure which are not
resolved simply by eliminating this and solving for that. The choice of
method must, then, be determined with reference to the end in view. But
the following general rules are of service:
(1) Get the completest possible expression = 0, or the least inclusive
possible expression == 1.
a + b + c+ . . . = 0 gives a = 0, 6 = 0, c = 0, . . . , a + 6 = 0, a + c = 0,
of Thought, p. 140.
The Classic, or Boole-Schroder, Algebra of Logic 163
etc. But a — 0 will not generally give a + b = 0, etc. Also, a == 1 gives
a + b = 1, a + . . . == 1, but a + b == 1 will not generally give a == 1.
(2) Reduce any number of equations, with which it is necessary to deal,
to a single equivalent equation, by first reducing each to the form in which
one member is 0 and then adding. The various constituent equations
can always be recovered if that be desirable, and the single equation gives
other derivatives also, besides being easier to manipulate. Do not forget
that it is possible so to combine equations that the result is less general
than the data. If we have a = 0 and 6 = 0, we have also a = b, or a b = 0,
or a + b = 0, according to the mode of combination. But a + b = 0 is
equivalent to the data, while the other two are less comprehensive.
A general method by which consequences of a given equation, in any
desired terms, may be derived, was formulated by Poretsky,12 and is, in
fact, a corollary of his Law of Forms, given above. We have seen that
this law may be formulated as the principle that if a = b, and therefore
a-b + -ab = 0 and a b + -a -b = 1 , then any t is such that a -b + -a b c t
and tcab + -a-b, or any t = the t which contains the "zero member"
of the set of equations equivalent to a = b, and is contained in the " whole
member" of this set. Now if x c t, u x c t, for any u whatever, and thus the
"zero member" of the Law of Forms may be multiplied by any arbitrarily
chosen u which we choose to introduce. Similarly, if tcy, then tcy + v,
and the "whole member" in the Law of Forms may be increased by the
addition of any arbitrarily chosen v. This gives the Law of Consequences.
7-6 If a = b, then t = (ab + -a -b + u) t + v (a -b + -a b) -/, where u and v
are arbitrary.
[7 • 1 • 12] If a = b, then a -b + -ab = 0 and ab + -a-b = 1.
Hence (a b + -a -b + it) t + v (a -b + -a b) -t = (1 + ?/) t + v-0--t -- t.
This law includes all the possible consequences of the given equation.
First, let us see that it is more general than the previous formulae of elimina
tion and solution. Given the equation A .r + B -x = 0, and choosing .1 B
for t. we should get the elimination resultant.
ItAx + B -x = 0, then AB = (-A x + -B -x + u) A B
Since u and v are both arbitrary and may assume the value 0, there
fore A B = 0.
12 Sept lois, etc., Chap. xn.
164 A Survey of Symbolic Logic
But this is only one of the unlimited expressions for A B which the law
gives. Letting u = 0, and v = 1, we have
A B = A -B x + -A B -x
Letting u = A and v = B, we have
AB = AB + -AB-x
And so on. But it will be found that every one of the equivalents of A B
which the law gives will be null.
Choosing x for our t, we should get the solution.
If A x + B -x = 0, then x = (-A x + -B -x + u) x + v (A x + B -x) -x
= (-A + u) x + v B -x.
Since u and v may both assume the value 0,
x = -A x, or x c -A (1)
And since u and v may both assume the value 1,
x = x + B -x, or B -x ex
But if B -x c x, then B -x = (B -.T) x = 0, or B c x (2)
Hence, (1) and (2), B cxc-A.
When u = 0 and v = 1, the Law of Consequences becomes simply the Law
of Forms. For these values in the above,
x = -A x + B -x
which is the form which Poretsky gives the solution for x.
The introduction of the arbitraries, u and v, in the Law of Consequences
extends the principle stated by the Law of Forms so that it covers not
only all equivalents of the given equation but also all the non-equivalent
inferences. As the explanation which precedes the proof suggests, this is
accomplished by allowing the limits of the function equated to t to be
expressed in all possible ways. If a = b, and therefore, by the Law of
Forms,
t = (a b + -a -b) t + (a -6 + -a b) -t
the lower limit of t, 0, is expressed as a -b + -a 6, and the upper limit of /,
1, is expressed as ab + -a -b. In the Law of Consequences, the lower
limit, 0, is expressed as v (a-b + -ab), that is, in all possible ways which
can be derived from its expression as a-b + -ab; and the upper limit, 1, is
expressed as a b + -a -6 + u, that is, in all possible ways which can be derived
from its expression as a b + -a -b. Since an expression of the form
t = (a b + -a -b) t + (a -b + -a b) -t
The Classic, or Boole-Schroder, Algebra of Logic 165
or of the form t = (a b + -a -b + u) t + v (a -b + -a b) -t
determines t only in the sense of thus expressing its limits, and the Law of
Consequences covers all possible ways of expressing these limits, it covers
all possible inferences from the given equation. The number of such
inferences is, of course, unlimited. The number expressible in terms of n
elements will be the number of derivatives from an equation with one
member 0 and the other member expanded with reference to n elements.
The number of constituent terms of this' expanded member will be 2n,
and the number of combinations formed from them will be 22". Therefore,
since pi + pz + p$ + . . . =0 gives pi = 0, pz = 0, ps = 0, etc., this is the
number of consequences of a given equation which are expressible in terms
of n elements.
As one illustration of this law, Poretsky gives the sixteen determinations
of a in terms of the three elements, a, b, and c, which can be derived from
the premises of the syllogism in Barbara: 13
If all a is b, a-b = 0,
and all 6 is c, b -c = 0,
then a-b + b -c = 0, and hence,
a = a (b + -c) = a (b + c) = a (-b + c) = a + 6 -c = a b = a (b c + -b -c)
= 5 _c + a (b c + _5 _c) = a c = b -c + a c = a (-b + c) + -a b -c = a b c
= b -c + ab c = a (b c + -b -c) + -a b -c = a c + -a b -c = a 6 c + -a b -c
The Inverse Problem of Consequences. — Just as the Law of Conse
quences expresses any inference from a = b by taking advantage of the fact
that if a-b + -ab = 0, then (a-b + -ab)v = 0, and if ab + -a-b = 1,
then a b + -a -b + u = 1 ; so the formula for any equation which will give
the inference a = b can be expressed by taking advantage of the fact that if
fl(a5 + _a_fr) = i} then ab + -a-b = 1, and if a-b + -ab + u = 0, then
a-b + -al> = o. We thus get Poretsky's Law of Causes, or as it would
be better translated, the Law of Sufficient Conditions.14
7-7 If for some value of u and some value of v
t = v (ab + -a -b) t + (a -b + -a b + u) -t,
then a = b.
If t = v(ab + -a-b)t+(a-b + -ab + u)-t, then [7-1, 5-72]
[v (a b + -a -b) t + (a -b + -a b + u) -t] -t = 0
— (a -^ + -a 1) + u) -t = (a-b + -a b) -t + u -i = 0
13 Ibid., pp. 98 /.
14 Ibid., Chap. xxm.
166 A Survey of Symbolic Logic
Hence (a -b + -a b) -t = 0 (1)
Hence also [5-7] t = v (a b + -a -b) t, and [4-9]
t--[v (a b + -a -b)] = 0 = t (-v + a -b + -a b) = t-v + (a -b + -ab) t
Hence [5 • 72] (a -b + -a b) t = 0 (2)
By (1) and (2), (a -b + -a b) (t + -t) = 0 = a -6 + -a 6.
Hence [7-1] a = b.
Both the Law of Consequences and the Law of Sufficient Conditions
are more general than the Law of Forms, which may be derived from either.
Important as are these contributions of Poretsky, the student must not
be misled into supposing that by their use any desired consequence or
sufficient condition of a given equation can be found automatically. The
only sense in which these laws give results automatically is the sense in
which they make it possible to exhaust the list of consequences or conditions
expressible in terms of a given set of elements. And since this process is
ordinarily too lengthy for practical purposes, these laws are of assistance
principally for testing results suggested by some subsidiary method or by
"intuition ". One has to discover for himself what values of the arbitrages
u and v will give the desired result.
V. FUNDAMENTAL LAWS OF THE THEORY OF INEQUATIONS
In this algebra, the assertory or copulative relations are = and c .
The denial of a = b may conveniently be symbolized in the customary way:
8-01 a 4= b is equivalent to "a - b is false ". Def.
We might use a symbol also for "a c b is false.". But since a c b is equiva
lent to a b = a and to a -b = 0, its negative may be represented by a b 4= a
or by a -b 4= 0. It is less necessary to have a separate symbolism for
"a c b is false ", since "a is not contained in b" is seldom met with in logic
except where a and b are mutually exclusive, — in which case a b = 0.
For every proposition of the form "If P is true, then Q is true ", there is
another, " If Q is false, then P is false ". This is the principle of the rednctio
ad absurdum,—oT the simplest form of it. In terms of the relations =
and =|=, the more important forms of this principle are:
(1) "If a = b, then c = d", gives also, "If c 4= d, then a 4= b ".
(2) "If a = 6, then c = d&ndh = k", gives also, "Ifc =(= <Z, then a 4= b",
and "If h 4= k, then a 4= b ".
(3) "If a = b and c - d, then h = k", gives also, "If a = b and h 4= k,
then c 4= d", and "If c = d and h 4= k, then a 4= b ".
The Classic, or Boole-Schroder, Algebra of Logic 167
(4) "a = b is equivalent to c = d" , gives also, "a 4= b is equivalent
toe 4= d".
(5) "a = b is equivalent to the set, c = <7, h = k, . ..," gives also,
"a 4= 6 is equivalent to 'Either c 4= </ or h 4= A', or ...'". 16
The general forms of these principles are themselves theorems of the
"calculus of propositions" — the application of this algebra to propositions.
But the calculus of propositions, as an applied logic, cannot be derived
from this algebra without a circle in the proof, for the reasoning in demon
stration of the theorems presupposes the logical laws of propositions at
every step. We must, then, regard these laws of the reductio ad absurdum,
like the principles of proof previously used, as given us by ordinary logic,
which mathematics generally presupposes. In later chapters,16 we shall
discuss another mode of developing mathematical logic — the logistic
method — which avoids the paradox of assuming the principles of logic in
order to prove them. For the present, our procedure may be viewed simply
as an application of the reductio ad absurdum in ways in which any mathe
matician feels free to make use of that principle.
Since the propositions concerning inequations follow immediately, for
the most part, from those concerning equations, proof will ordinarily be
unnecessary.
Elementary Theorems. — The more important of the elementary propo
sitions are as follows:
8-1 If ac 4= be, then a 4= b.
[2-1]
8-12 If a + c =t= b + c, then a 4= b.
[3-37]
8-13 a 4= b is equivalent to -a 4" -b.
[3-2]
8-14 a + b 4= b, a b 4= a, -a + b 4= 1, and a -b 4= 0 are all equivalent.
[4-9]
8-15 If a + b = x and b 4= x, then a 4= 0
[5-7]
8-151 If a = 0 and b 4= x, then a + b 4= x.
[5-7]
8-16 If a b = x and b 4= .r, then a 4= 1.
[5-71]
15 "Either ... or ..." is here to be interpreted as not excluding the possibility that
both should be true.
1(5 Chap, iv, Sect, vi, and Chap. v.
168 A Survey of Symbolic Logic
8-161 If a = 1 and b 4= x, then a b 4= x.
[5-71]
8-17 If a + b 4= 0 and a = 0, then b 4= 0.
[5-72]
8-18 If a b 4= 1 and a = 1, then b 4= 1.
[5-73]
8-17 allows us to drop null terms from any sum 4= 0- In this, it gives
a rule by which an equation and an inequation may be combined. Suppose,
for example, a + b 4= 0 and x = 0.
a + b = (a + b)(x + -x) =ax+bx+a-x+b -x.
Hence ax + bx + a-x + b-x 4= 0.
But if x = 0, then a x = 0 and b x = 0.
Hence [8 • 17] a-x + b -x 4= 0.
8-2 If a 4= 0, then a + b 4= 0.
[5-72]
8-21 If a 4= 1, then 064=!.
[5-73]
8-22 If a 6 4= 0, then a 4= 0 and b 4= 0.
[1-5]
8-23 If 0 + 6 4= 1, then a 4= 1 and b 4= 1.
[4-5]
8-24 If a b 4- x and a = x, then b 4= x.
[1-2]
8-25 If a 4= 0 and a c6, then 64-0.
[1-9] If ac6, then a b = a.
Hence if a 4= 0 and a c 6, then a 6 4= 0.
Hence [8-22] 6 + 0.
8-26 a + 6 4- 0 is equivalent to "Either a 4= 0 or 6 4= 0 ".
[5-72]
8-261 ai + a2 + a3 + . . . 4= 0 is equivalent to "Either ax 4= 0 or a2 4= 0 or
as 4= 0, or ... ".
8-27 a 6 4= 1 is equivalent to "Either a 4= 1 or 6 4= 1 ".
[5-73]
8-271 aia2«3 . +1 is equivalent to "Either ai 4= 1 or a2 4= 1 or
«3 + 1 or ... ".
The difference between 8-26 and 8-27 and their analogues for equa
tions— 5-72 a + b = 0 is equivalent to the pair, a = 0 and 6 = 0, and
The Classic, or Boole-Schroder, Algebra of Logic 169
5-73 a b = 1 is equivalent to the pair, a == 1 and b = 1 — points to a neces
sary difference between the treatment of equations and the treatment of
inequations. Two or more equations may always be combined into an
equivalent equation; two or more inequations cannot be combined into
an equivalent inequation. But, by 8-2, a+ b 4= 0 is a consequence of the
pair, a 4= 0 and b =|= 0.
Equivalent Inequations of Different Forms. — The laws of the equiva
lence of inequations follow immediately from their analogues for equations.
8-3 a 4= b is equivalent to a -b + -a b 41 0.
[7-1]
8-31 a 4= 1 is equivalent to -a =h 0-
[7-12]
8-32 If &(xi, £2, ... xn) and V(xi, .r2, . . . xn) be any two functions of
the same variables, then
, X2, . . . Xn) 4=
is equivalent to F(XI, .T2, . . . xn) 4= 0? where F is a function of these same
variables and such that, if Ai, Az, A3, etc., be the coefficients in $ and
Bif B2, B3, etc., be the coefficients of the corresponding terms in ^, then
the coefficients of the corresponding terms in F will be Ai-Bi + -AiBi,
A, -B2 + -Az B2, A3 -B3 + -At B,, etc.
[7-13]
Poretsky 's Law of Forms for inequations will be :
8-33 a 4= 0 is equivalent to t 3= a -t+ -at.
[7-15]
Or in more general form :
8-34 a 4= b is equivalent to t =h (ab + -a -b) t + (a -b + -a b) -t.
[7-16]
Elimination. — The laws governing the elimination of elements from an
inequation are not related to the corresponding laws governing equations
by the reductio ad absurdum. But these laws follow from the same theorems
concerning the limits of functions.
8-4 It A x + B -x 4= 0, then A + B + 0.
[6-3\Ax + B -x cA+B. Hence [8 • 25] Q.E.D.
8-41 If the coefficients in anv function of n variables, F(XI, xz, . . . xn),
170 A Survey of Symbolic Logic
be Ci, C2, To, etc., and if F(x1} .r2, - . . .rn) =f= 0, then
ZC=NO
[0-32] F(xlf .r«, . . . a-n) cC. Hence [8-25] Q.E.D.
Thus, to eliminate any number of elements from an inequation with
one member 0, reduce the other member to the form of a normal function
of the elements to be eliminated. The elimination is then secured by
putting =f= 0 the sum of the coefficients. The form of elimination resultants
for inequations of other types follows immediately from the above. It is
obvious that they will be analogous to the elimination resultants of equa
tions as follows: To get the elimination resultant of any inequation, take
the elimination resultant of the corresponding equation and replace = by =f= ,
and x by + .
A universal proposition in logic is represented by an equation: "All
a is b " by a -b = 0, " No a is b " by a b = 0. Since a particular proposition
is always the contradictory of some universal, any particular proposition
may be represented by an inequation: "Some a is b" by a b =f= 0, "Some
a is not b" by a -b =j= 0. The elimination of the "middle" term from an
equation which represents the combination of two universal premises
gives the equation which represents the universal conclusion. But elimina
tion of terms from inequations does not represent an analogous logical
process. Two particulars give no conclusion: a particular conclusion
requires one universal premise. The drawing of a particular conclusion is
represented by a process which combines an equation with an inequation,
by 8-17, and then simplifies the result, by 8-22. For example,
All aisb, a-b = 0. .'. a -b c = 0.
Some a is c, a c =J= 0. .'. a b c + a -b c =h 0.
.'. abc 4= 0. [8-17]
Some fr is c. .'. 6c4=0. [8-22]
" Solution " of an Inequation.— An inequation may be said to have a
solution in the sense that for any inequation involving x an equivalent
inequation one member of which is x can always be found.
8-5 A x + B -x 4= 0 is equivalent to x 4= -A x + B -x.
[7-23]
8-51 A x + B -x 4= 0 is equivalent to "Either B -x ^ 0 or A x ^ 0 ",—
i. e., to "Either B ex is false or xc-A is false ".
[7-21
The Classic, or Boole-Schroder, Algebra of Logic 171
Neither of these "solutions" determines x even within limits. " B ex
is false " does not mean " B is excluded from x"; it means only " B is not
wholly within x". " Either Bex is false or xc-A is false" does not
determine either an upper or a lower limit of x; and limits x only by ex
cluding B + u-A from the range of its possible values. Thus "solutions"
of inequations are of small significance.
Consequences and Sufficient Conditions of an Inequation. — By Poret-
sky's method, the formula for any consequence of a given inequation follows
from the Law of Sufficient Conditions for equations.17 If for some value
of u and some value of v,
t = x (a b + -a -6) t + (a -b + -a b + u) -t
then a = b. Consequently, we have by the reductio ad absurdum:
8-52 If a 4= b, then t ^= v (a b + -a -b) t + (a -b + -a b + u) -t, where u
and v are arbitrary.
[7-7]
The formula for the sufficient conditions of an inequation similarly fol
lows from the Law of Consequences for equations. If a = b, then
t = (a b + -a -b + u) t + v (a -b + -a b) -t
where u and v are arbitrary. Consequently, by the reductio ad absurdum:
8-53 If for some value of u and some value of v,
t 3= (a b + -a -b + u) t + v (a -b + -a b) -t
then a =|= b.
[7-6]
System of an Equation and an Inequation. — If we have an equation in
one unknown, x, and an inequation which involves x, these may be combined
in either of two ways: (1) each may be reduced to the form in which one
member is 0 and expanded with reference to all the elements involved in
either. Then all the terms which are common to the two may, by 8 • 17,
be dropped from the inequation; (2) the equation may be solved for x,
and this value substituted for x in the inequation.
8-6 If A x + B -x = 0 and Cx + D -x * 0, then -.1 C x + -BD -.r 4= 0.
[5-8] If Cx + D-x =(= 0, then
ACx + -ACx + BD-x + -B D -.r + 0
17 See Poretsky, Theorie des non-egaliUs logiques, Chaps. 71, 76.
172 A Survey of Symbolic Logic
[5-72] If Ax + B-x = 0, then A x = 0 and B -x = 0, and hence
A C x = 0 and B D -x = 0.
Hence [8 • 17] -.4 C x + -B D -x 4= 0.
The result here is not equivalent to the data, since — for one reason—
the equation A C x + B D -x = 0 is not equivalent to A x + B -x = 0.
Nevertheless this mode of combination is the one most frequently useful.
8-61 The condition that the equation A x + B -x = 0 and the inequation
C x + D -x 4= 0 may be regarded as simultaneous is, A B = 0 and -A C
+ -B D 4= 0, and the determination of x which they give is
x 4= (-A -C + A-D)x + (BC + -B D) -x
[7-23] A x + B -x = 0 is equivalent to x = -A x + B -x. Substi
tuting this value of x in the inequation,
C (-A x + B -x) + D (A x + -B -x) 4= 0
or (-A C + A D) x+(B C + -BD) -x 4= 0.
[8-4] A condition of this inequation is
4=0,
or (-A+B) C+(A + -B) D 4= 0.
But the equation A x + B -x = 0 requires that ^45 = 0, and hence
that -A + B = -A and -B + A = -B.
Hence if the equation be possible and A B = 0, the condition of the
inequation reduces to -A C + -B D 4= 0.
[8-4] If the original inequation be possible, then C + D 4= 0. But
this condition is already present in -A C + -B D 4= 0, since -A C c C
and hence [8-25] if -.4(74=0, then C 4= 0, and -BDcD and
hence if -B D 4= 0, then D 4= 0, while [8-26] C + D 4= 0 is equivalent
to " Either C 4= 0 or D 4= 0 ", and -A C + -B D 4= 0 is equivalent to
"Either -A C 4= 0 or -B D 4= 0 ".
Hence the entire condition of the system is expressed by
A B = 0 and -A C + -B D 4= 0
And [8 • 5] the solution of the inequation,
(-AC + AD)x + (BC + -BD)-x 4= 0, is
x 4= (-A-C + A-D)x + (BC + -BD)-x
This method gives the most complete determination of x, in the form of
an inequation, afforded by the data.
The Classic, or Boole-Schroder, Algebra of Logic 173
VI. NOTE ON THE INVERSE OPERATIONS, "SUBTRACTION" AND
"DIVISION"
It is possible to define "subtraction" { — } and "division" { : j in the
algebra. Let a — b be x such that b + x = a. And let a : b be y such
that by = a. However, these inverse operations are more trouble than
they are worth, and should not be admitted to the system.
In the first place, it is not possible to give these relations a general
meaning. We cannot have in the algebra: (1) If a and 6 are elements
in A", then a : b is an element in K; nor (2) If a and 6 are elements in K,
then a — b is an element in K. If a : b is an element, y, then for some y
it must be true that b y = a. But if b y = a, then, by 2-2, a cb y and,
by 5-2, a cb. Thus if a and b be so 'chosen that a cb is false, then a : b
cannot be any element in K. To give a : b a general meaning, it would
be required that every element be contained in every element — that is,
that all elements in K be identical. Similarly, if a — b be an element,
x, in A, then for some x, it must be true that 6 + x = a. But if b + x = a,
then, by 2-2, b + xca and, by 5-21, be a. Thus if a and b be so chosen
that b c a is false, then a — b cannot be any element in A".
Again, a — b and a : b are ambiguous. It might be expected that,
since a + -a = 1, the value of 1 — a would be unambiguously -a. But
1 — a = x is satisfied by any x such that -a c x. For 1 — a = x is equiva
lent to x + a = 1, which is equivalent to
-(x + a) = -1 = 0 = -a -x
And -a -x = 0 is equivalent to -a c x. Similarly, it might be expected
that, since a -a = 0, the value of 0 : a would be unambiguously -a. But
0 : a = y, or a y = 0, is satisfied by any y such that y c -a. a y = 0 and
y c -a are equivalent.
Finally, these relations can always be otherwise expressed. The value
of a : b is the value of y in the equation, b y = a. b y = a is equivalent to
-a b y + a -b + a -y = 0
The equation of condition here is a -b = 0. And the solution, on this
condition, is
y = a + u (a + -b) = ab + u-a-b, where u is undetermined.
The value of a — b is the value of x in the equation, b + x = a. b + x = a
is equivalent to
-a b + -a x + a -b -x = 0
174 A Survey of Symbolic Logic
The equation of condition here is, -a b = 0. And the solution, on this
condition, is
x = a -b + v a = a -b + v a 6, where v is undetermined.
In each case, the equation of condition gives the limitation of the meaning
of the expression, and the solution expresses the range of its possible values.
CHAPTER III
APPLICATIONS OF THE BOOLE-SCHRODER ALGEBRA
There are four applications of the classic algebra of logic which are
commonly considered: (1) to spatial entities, (2) to the logical relations
of classes, (3) to the logical relations of propositions, (4) to the logic of
relations.
The application to spatial entities may be made to continuous and
discontinuous segments of a line, or to continuous and discontinuous regions
in a plane, or to continuous and discontinuous regions in space of any
dimensions. Segments of a line and regions in a plane have both been
used as diagrams for the relations of classes and of propositions, but the
application to regions in a plane gives the more workable diagrams, for
obvious reasons. And since it is only for diagrammatic purposes that
the application of the algebra to spatial entities has any importance, we
shall confine our attention to regions in a plane.
I. DIAGRAMS FOR THE LOGICAL RELATIONS OF CLASSES
For diagrammatic purposes, the elements of the algebra, a, b, c, etc.,
will denote continuous or discontinuous regions in a given plane, or in a
circumscribed portion of a plane. 1 represents the plane (or circumscribed
portion) itself. 0 is the null-region which is supposed to be contained in
every region. For any given region, a, -a denotes the plane exclusive of
a, — i. e., not-a. The "product", a xb or a b, is that region which is com
mon to a and b. If a and 6 do not "overlap ", then a b is the null-region, 0.
The "sum", a + b, denotes the region which is either a or b (or both). In
determining a + b, the common region, a b, is not, of course, counted twice
over.
a + b = a -b + a b + -a b.
This is a difference between + in the Boole-Schroder Algebra and the +
of arithmetic. The equation, a = b, signifies that a and b denote the same
region. ac6 signifies that a lies wholly within b, that a is included or
contained in b. It should be noted that whenever a = b, a c b and b c a.
Also, a c a holds always. Thus the relation c is analogous not to < in
arithmetic but to — .
175
176 A Survey of Symbolic Logic
While the laws of this algebra hold for regions, thus denoted, however
those regions may be distributed in the plane, not every supposition about
their distribution is equally convenient as a diagram for the relations of
classes. All will be familiar with Euler's diagrams, invented a century
earlier than Boole's algebra. "All a is 6" is represented by a circle a
wholly within a circle b; " No a is b " by two circles, a and b, which nowhere
intersect; "Some a is 6" and "Some a is not 6" by intersecting circles,
sometimes with an asterisk to indicate that division of the diagram which
represents the proposition. The defects of this style of diagram are obvious :
x-*>c^\ ^-^^\
cuto;
All a is 6 No a is b Some a is & Some a is not 6
FIG. 1
the representation goes beyond the relation of classes indicated by the propo
sition. In the case of "All a is &", the circle a falls within b in such wise
as to suggest that we may infer "Some b is not a", but this inference is
not valid. The representation of "No a is b" similarly suggests "Some
things are neither a nor b", which also is unwarranted. With these dia
grams, there is no way of indicating whether a given region is null/ But
the general assumption that no region of the diagram is null leads to the
misinterpretations mentioned, and to others which are similar. Yet
Euler's diagrams were in general use until the invention of Venn, and are
still doing service in some quarters.
The Venn diagrams were invented specifically to represent the relations
of logical classes as treated in the Boole-Schroder Algebra.1 The principle
of these diagrams is that classes be represented by regions in such relation
to one another that all the possible logical relations of these classes can be
indicated in the same diagram. That is, the diagram initially leaves room
for any possible relation of the classes, and the actual or given relation, can
then be specified by indicating that some particular region is null or is not-
null. Initially the diagram represents simply the "universe of discourse",
or 1. For one element, a, 1 = a + -a.2 For two elements, a and b,
1 = (a + -a) (b + -b) = a b + a -b + -a b + -a -b
1 See Venn, Symbolic Logic, Chap. v. The first edition of this book appeared before
Schroder's Algebra der Logik, but Venn adopts the most important alteration of Boole's
original algebra — the non-exclusive interpretation of a + b.
2 See above, Chap, n, propositions 4-8 and 5-92.
Applications of the Boole-Schroder Algebra 177
For three elements, a, b, and c,
1 = (a + -a) (b + -b) (c + -c) = a b c + a b -c + a -b c + -a b c + a -b -c
+ -a b -c + -a -b c + -a -6 -c
Thus the " universe of discourse" for any number of elements, n, must
correspond to a diagram of 2r( divisions, each representing a term in the
expansion of 1. If the area within the square in the diagram represent
-a
-a-b
FIG. 2
the universe, and the area within the circle represent the element a, then
the remainder of the square will represent its negative, -a. If another
element, 6, is to be introduced into the same universe, then b may be repre
sented by another circle whose periphery cuts the first. The divisions,
(1) into a and, -a, (2) into b and -b, will thus be cross-divisions in the uni
verse. If a and b be classes, this arrangement represents all the possible
subclasses in the universe; — a b, those things which are both a and b;
a-b, those things which are a but not b; -ab, those things which are 6
but not a; -a -6, those things which are neither a nor b. The area which
represents the product, a b, will readily be located. We have enclosed
by a broken line, in figure 2, the area which represents a + b.
The negative of any entity is always the plane exclusive of that entity.
For example, -(a b + -a -6), in the above, will be the sum of the other
two divisions of the diagram, a -b + -a 6.
If it be desired to introduce a third element, c, into the universe, it is
necessary to cut each one of the previous subdivisions into two — one
part which shall be in c and one part which shall be outside c. This can be
be accomplished by introducing a third circle, thus
It is not really necessary to draw the square, 1, since the area given to the
figure, or the whole page, may as well be taken to represent the universe.
But when the square is omitted, it must be remembered that the unenclosed
13
178
A Survey of Symbolic Logic
area outside all the lines of the figure is a subdivision of the universe—
the entity -a, or -a -ft, or -a -b -c, etc., according to the number of elements
involved.
-a-b-c
FIG. 3
If a fourth element, d, be introduced, it is no longer possible to repre
sent each element by a circle, since a fourth circle could not be introduced
in figure 3 so as to cut each previous subdivision into two parts — one part
in d and one part outside d. But this can be done with ellipses.3 Each
-a-b-c-d
FIG. 4
3 We have deformed the ellipses slightly and have indicated the two points of junction.
This helps somewhat in drawing the diagram, which is most easily done as follows: First,
draw the upright ellipse, a. Mark a point at the base of it and one on the left. Next,
Applications of the Boole-Schroder Algebra
179
one of the subdivisions in figure 4 can be "named" by noting whether it
is in or outside of each of the ellipses in turn. Thus the area indicated by
6 is a b c -d, and the area indicated by 12 is -a -b c d. With a diagram of
four elements, it requires care, at first, to specify such regions as a + c,
a c + b d, b + -d. These can always be determined with certainty by
developing each term of the expression with reference to the missing ele
ments.4 Thus
ac + b d = ac (b + -b) (d + -d) + b d (a + -a) (c + -c)
= a b c d + a b c -d + a -b c d + a -b c -d + a b -c d + -a b c d + -a b -c d
The terms of this sum, in the order given, are represented in figure 4 by the
divisions numbered 10, 6, 9, 5, 14, 11, 15. Hence ac + bd is the region
which combines these. With a little practice, one may identify such
regions without this tedious procedure. Such an area as b + -d is more
easily identified by inspection: it comprises 2, 3, 6, 7, 10, 11, 14, 15, and
1, 4, 5, 8.
Into this diagram for four elements, it is possible to introduce a fifth,
e, if we let e be the region between the broken lines in figure 5. The principle
of the "square diagram" (figure 6) is the same as Venn's: it represents all
FIG. 5
draw the horizontal ellipse, d, from one of these points to the other, so that the line con
necting the two points is common to a and d. Then, draw ellipse 6 from and returning to
the base point, and ellipse c from and returning to the point on the left. If not done in
this way, the first attempts are likely to give twelve or fourteen subdivisions instead of
the required sixteen.
4 See Chap, n, 5-91.
180
A Survey of Symbolic Logic
the subclasses in a universe of the specified number of elements. No
diagram is really convenient for more than four elements, but such are
-a
-a
a-b
-a b
-a -b
FIG. 6
frequently needed. The most convenient are those made by modifying
slightly the square diagram of four terms, at the right in figure 6.5 Figure 7
-a
-b
b<
-e
h
-h
FIG. 7
-e
d
gives, by this method, the diagrams for five and for six elements. We give
also the diagram for seven (figure 8) since this is frequently useful and
not easy to make in any other way.
The manner in which any function in the algebra may be specified in a
diagram of the proper number of divisions, has already been explained.
We must now consider how any asserted relation of elements — any inclu-
6 See Lewis Carroll, Symbolic Logic, for the particular form of the square diagram which
we adopt. Mr. Dodgson is able, by this method, to give diagrams for as many as 10 terms,
1024 subdivisions (p. 176).
Applications of the Boole-Schroder Algebra
181
sion, a c b, or any equation, a = b, or inequation, a =)= b — may be repre
sented. Any such relation, or any set of such relations, can be completely
specified in these diagrams by taking advantage of the fact that they
can always be reduced either to the form of an expression = 0 or to the
form of an expression 4= 0- Any inclusion, a c b, is equivalent to an equa
tion, a -6 = O.6 And every equation of the form a = b is equivalent to
one of the form a -b + -a b = O.7 Thus any inclusion or equation can be
represented by some expression ••= 0. Similarly, any inequation of the
form a --r b is equivalent to one of the form a-b + -ab =H O.8 Thus any
asserted relation whatever can be specified by indicating that some region
(continuous or discontinuous) either is null, { = Oj, or is not-null, j 41 0}.
We can illustrate this, and at the same time indicate the manner in
which such diagrams are useful, by applying the method to a few syllogisms.
Given: All a is 6,
. and All b is c,
6 See Chap, n, 4-9.
7 See Chap, n, 6-4.
8 See Chap, n, 7-1.
a cb,
b cc,
a -b = 0.
b -c = 0.
182
A Survey of Symbolic Logic
We have here indicated (figure 9) that a -b — the a which is not b — is null
by striking it out (with horizontal lines). Similarly, wTe have indicated
that all b is c by striking out b -c (with vertical lines). Together, the two
operations have eliminated the whole of a -c, thus indicating that a -c = 0,
or "All a is c ".
FIG. 9
For purposes of comparison, we may derive this same conclusion by
algebraic processes.9
Since a -b = 0 = a -b (c + -c) = a -b c + a -b -c,
and b -c = 0 = b -c (a + -a) = a b -c + -a b -c,
therefore, a -b c + a -b -c + a b -c + -a b -c = 0,
and [5-72] a b -c + a -b -c = 0 = a -c (b + -b) = a -c.
The equation in the third line, which combines the two premises, states
exactly the same facts which are represented in the diagram. The last
equation gives the conclusion, which results from eliminating the middle
term, b. Since a diagram will not perform an elimination, we must there
"look for" the conclusion.
One more illustration of this kind :
Given: All a is b, a -b = 0.
and No b is c, b c = 0.
The first premise is indicated (figure 10) by striking out the area a -b (with
horizontal lines), the second by striking out be (with vertical lines). To
gether, these operations have struck out the whole of a c, giving the con
clusion ac = 0, or "No a is c".
9 Throughout this chapter, references in square brackets give the number of the the
orem in Chap, u by which any unobvious step in proof is taken.
Applications of the Boole-Schroder Algebra
183
In a given diagram where all the possible classes or regions in the uni
verse are initially represented, as they are by this method of diagramming,
we cannot presume that a given subdivision is null or is not-null. The
actual state of affairs may require that some regions be null, or that some
be not-null, or that some be null and others not. Consequently, even
when we have struck out the regions which are null, we cannot presume
that all the regions not struck out are no/-null. This would be going beyond
the premises. All we can say, when we have struck out the null-regions,
is that, so far as the premises represented are concerned, any region not
struck out may be not-null. If, then, we wish to represent the fact that a
given region is definitely not-null — that a given class has members, that
there is some expression 4= 0 — we must indicate this by some distinctive
mark in the diagram. For this purpose, it is convenient to use asterisks.
That a b =}= 0, may be indicated by an asterisk in the region a b. But here
a further difficulty arises. If the diagram involve more than two elements,
say, a, b, and c, the region a b will be divided into two parts, a b c and
a b -c. Now the inequation, a b 4= 0, does not tell us that a b c =}= 0, and
it does not tell us that a 6 -c 4= 0- It tells us only that a b c + a b -c =|= 0.
If, then, we wish to indicate a b 4= 0 by an asterisk in the region a b, we
shall not be warranted in putting it either inside the circle c or outside c.
It belongs in one or the other or both — that is all we know. Hence it is
convenient to indicate a b 4= 0 by placing an asterisk in each of the divisions
of a b and connecting them by a broken line, to signify that at least one of
these regions is not-null (figure 11).
ab =¥0
FIG. 11
We shall show later that a particular proposition is best interpreted by
an inequation; "Some a is 6", the class a b has members, by ab * 0.
184
A Survey of Symbolic Logic
Suppose, then, we have:
Given: All a is b, a-b = 0.
and Some a is c, a c H= 0.
The conclusion, "Some b is c", is indicated (figure 12) by the fact that
one of the two connected asterisks must remain — the whole region a b c
+ a -be cannot be null. But one of them, in a -b c, is struck out in indi
cating the other premise, a -6 = 0. Thus a b c =}= 0, and hence a c 4= 0.
FIG. 12
The entire state of affairs in a universe of discourse may be represented
by striking out certain regions, indicating by asterisks that certain regions
are not-null, and remembering that any region which is neither struck
out nor occupied by an asterisk is in doubt. Also, the separate subdivisions
of a region occupied by connected asterisks are in doubt unless all but one
of these connected asterisks occupy regions which are struck out. And
any regions which are left in doubt by a given set of premises might, of
course, be made specifically null or not-null by an additional premise.
In complicated problems, the use of the diagram is often simpler and
more illuminating than the use of transformations, eliminations, and solu
tions in the algebra. All the information to be derived from such opera
tions, the diagram gives (for one who can "see" it) at a glance. Further
illustrations will be unnecessary here, since we shall give diagrams in con
nection with the problems of the next section.
II. THE APPLICATION TO CLASSES
The interpretation of the algebra for logical classes has already been
explained.10 a, b, c, etc., are to denote classes taken in extension; that is
10 Chap, n, pp. 121-22.
Applications of the Boole-Schroder Algebra 185
to say, c signifies, not a class-concept, but the aggregate of all the objects
denoted by some class-concept. Thus if a = b, the concept of the class a
may not be a synonym for the concept of the class b, but the classes a and b
must consist of the same members, have the same extension, acb sig
nifies that every member of the class a is also a member of the class b.
The " product", a b, denotes the class of those things which are both mem
bers of a and members of b. The "sum ", a + 6, denotes the class of those
things which are either members of a or members of b (or members of both).
0 denotes the null-class, or class without members. Various concepts may
denote an empty class — "immortal men", "feathered invertebrates",
"Julius Caesar's twin," etc. — but all such terms have the same extension;
they denote nothing existent. Thus, since classes are taken in extension,
there is but one null-class, 0. Since it is a law of the algebra that, for
every x, 0 ex, we must accept, in this connection, the convention that
the null-class is contained in every class. All the immortal men are mem
bers of any class, since there are no such. 1 represents the class "every
thing ", the "universe of discourse ", or simply the "universe ". This term
is pretty well understood. But it may be defined as follows: if an be any
member of the class a, and A" represent the class-concept of the class x,
then the "universe of discourse" is the class of all the classes, x, such that
"an is an X" is either true or false. If "The fixed stars are blind" is
neither true nor false, then "fixed stars" and the class "blind" do not
belong to the same universe of discourse.
The negative of a, -a, is a class such that a and -a have no members in
common, and a and -a between them comprise everything in the universe
of discourse: a -a = 0, "Nothing is both a and not-a", and « + -a == 1,
"Everything is either a or -a".
Since inclusions, a c 6, equations, a = b, and inequations, a 4= b, repre
sent relations which are asserted to hold between classes, they are capable
of being interpreted as logical propositions. And the operations of the
algebra — transformations, eliminations, and solutions — are capable of
interpretation as processes of reasoning. It would hardly be correct to
say that the operations of the algebra represent the processes of reasoning
from given premises to conclusions: they do indeed represent processes
of reasoning, but they seldom attain the result by just those operations
which are supposed to characterize the customary processes of thinking.
In fact, it is the greater generality of the symbolic operations which makes
their application to reasoning valuable. *
186 A Survey of Symbolic Logic
The representation of propositions by inclusions, equations, and in
equations, and the interpretation of inclusions, equations, and inequations
in the algebra as propositions, offers certain difficulties, due to the fact
that the algebra represents relations of extension only, while ordinary logical
propositions quite frequently concern relations of intension. In discussing
the representation of the four typical propositions, we shall be obliged to
consider some of these problems of interpretation.
The universal affirmative, " All a is 6", has been variously represented as,
(1) a = a b,
(2) fl-c6,
(3) a = v b, where v is undetermined,
(4) a -b = 0.
All of these are equivalent.11 The only possible doubt concerns (3) a = v b,
where v is undetermined. But its equivalence to the others may be demon
strated as follows:
[7- 1] a = v b is equivalent to a--(v b) + -a v b = 0.
But a • -(v b) + -a v b = a (-v + -b) + -a v b = a -v + a -b + -a v b.
Hence [5-72] if a = vb, then a -b = 0.
And if a = a b} then for some value of v (i. e., v = a), a = v b.
These equivalents of "All a is 6" would most naturally be read:
(1) The a's are identical with those things which are a's and 6's both.
(2) a is contained in b: every member of a is also a member of b.
(3) The class a is identical with some (undetermined) portion of the
class 6.
(4) The class of those things which are members of a but not members
of 6 is null.
If we examine any one of these symbolic expressions of "All a is 6",
we shall discover that not only may it hold when a = 0, but it always
holds when a = 0. 0 = 0-6, 0 c 6, and 0--6 = 0, will be true for every
element b. And "0 = 0 b for some value of v" is always true— for v = 0.
Since a = 0 means that a has no members, it is thus clear that the algebra
requires that "All a is 6" be true whenever no members of a exist. The
actual use of language is ambiguous on this point. We should hardly say
that "All sea serpents have red wings, because there aren't any sea ser
pents"; yet we understand the hero of the novel who asserts "Whoever
11 See Chap. n. 4-9. %
Applications of the Boole-Schroder Algebra 187
enters here must pass over my dead body". This hero does not mean to
assert that any one will enter the defended portal over his body: his desire
is that the class of those who enter shall be null. The difference of the
two cases is this: the concept "sea serpent" does not necessarily involve the
concept "having red wings", while the concept of "those who enter the
portal" — as conceived by the hero — does involve the concept of passing
over his body. We readily accept and understand the inclusion of an
empty class in some other when the concept of the one involves the concept
of the other — when the relation is one of intension. But in this sense, an
empty class is not contained in any and every class, but in some only. In
order to understand this law of the algebra, "For every .r, 0 ex", we must
bear in mind two things: (1) that the algebra treats of relations in extension
only, and (2) that ordinary language frequently concerns relations of
intension, and is usually confined to relations of intension where a null
class is involved. The law does not accord with the ordinary use of language.
This is, however, no observation upon its truth, for it is a necessary law
of the relation of classes in extension. It is an immediate consequence of
the principle, "For every y, y cl", that is, "All members of any class, y,
are also members of the class of all things". One cannot accept this last
without accepting, by implication, the principle that, in extension, the null-
class is contained in every class.
The interpretation of propositions in which no null-class is involved is
not subject to any corresponding difficulty, both because in such cases the
relations predicated are frequently thought of in extension and because
the relation of classes in extension is entirely analogous to their relation in
intension except where the class 0 or the class 1 is involved. But the
interpretation of the algebra must, in all cases, be confined to extension.
In brief: "All a is 6" must always be interpreted in the algebra as stating
a relation of classes in extension, not of class-concepts, and this requires
that, whenever a is an empty class, "All a is 6" should be true.
The proposition, "No a is b", is represented by ab = 0— "Nothing is
both a and 6", or "Those things which are members of a and of b both,
do not exist". Since "No a is 6" is equivalent to "All a is not-6", it may
also be represented by a -b = -b, a c-6, 6c-a, or a = v -b, where v is
undetermined. In the case of this proposition, there is no discrepancy
between the algebra and the ordinary use of language.
The representation of particular propositions has been a problem to
symbolic logicians, partly because they have not clearly conceived the
188 A Survey of Symbolic Logic
relations of classes and have tried to stretch the algebra to cover traditional
relations which hold in -intension only. If "Some a is 6" be so interpreted
that it is false when the class a has no members, then "Some a is 6" will
not follow from "All a is 6", for "All a is 6" is true whenever a = 0. But
on the other hand, if " Some a is b " be true when a = 0, we have two diffi
culties: (1) this does not accord with ordinary usage, and (2) "Some a is 6"
will not, in that case, contradict "No a is 6". For whenever there are no
members of a (when a = 0), "No a is 6" (a b = 0) will be true. Hence if
"Some a is 6" can be true when a = 0, then "Some a is b" and "No a is 6"
can both be true at once. The solution of the difficulty lies in observing
that "Some a is b" as a relation of extension requires that there be some a—
that at least one member of the class a exist. Hence, when propositions
are interpreted in extension, "Some a is 6" does not follow from "All a
is 6", precisely because whenever a = 0, "All a is 6" will be true. But
"Some a is b" does follow from "All a is 6, and members of a exist".
To interpret properly "Some a is 6", we need only remember that it
is the contradictory of "No a is 6". Since "No a is 6" is interpreted by
a b = 0, "Some a is 6" will be a b =(= 0, that is, "The class of things which
are members of a and of b both is not-null".
It is surprising what blunders have been committed in the representation
of particular propositions. "Some x is y" has been symbolized by x y = v,
where v is undetermined, and by u x = v y, where u and v are undetermined.
Both of these are incorrect, and for the same reason: An "undetermined"
element may have the value 0 or the value 1 or any other value. Conse
quently, both these equations assert precisely nothing at all. They are
both of them true a priori, true of every x and y and in all cases. For
them to be significant, u and v must not admit the value 0. But in that
case they are equivalent to x y =J= 0, which is much simpler and obeys well-
defined laws which are consonant with its meaning.
Since we are to symbolize "All a is b" by a -b = 0, it is clear that its
contradictory, "Some a is not 6", will be a -b =|= 0.
To sum up, then: the four typical propositions will be symbolized as
follows:
A. All a is b, a -b = 0.
E. No a is b, ab = 0.
I. Some a is b, a b =|= 0.
O. Some a is not b, a -b =}= 0.
Each of these four has various equivalents:12
12 See Chap. 11, 4 • 9 and 8 • 14.
Applications of the Boole-Schroder Algebra 189
A. a -b = 0, a = a b, -a + b = 1, -a + -b = -a, a c b, and -b c-a are
all equivalent.
E. a b = 0, a = a -b, -a + -b = 1, -a + b = -a, ac-6, and be -a are
all equivalent.
I. a b 41 0, a 4= a -&, -a + -6 41 1, and -a + 6 4s -a are all equivalent.
O. o -6 =}= 0, a 4= a 6, -a + 6 41 1, and -a + -b 4= -a are all equivalent.
The reader will easily translate these equivalent forms for himself.
With these symbolic representations of A, E, I and O, let us investigate
the relation of propositions traditionally referred to under the topics,
"The Square of Opposition", and "Immediate Inference".
That the traditional relation of the two pairs of contradictories holds,
is at once obvious. If a -b = 0 is true, then a -6 =f= 0 is false; if a -b = 0
is false, then a -b =)= 0 is true. Similarly for the pair, a b = 0 and a b 4= 0.
The relation of contraries is defined: Two propositions such that both
may be false but both cannot be true are "contraries". This relation is
traditionally asserted to hold between A and E. It does not hold in ex
tension: it fails to hold in the algebra precisely whenever the subject of
the two propositions is a null-class. If a = 0, then a -b = 0 and a b = O.13
That is to say, if no members of a exist, then from the point of view of
extension, "All a is 6" and "No a is 6" are both true. But if it be assumed
or stated that the class a has members (a =|= 0), then the relation holds.
a = a (b + -b) = a b + a -b.
Hence if a 4= 0, then a b + a -b 4= 0.
[8-17] If a b + a -b 4= 0 and a -b = 0, then a b 4= 0. (1)
And if a b + a -b =h 0 and a b = 0, then a -b 4= 0. (2)
We may read the last two lines :
(1) If there are members of the class a and all a is b, then "No a is 6"
is false.
(2) If there are members of the class a and no a is b, then "All a is 6"
is false.
By tradition, the particular affirmative should follow from the universal
affirmative, the particular negative from the universal negative. As has
been pointed out, this relation fails to hold when a = 0. But it holds when
ever a 4= 0. WTe can read a b 4= 0, in (1) above, as "Some a is 6" instead
of "'No a is b' is false", and a -b 4= 0, in (2), as "Some a is not 6" instead
of " ' All a is b ' is false ". We then have :
13 See Chap, n, 1-5.
190 A Survey of Symbolic Logic
(1) If there are members of a, and all a is b, then some a is b.
(2) If there are members of a, and no a is b, then some a is not b.
" Subcontraries " are propositions such that both cannot be false but
both may be true. Traditionally "Some a is 6" and "Some a is not 6"
are subcontraries. But whenever a = 0, a b 4= 0 and a -6 =j= 0 are both
false, and the relation fails to hold. When a ^ 0, it holds. Since a b = 0
is "'Some a is b' is false ", and a -b = 0 is "'Some a is not b1 is false", we
can read (1) and (2) above:
(1) If there are members of a, and "Some a is 6" is false, then some a
is not b.
(2) If there are members of a and "Some a is not 6" is false, then some
a is 6.
To sum up, then: the traditional relations of the "square of opposition"
hold in the algebra whenever the subject of the four propositions denotes a
class which has members. When the subject denotes a null-class, only
the relation of the contradictories holds. The two universal propositions
are, in that case, both true, and the two particular propositions both false.
The subject of immediate inference is not so well crystallized by tradi
tion, and for the good reason that it runs against this very difficulty of the
class without members. For instance, the following principles would all
be accepted by some logicians :
"No a is 6" gives "Xo b is a".
"No b is a" gives "All b is not-a".
"All b is not-a" gives "Some b is not-a".
"Some b is not-a" gives "Some not-a is 6".
Hence "Xo a is 6" gives "Some not-a is 6".
'Xo cows (a) are inflexed gasteropods (6) " implies "Some non-cows are
inflexed gasteropods": "Xo mathematician (a) has squared the circle (b) "
implies "Some non-mathematicians have squared the circle". These infer
ences are invalid precisely because the class b — inflexed gasteropods, suc
cessful circle-squarers — is an empty class; and because it was presumed
that "All b is not-a" gives "Some b is not-a". Those who consider the
algebraic treatment of null-classes to be arbitrary will do well to consider
the logical situation just outlined with some care. The inference of any
particular proposition from the corresponding universal requires the
assumption that either the class denoted by the subject of the particular
proposition or the class denoted by its predicate ("not-6" regarded as the
predicate of "Some a is not 6") is a class which has members,
Applications of the Boole-Schroder Algebra
191
The " conversion" of the universal negative and of the particular affirma
tive is validated by the law ab = b a. " Xo a is b ", a b = 0, gives b a = 0,
"No b is a". And "Some a is 6", a 6 4= 0, gives b a 4= 0, "Some b is a".
Also, "Some a is not 6", a -6 41 0, gives -6 a 4= 0, "Some not-6 is a".
The "converse" of the universal affirmative is simply the "converse"
of the corresponding particular, the inference of which from the universal
has already been discussed.
What are called "obverses" — i. e., two equivalent propositions with
the same subject and such that the predicate of one is the negative of the
predicate of the other — are merely alternative readings of the same equation,
or depend upon the law, -(-a) = a14. Since x y = 0 is " Xo x is y ", a -b = 0,
which is "All a is 6", is also "Xo a is not-6". And since a b = 0 is equiva
lent to a -(-b) = 0, "Xo a is 6" is equivalent to "All a is not-6".
A convenient diagram for immediate inferences can be made by putting
S (subject) and P (predicate) in the center of the circles assigned to them,
-S between the two divisions of -*S, and -P between its two constituent
divisions. The eight arrows indicate the various ways in which the dia-
(Mverl Prop,
Converse
-S-P
FIG. 13
gram may be read, and thus suggest all the immediate inferences which
are valid. For example, the arrow marked "converse" indicates the two
terms which will appear in the converse of the given proposition and the
order in which they occur. In this diagram, we must specify the null and
14 See Chap, n, 2-8.
192
A Survey of Symbolic Logic
not-null regions indicated by the given proposition. And we may — if we
wish — add the qualification that the classes, S and P, have members.
If "No £ is P", and S and P have members:
SP =
P 4= o
5.
-4.
-S-P
7.
6.
FIG. 14
Reading the diagram of figure 14 in the various possible ways, we have:
1. No S is P, and 1. Some S is not P. (According as we read what
is indicated by the fact that S P is null, or what is indicated by the fact
that S -P is not-null.)
2. All S is not-P, and 2. Some S is not-P.
3. All P is not-S, and 3. Some P is not-S.
4. No P is S, and 4. Some P is not-S.
5. Wanting.
6. Some not-S is P.
7. Some not-P is S.
8. Wanting.
Applications of the Boole-Schroder Algebra
193
Similarly, if "All S is P", and S and P have members:
S -P = 0, S + 0, P 4= 0
f. 5.
' *
FIG. 15
Reading from the diagram (figure 15), we have:
1. All S is P, and 1. Some S is P.
2. No S is not-P.
3. Wanting.
4. Some P is S.
5. Some not-$ is not-P.
6. Wanting.
7. No not-P is S.
8. All not-P is not-S, and 8. Some not-P is not-S.
The whole subject of immediate inference is so simple as to be almost
trivial. Yet in the clearing of certain difficulties concerning null-classes
the algebra has done a real service here.
The algebraic processes which give the results of syllogistic reasoning
have already been illustrated. But in those examples we carried out the
14
194 A Survey of Symbolic Logic
operations at unnecessary lengths in order to illustrate their connection
with the diagrams. The premises of any syllogism give information which
concerns, altogether, three classes. The object is to draw a conclusion
which gives as much of this information as can be stated independently of
the "middle" term. This is exactly the kind of result which elimination
gives in the algebra. And elimination is very simple. The result of
eliminating x from A x + B -x — 0 is A B = O.15 Whenever the conclusion
of a syllogism is universal, it may be obtained by combining the premises
in a single equation one member of which is 0, and eliminating the "middle"
term. For example:
No x is y, x y = 0.
All z is x, z -x = 0.
Combining these, x y + z -x = 0.
Eliminating x, y z = 0.
Hence the valid conclusion is "No y is z", or "No z is y".
Any syllogism with a universal conclusion may also be symbolized so
that the conclusion follows from the law, "If a c6 and b cc, then ace".
By this method, the laws, -(-a) = a and "If a c b, then -b c-a", are some
times required also.16 For example:
No x is y, x c -y.
All z is x, z ex.
Hence z c-y, or "No z is y", and y c-z, or "No y is z".
There is no need to treat further examples of syllogisms with universal
conclusions : they are all alike, as far as the algebra is concerned. Of course,
there are other ways of representing the premises and of getting the con
clusion, but the above are the simplest.
When a syllogism has a particular premise, and therefore a particular
conclusion, the process is somewhat different. Here we have given one
equation 1=0) and one inequation {+0j. We proceed as follows:
(1) expand the inequation by introducing the third element; (2) multiply
the equation by the element not appearing in it; (3) make use of the prin
ciple, " If a + b =f= 0 and a = 0, then b 4= 0", to obtain an inequation with
only one term in the literal member; (4) eliminate the element representing
the "middle term" from this inequation. Take, for example, A 1 1 in
15 See Chap, n, 7-4.
16 See Chap, n, 2-8 and 3-1.
Applications of the Boole-Schroder Algebra 195
the third figure:
All x is z, x -z = 0.
Some x is y, x y =}= 0.
x y = x y (z + -z) = x y z + x y -z. Hence, x y z + x y -z =f= 0.
[1-5] Since x -z = 0. x y -z = 0.
[8 • 17] Since xy z + xy -z 4= 0 and £ # -.r = 0, therefore x y z =f= 0.
Hence [8-22] yz 4= 0, or "Some y is a".
An exactly similar process gives the conclusion for every syllogism with a
particular premise.
We have omitted, so far, any consideration of syllogisms with both
premises universal and a particular conclusion — those with "weakened"
conclusions, and A A I and E A 0 in the third and fourth figures. These
are all invalid as general forms of reasoning. They involve the difficulty
which is now familiar: a universal does not give a particular without an
added assumption that some class has members. If we add to the premises
of such syllogisms the assumption that the class denoted by the middle
term is a class with members, this makes the conclusion valid. Take, for
example, A A I in the third figure:
All x is y, x-y = 0, and x has members, x =|= 0.
All x is z, x -z = 0.
Since x =k Q, xy + x -y 4= 0, and since x -y = 0, x y 41 0.
Hence x y z + x y -z 4= 0. (1)
Since x -z = 0, x y -z = 0. (2)
By (1) and (2), x y z 4 0, and hence y z =f= 0, or "Some y is z".
Syllogisms of this form are generally considered valid because of a tacit
assumption that we are dealing with things which exist. In symbolic
reasoning, or any other which is rigorous, any such assumption must be
made explicit.
An alternative treatment of the syllogism is due to Mrs. Ladd-Franklin.17
If we take the two premises of any syllogism and the contradictory of its
conclusion, we have what may be called an "inconsistent triad" — three
propositions such that if any two of them be true, the third must be false.
For if the two premises be true, the conclusion must be true and its con-
17 See "On the Algebra of Logic", in Studies in Logic by members of Johns Hopkins
University, ed. by Peirce; also articles listed in Bibl. We do not follow Mrs. Franklin's
symbolism but give her theory in a modified form, due to Josiah Royce.
196 A Survey of Symbolic Logic '
tradictory false. And if the contradictory of the conclusion be true, i. e.,
if the conclusion be false, and either of the premises true, then the other
premise must be false. As a consequence, every inconsistent triad corre
sponds to three valid syllogisms. Any two members of the triad give the
contradictory of the third as a conclusion. For example:
Inconsistent Triad
1. All a: is y
2. All y is z
3. Some x is not z.
Valid Syllogisms
1. Allan's?/ 1. M\ x is y 2. All y is 2
2. All y is z 3. Some x is not z 3. Some x is not 2
.'. All x is 2. .'. Some y is not z. .'. Some x is not y.
Omitting the cases in which two universal premises are supposed to
give a particular conclusion, since these really have three premises and
are not syllogisms, the inconsistent triad formed from any valid syllogism
will consist of two universals and one particular. For two universals will
give a universal conclusion, whose contradictory will be a particular; while
if one premise be particular, the conclusion will be particular, and its
contradictory will be the second universal. Representing universals and
particulars as we have done, this means that if we symbolize any incon
sistent triad, we shall have two equations { = Oj and one inequation { ={= 0}.
And the two universals { = Oj must give the contradictory of the particular
as a conclusion. This means that the contradictory of the particular
must be expressible as the elimination resultant of an equation of the form
a x + b -x = 0, because we have found all conclusions from two universals
to be thus obtainable. Hence the two universals of any inconsistent triad
will be of the form a x = 0 and b -x = 0 respectively. The elimination
resultant of a x + b -x = 0 is a b = 0, whose contradictory will be a b ^ 0.
Hence every inconsistent triad will have the form :
a x = 0, b -x = 0, a b =f 0
where a and b are any terms whatever positive or negative, and x is any
positive term.
The validity of any syllogism m-ay be tested by expressing its proposi
tions in the form suggested, contradicting its conclusion by changing it
from {=0} to {=t=.0} or the reverse, and comparing the resulting triad
Applications of the Boole-Schroder Algebra 197
with the above form. And the conclusion of any syllogism may be got by
considering how the triad must be completed to have the required form.
Thus, if the two premises are
No x is y, x y = 0
and All not-z is y, -z -y = 0
the conclusion must be universal. The particular required to complete
the triad is x -z 4= 0. Hence the conclusion is x -z = 0, or "All x is z".
(Incidentally it may be remarked that this valid syllogism is in no one of
the Aristotelian moods.) Again, if the premises should be x y = 0 and
yz = 0, no conclusion is possible, because these two cannot belong to the
same inconsistent triad.
We can, then, frame a single canon for all strictly valid syllogistic reason
ing: The premises and the contradictory of the conclusion, expressed in
symbolic form, { = 0} or { 4= 0}, must form a triad such that
(1) There are two universals { = 0} and one particular {4= 0).
(2) The two universals have a term in common, which is once positive
and once negative.
(3) The particular puts 4= 0 the product of the coefficients of the com
mon term in the two universals.
A few experiments with traditional syllogisms will make this matter clear
to the reader. The validity of this canon depends solely upon the nature
of the syllogism— three terms, three propositions— and upon the law of
elimination resultants, "If a .r + b -x = 0, then a b = 0".
Reasoning which involves conditional propositions— hypothetical argu
ments, dilemmas, etc.— may be treated by the same process, if we first
reduce them to syllogistic form. For example, we may translate "If A
is J5, then C is D" by "All x is y", where x is the class of cases in which
A is B, and y the class of cases in which C is D—i. e., " All cases in which .1
is B are cases in which C is D". And we may translate "But A is B"
by " All 2 is x ", where z is the case or class of cases under discussion. Thus
the hypothetical argument: "If . 1 is B, C is D. But ,1 is B. Therefore,
C is Z)", is represented by the syllogism:
"All cases in which A is B are cases in which C is D.
" But all the cases in question are cases in which A is B.
"Hence all the cases in question are cases in which C is Z)."
And all other arguments of this type are reducible to syllogisms in some
similar fashion. Thus the symbolic treatment of the syllogism extends to
198 A Survey of Symbolic Logic
them also. But conditional reasoning is more easily and simply treated
by another interpretation of the algebra — the interpretation for propositions.
The chief value of the algebra, as an instrument of reasoning, lies in
its liberating us from the limitation to syllogisms, hypothetical arguments,
dilemmas, and the other modes of traditional logic. Many who object to
the narrowness of formal logic still do not realize how arbitrary (from the
logical point of view) its limitations are. The reasons for the syllogism,
etc., are not logical but psychological. It may be worth while to exemplify
this fact. We shall offer two illustrations designed to show, each in a
different way, a wide range of logical possibilities undreamt of in formal
logic. The first of these turns upon the properties of a triadic relation
whose significance was first pointed out by Mr. A. B. Kempe.18
It is characteristically human to think in terms of dyadic relations:
we habitually break up a triadic relation into a pair of dyads. In fact, so
ingrained is this disposition that some will be sure to object that a triadic
relation is a pair of dyads. It would be exactly as logical to maintain that
all dyadic relations are triads with a null member. Either statement is
correct enough : the difference is simply one of point of view — psychological
preference. If there should be inhabitants of Mars whose logical sense
coincided with our own, so that any conclusion which seemed valid to us
would seem valid to them, and vice versa, but whose psychology otherwise
differed from ours, these Martians might have an equally fundamental
prejudice in favor of triadic relations. We can point out one such which
they might regard as the elementary relation of logic — as we regard equality
or inclusion. In terms of this triadic relation, all their reasoning might
be carried out with complete success.
Let us symbolize by (ac/b), a-bc + -ab-c = 0. This relation may be
diagrammed as in figure 16, since a-bc + -ab-c = 0 is equivalent to
accbc(a + c). (Note that (ac/b) and (ca/b) are equivalent, since a -be
+ -ab -c is symmetrical with respect to a and c.)
This relation (ac/b) represents precisely the information which we
habitually discard in drawing a syllogistic conclusion from two universal
premises. If all a is b and all b is c, we have
a -b = 0 and 6 -c = 0
Hence a -b (c + -c) + (a + -a) b -c = 0,
18 See his paper "On the Relation of the Logical Theory of Classes and the Geometrical
Theory of Points/', Proc. London Math. Soc., xxi, 147-82. But the use we here make of
this relation is due to Josiah Royce. For a further discussion of Kempe's triadic relation,
see below, Chap, vi, Sect. iv.
Applications of the Boole-Schroder Algebra
199
Or, a -6 c + a -b -c + a b -c + -a b -c = 0.
[5-72] This equation is equivalent to the pair,
(1) a -b -c + a b -c = a -c (b + -b) = a -c = 0,
and (2) a -b c + -a b -c = 0.
(1) is the syllogistic conclusion, "All a is c"; (2) is (ac/b). Perhaps most
of us would feel that a syllogistic conclusion states all the information
given by the premises: the Martians might equally well feel that precisely
FIG. 16
what we overlook is the only thing worth mentioning. And yet with this
curious "illogical" prejudice, they would still be capable of understanding
and of getting for themselves any conclusion which a syllogism or a hypo
thetical argument can give, and many others which are only very awkwardly
stateable in terms of our formal logic. Our relation, a c6, or "All a is 6",
would be, in their terms, (Ob/ a). (Qb/a) is equivalent to
1 • a -6 + 0 • -a 6 = 0 = a -b
Hence the syllogism in Barbara would be " (06/a) and (Oc/6), hence (Oc/a) ".
This would, in fact, be only a special case of a more general principle which
is one of those we may suppose the Martians would ordinarily rely upon
for inference: "If (xb/a) and (xc/b), then (arc/a)". That this general
principle holds, is proved as follows:
(xb/a) is -x a -6 + x -a b = 0
(xc/b) is -x b -c + x -b c = 0
These two together give :
-x a-b (e + -c) + x -a b (c + -c) + -x b -c (a + -a) + x -b c (a + -a) = 0,
or? -x a _5 c + -x a -b -c + x -a b c + x -a b -c + -x a b -c + -x -a b -c
+ x a -b c + x -a -6 c = 0.
200 A Survey of Symbolic Logic
[5-72] This equation is equivalent to the pair,
(1) -.r a b -c + -x a -b -c + x -ab c + x -a -b c
= -x a-c (b + -b) + x -a c (b + -b)
= -x a -c + a* -a c = 0.
(2) x -a b -c + -x -a b -c + x a -b c + -x a -b c
= -a b -c (x + -x) + a -b c (x + -x)
= -a b -c + a -b c = 0.
(1) is (xc/a), of which our syllogistic conclusion is a special case; (2) is a
similar valid conclusion, though one which we never draw and have no
language to express.
Thus these Martians could deal with and understand our formal logic
by treating our dyads as triads with one member null. In somewhat
similar fashion, hypothetical propositions, the relation of equality, syllo
gisms with a particular premise, dilemmas, etc., are all capable of state
ment in terms of the relation (ac/b). As a fact, this relation is much more
powerful than any dyad for purposes of reasoning. Anyone who will
trouble to study its properties will be convinced that the only sound reason
for not using it, instead of our dyads, is the psychological difficulty of
keeping in mind at once two triads with two members in common but
differently placed, and a third member which is different in the two. Our
attention-span is too small. But the operations of the algebra are inde
pendent of such purely psychological limitations — that is to say, a process
too complicated for us in any other form becomes sufficiently simple to be
clear in the algebra. The algebra has a generality and scope which " formal "
logic cannot attain.
This illustration has indicated the possibility of entirely valid non-
traditional modes of reasoning. We shall now exemplify the fact that by
modes which are not so remote from familiar processes of reasoning, any
number of non-traditional conclusions can be drawn. For this purpose,
we make use of Poretsky's Law of Forms:19
x = 0 is equivalent to t = t -x + -t x
This law is evident enough: if x = 0, then for any t, t-x = t-1 = t, and
-tx = -t-0 = 0, while £ + 0 = t. Let us now take the syllogistic premises,
"All a is 6" and "All 6 is c", and see what sort of results can be derived
from them by this law.
All a is b, a-b = 0.
All b is c, b-c = 0.
19 See Chap, n, 7-15 and 7-16.
Applications of the Boole-Schroder Algebra 201
Combining these, a-b + b -c = 0.
And [3-4-41] -(a-b + b-c) = -(a -b) --(b -c) = (-a + b)(-b + c)
= -a -b + -a c + b c.
Let us make substitutions, in terms of a, 6, and c, for the t of this formula.
a + b = (a + b) (-a -b + -a c + b c) + -a -b (a-b + b -c)
= abc+-abc + bc = be
What is either a or 6 is identical with that which is both b and c. This is a
non-syllogistic conclusion from "All a is 6 and all b is c". Other such
conclusions may be got by similar substitutions in the formula.
a + c = (a + c) (-a -b + -a c + b c) + -a -c (a -b + b -c)
= a b c + -a -b c + -a c + b c + -a b -c = a b c + -a (b + c) .
What is either a or c is identical with that which is a, b, and c, all three, or
is not a and either b or c.
-5 c = -b c (-a -b + -a c + b c) + (b + -c) (a -b + b -c)
= -a -b c + b -c + a -b -c = -a-b c + (a + b) -c
That which is b but not c is identical with what is c but neither a nor b
or is either a or b but not c. The number of such conclusions to be got from
the premises, "All a is 6" and "All b is c", is limited only by the number of
functions which can be formed with a, b, and c, and the limitation to sub
stitutions in terms of these is, of course, arbitrary. By this method, the
number of conclusions which can be drawn from given premises is entirely
unlimited.
In concluding this discussion of the application of the algebra to the
logic of classes, we may give a few examples in which problems more involved
than those usually dealt with by formal logic are solved. The examples
chosen are mostly taken from other sources, and some of them, like the
first, are fairly historic.
Example I.20
A certain club has the following rules: (a) The financial committee
shall be chosen from among the general committee; (b) No one shall be a
member both of the general and library committees unless he be also on
the financial committee; (c) Xo member of the library committee shall be
on the financial committee.
Simplify the rules.
10 See Venn, Symbolic Logic, ed. 2, p. 331.
202 A Survey of Symbolic Logic
Let / = member of financial committee.
g = " " general
/ = " " library
The premises then become:
(a) fcg, or / -g = 0.
(6) (gl)cf, or -fgl = 0.
(c) // = 0.
We can discover by diagramming whether there is redundancy here. In
figure 17, (a) is indicated by vertical lines, (6) by horizontal, (c) by oblique.
(a) and (c) both predicate the non-existence of / -g I. To simplify the
rules, unite (a), (6), and (c) in a single equation:
-g) = f -g + -f g l + fg l+f-g I
[5-91] =f-g+(-f+f)gl=f-g + gl = 0.
And [5-72] this is equivalent to the pair, f -g = 0 and g I = 0.
Thus the simplified rules will be :
(a'} The financial committee shall be chosen from among the general
committee.
(6') No member of the general committee shall be on the library com
mittee.
Applications of the Boole-Schroder Algebra
203
Example 2.21
The members of a certain collection are classified in three ways — as
a's or not, as 6's or not, and as c's or not. It is then found that the class 6
is made up precisely of the a's which are not c's and the c's which are not a's.
How is the class c constituted?
Given: b = a -c + -a c. To solve for c.22
b = b (c + -c) = 6 c + b -c.
Hence, b c + b -c = a -c + -a c.
Hence [7-27] a -6 + -a b c c c a -b + -a b.
Or [2-2] c = a-b + -ab.
The c's comprise the a's which are not 6's and the 6's which are not a's.
Another solution of this problem would be given by reducing b = a -c
+ -ac to the form { = 0} and using the diagram.
[7 • 1] b = a -c + -a c is equivalent to
6 --(a -c + -a c) + -b (a -c + -a c) = 0
And [6 • 4] -(a -c + -a c) = a c + -a -c.
Hence, a b c + -a 6 -c + a -6 -c + -a -6 c = 0.
We observe here (figure 18) not only that c = a -b + -a b, but that the
FIG. 18
relation of a, 6, and c, stated by the premise is totally symmetrical, so that
we have also a = b -c + -6 c.
21 Adapted from one of Venn's, first printed in an article on "Boole's System of Logic",
Mind, i (1876), p. 487.
22 This proof will be intelligible if the reader understands the solution formula referred to.
204
A Survey of Symbolic Logic
Example 3.23
If x that is not a is the same as 6, and a that is not x is the same as c,
what is x in terms of a, 6, and c?
Given: b = -ax and c = a -.r. To solve for .r.
[7-1] 6 = -a .if is equivalent to
-(-a .T) 6 + -a -b x = 0 = (a + -.r) 6 + -a -6 a:
= a b + b -x + -a -6 .r = 0 (1)
And c = a -x is equivalent to
-(a -x) c + a -c -x = 0 = (-a + .T) c + a-c -x
= -a c + c x + a -c -x = 0 (2)
Combining (1) and (2),
a b + -a c + (-a -b + c) x + (b + a -c) -x = 0 (3)
Hence [5 • 72] (-a -b + c) x + (b + a -c) -# = 0 (4)
[7-221] This gives the equation of condition,
(-a -b + c)(6 + a -c) = 6 c = 0 (5)
[7-2] The solution of (4) is
(b + a -c) c # c -(-a -6 + c)
And by (5),
-(-a -6 + c) = -(-a -6 + c) + 6 c = (a + b) -c + b c
= a -c + b (c + -c) = b + a -c
Hence [2-2] x = b + a -c.
FIG. 19
2S See Lambert, Logische Abhandlungen, i, 14.
Applications of the Boole-Schroder Algebra
205
This solution is verified by the diagram (figure 19) of equation (3), which
combines all the data. Lambert gives the solution as
x — (a + b) -c
This also is verified by the diagram.
Example 4.24
What is the precise point at issue between two disputants, one of whom,
A, asserts that space should be defined as three-way spread having points
as elements, while the other, E, insists that space should be defined as
three-way spread, and admits that space has points as elements.
Let s = space,
t = three-way spread,
p = having points as elements.
A asserts: s = t p. B states: s = t and s cp.
s = t p is equivalent to
s--(tp)+-stp = 0 = s -t + s -p + -s t p = 0 (1)
s c p is equivalent to s -p = 0 (2)
And s = t is equivalent to s -t + -s t = 0 (3)
(2) and (3) together are equivalent to
s -t + s -p + -s t = 0 (4)
(1) represents .4's assertion, and (4) represents #'s. The difference between
FIG. 20
the two is that between -s tp = 0 and -* * = 0. (See figure 20.)
-s t = -s t p + -s t -p
24 Quoted from Jevons by Mrs. Ladd-Franklin, loc, tit., p. 52.
206 A Survey of Symbolic Logic
The difference is, then, that B asserts -st-p = 0, while A does not. It
would be easy to misinterpret this issue, -st-p = 0 is t-pcs, "Three-
way spread not having points as elements, is space ". But B cannot sig
nificantly assert this, for he has denied the existence of any space not having
points as elements. Both assert s = t p. The real difference is this: B
definitely asserts that all three-way spread has points as elements and is
space, while A has left open the possibility that there should be three-way
spread not having points as elements which should not be space.
Example 5.
Amongst the objects in a small boy's pocket are some bits of metal
which he regards as useful. But all the bits of metal which are not heavy
enough to sink a fishline are bent. And he considers no bent object useful
unless it is either heavy enough to sink a fishline or is not metal. And the
only objects heavy enough to sink a fishline, which he regards as useful,
are bits of metal that are bent. Specifically what has he in his pocket which
he regards as useful?
Let x = bits of metal,
y = objects he regards as useful,
z = things heavy enough to sink a fishline,
w = bent objects.
Symbolizing the propositions in the order stated, we have
xy * 0
x -z c w, or x -z -w = 0
y w c (z + -.i'), or x y -z w = 0
zy ex ID, or -x y z + y z -w = 0
Expanding the inequation with reference to z and w,
x y z w + x y z -w + x y -z w + x y -z -w =H 0
Combining the equations,
x -z -w (y + -y) + x y -z w + -x y z (w + -w) + y z -w (x + -x) = 0
or x y -z -w + x -y -z -w + x y -z w + -x y z w + -x y z-w + x y z -w = 0
All the terms of the inequation appear also in this equation, with the
exception of x y z w. Hence, by 8- 17, x y z w 4= 0. The small boy has
Applications of the Boole-Schroder Algebra
207
some bent bits of metal heavy enough to sink a fishline, which he considers
useful. This appears in the diagram (figure 21) by the fact that while
FIG. 21
some subdivision of x y must be not-null, all of these but x y z w is null.
It appears also that anything else he may have which he considers useful
may or may not be bent but is not metal.
Example 6.25
The annelida consist of all invertebrate animals having red blood in a
double system of circulating vessels. And all annelida are soft-bodied,
and either naked or enclosed in a tube. Suppose we wish to obtain the
relation in which soft-bodied animals enclosed in tubes are placed (by virtue
of the premises) with respect to the possession of red blood, of an external
covering, and of a vertebral column.
Let a = annelida,
s = soft-bodied animals,
n = naked,
/ = enclosed in a tube,
i = invertebrate,
r = having red blood, etc.
Given: a = i r and a cs (n + t), with the implied condition, n t = 0. To
eliminate a and find an expression for s t.
25 See Boole, Laws of Thought, pp. 144-46.
208
A Survey of Symbolic Logic
a = i r is equivalent to
-(i r) a + -a i r = a -i + a -r + -a i r = 0 (1)
a cs (n+t) is equivalent to a --(s n + s t) = 0.
-0 n + s t) = -(s ii) - -(s t) = (-s + -n) (-s + -0 = -s + -n -t.
Hence, a-s + a-n-t = 0 (2)
Combining (1) and (2) and n t = 0,
a -i + a -r + -a i r + a -s + a -n -t + n t = 0 (3)
Eliminating a, by 7 • 4,
(-i + -r + -s + -7i -2 + n t) (i r + n t) = n t + i r -s + i r -n -t = 0
The solution of this equation for s is26 i r cs.
And its solution for tisir-nctc-n.
Hence [5-3] ir-nc.stc.-n, or s t = ir-n+u--n, where u is un
determined.
The soft-bodied animals enclosed in a tube consist of the invertebrates
FIG. 22
26 See Chap, n, Sect, iv, "Symmetrical and Unsymmetrical Constituents of an Equa
tion".
Applications of the Boole-Schroder Algebra 209
which have red blood in a double system of circulating vessels and a body
covering, together with an undetermined additional class (which may be
null) of other animals which have a body covering. This solution may be
verified by the diagram of equation (3) (figure 22). In this diagram, s t is
the square formed by the two crossed rectangles. The lower half of this
inner square exhibits the solution. Note that the qualification, -n, in
i r -n c s t, is necessary. In the top row is a single undeleted area repre
senting a portion of i r (n) which is not contained in s t.
Example 7.27
Demonstrate that from the premises "All a is either b or c", and
"All c is a", no conclusion can be drawn which involves only two of the
classes, a, 6, and c.
Given: a c (b + c) and c c a.
To prove that the elimination of any one element gives a result which
is either indeterminate or contained in one or other of the premises.
a c (b + c) is equivalent to a -6 -c = 0.
And c c a is equivalent to -a c = 0.
Combining these, a -b -c + -a c = 0.
Eliminating a [7-4], (-b -c) c = 0, which is the identity, 0 = 0.
Eliminating c, (a -b) -a = 0, or 0 = 0.
Eliminating b, (-a c + a -c) -a c = -a c = 0, which is the second
premise.
Example 8.
A set of balls are all of them spotted with one or more of the colors, red,
green, and blue, and are numbered. And all the balls spotted with red are
also spotted with blue. All the odd-numbered blue balls, and all the even
numbered balls which are not both red and green, are on the table. De
scribe the balls not on the table.
Let e = even-numbered, -e = odd-numbered,
r = spotted with red,
b = spotted with blue,
g = spotted with green,
t = balls on the table.
Given: (1) -r -b -g = 0.
27 See De Morgan, Formal Logic, p. 123.
15
210
A Survey of Symbolic Logic
(2) r -6 = 0.
(3) [-eb + e-(rg)]ct, or (-e b + e -r + e -g) -t = 0.
To find an expression, x, such that -t c x, or -t x = -t. Such an expression
should be as brief as possible. Consequently we must develop -t with
respect to e, r, b, and g, and eliminate all null terms. (An alternative
method would be to solve for -t, but the procedure suggested is briefer.)
-t = -t(e + -e) (r + -r) (b + -6) (g + -g)
= -t(erbg+erb-g+er-bg + e-rbg + -erbg + er-b-g
+ e -r b -g + -e r b -g + e -r -b g + -e -r b g + -e r -b g
+ e -r -b -g + -e r -b -g + -e -r b -g + -e -r -6 g + -e -r -b -g) (4)
From (1), (2), and (3),
-t (-e b + e -r + e -g + r -b + -r -b -g) = 0 (5)
Eliminating from (4) terms involved in (5),
-t = -t (e r b g + -e -r -b g), or -t c (e r b g + -e -r -b g)
All the balls not on the table are even-numbered and spotted with all three
colors or odd-numbered and spotted with green only.
g...
Applications of the Boole-Schroder Algebra 211
In the diagram (figure 23), equation (1) is indicated by vertical lines,
(2) by oblique, (3) by horizontal.
Example 9.28
Suppose that an analysis of the properties of a particular class of sub
stances has led to the following general conclusions:
1st. That wherever the properties a and b are combined, either the
property c, or the property d, is present also; but they are not jointly present.
2d. That wherever the properties b and c are combined, the properties
a and d are either both present with them, or both absent.
3d. That wherever the properties a and b are both absent, the proper
ties c and d are both absent also; and vice versa, where the properties
c and d are both absent, a and b are both absent also.
Let it then be required from the above to determine what may be con
cluded in any particular instance from the presence of the property a with
respect to the presence or absence of the properties b and c, paying no
regard to the property d.
Given: (1) a b c (c -d + -c d).
(2) bcc(ad + -a-d).
(3) -a -b = -c -d.
To eliminate d and solve for a.
(1) is equivalent to a b--(c -d + -c d) = 0.
(2) is equivalent to b c--(a d + -a -d) = 0.
But [6-4] -(c-d + -cd) =cd + -c-d,
and -(a d + -a -d) = -a d + a -d.
Hence we have, a b (c d + -c -d) =abcd+ab-c-d = Q (4)
and b c (-a d+ a -d) =-abcd+abc-d = () (5)
(3) is equivalent to
-a -b (c + d) + (a + b) -c -d
= -a -b c + -a -b d + a -c -d + b -c -d = 0 (0)
Combining (4), (5), and (6), and giving the result the form of a
function of d,
(-a -b c + -a -6 + a b c + -a b c) d
+ (-a -b c + a -c + b -c + a b -c + a b c) -d = 0
28 See Boole, Laws of Thought, pp. 118-20. For furfher problems, see Mrs. Larld-
Franklin, loc. cit., pp. 51-61, Venn, Symbolic Logic, Chap, xm, and Schroder, Algebra dcr
Logik: Vol. I, Dreizehnte Vorlesung.
212 A Survey of Symbolic Logic
Or, simplifying, by 5-4 and 5-91,
(_a -fc + b c) d + (-a -b c + a -c + b -c + a b c) -d = 0
Hence [7 • 4] eliminating d,
(_a -I) + I) c) (-a -b c + a-c + b -c + ab c) = -a-b c + ab c = 0
Solving this equation for a [7-2], -b c c a c (-b + -c).
The property a is always present when c is present and b absent, and when
ever a is present, either b is absent or c is absent.
The diagram (figure 24) combines equations (4), (5), and (6).
FIG. 24
As Boole correctly claimed, the most powerful application of this algebra
is to problems of probability. But for this, additional laws which do not
belong to the system are, of course, required. Hence we omit it. Some
thing of what the algebra will do toward the solution of such problems will
be evident if the reader imagine our Example 8 as giving numerically the
proportion of balls spotted with red, with blue, and with green, and the
quaesitum to be "If a ball not on the table be chosen at random, what is
the probability that it will be spotted with all three colors? that it wrill be
spotted with green?" The algebra alone, without any additional laws,
answers the last question. As the reader will observe from the solution,
all the balls not on the ta*ble are spotted with green.
Applications of the Boole-Schroder Algebra 213
III. THE APPLICATION TO PROPOSITIONS
If, in our postulates, a, b, c, etc., represent propositions, and the "prod
uct", a b, represent the proposition which asserts a and b both, then we
have another interpretation of the algebra. Since a+b is the negative of
-a -6, a + b will represent "It is false that a and b are both false", or
"At least one of the two, a and 6, is true". It has been customary to read
a + b, " Either a or 6", or " Either a is true or b is true ". But this is some
what misleading, since "Either ... or ..." frequently denotes, in
ordinary use, a relation which is to be understood in intension, while this
algebra is incapable of representing relations of intension. For instance,
we should hardly affirm "Either parallels meet at finite intervals or all
men are mortal". We might well say that the "Either . . . or . . . "
relation here predicated fails to hold because the two propositions are
irrelevant. But at least one of the two, " Parallels meet at finite intervals"
and "All men are mortal", is a true proposition. The relation denoted
by + in the algebra holds between them. Hence, if we render a + b by
"Either a or 6", we must bear in mind that no necessary connection of a
and 6, no relation of "relevance" or "logical import", is intended.
The negative of a, -a, will be its contradictory, or the proposition "a is
false". It might be thought that -a should symbolize the "contrary"
of a as well, — that if a be "All men are mortal ", then " Xo men are mortal "
should be -a. But if the contrary as well as the contradictory be denoted
by -a, then -a will be an ambiguous function of a, whereas the algebra
requires that -a be unique. 29
The interpretation of 0 and 1 is most easily made clear by considering
the connection between the interpretation of the algebra for propositions
and its interpretation for classes. The prepositional sign, a, may equally
well be taken to represent the class of cases in which the proposition a is
true, a b will then represent the class of cases in which a and b are both
true; -a, the class of cases in which a is false, and so on. The " universe", 1,
will be the class of all cases, or all "actual" cases, or the universe of facts.
Thus a = 1 represents "The cases in which a is true are all cases", or
"a is true in point of fact", or simply "a is true". Similarly 0 is the class
of no cases, and a = 0 will mean "a is true in no case", or "a is false".
It might well be asked: May not a, b, c, etc., represent statements which
are sometimes true and sometimes false, such as "Today is Monday"
or "The die shows an ace"? May not a symbolize the cases in which a is
29 See Chap, n, 3-3.
214 A Survey of Symbolic Logic
true, and these be not all but only some of the cases? And should not
a = 1 be read "a is always true", as distinguished from the less com
prehensive statement, "a is true"? The answer is that the interpretation
thus suggested can be made and that Boole actually made it in his chapters
on "Secondary Propositions".30 But symbolic logicians have come to
distinguish between assertions which are sometimes true and sometimes
false and propositions. In the sense in which "Today is Monday" is
sometimes true and sometimes false, it is called a propositional function
and not a proposition. There are two principal objections to interpreting
the Boole-Schroder Algebra as a logic of propositional functions. In the
first place, the logic of propositional functions is much more complex than
this algebra, and in the second place, it is much more useful to restrict the
algebra to propositions by the additional law "If a H= 0, then a = 1, and
if a =^ 1, then a = 0", and avoid any confusion of propositions with asser
tions which are sometimes true and sometimes false. In the next chapter,
we shall investigate the consequences of this law, which holds for proposi
tions but not for classes or for propositional functions. We need not pre
sume this law at present : the Boole-Schroder Algebra, exactly as presented
in the last chapter, is applicable throughout to propositions. But we shall
remember that a proposition is either always true or never true : if a proposi
tion is true at all, it is always true. Hence in the interpretation of the
algebra for propositions, a == 1 means "a is true" or "a is always true"
indifferently — the two are synonymous. And a = 0 means either "a is
false" or "a is always false".
The relation a c b, since it is equivalent to a -b = 0, may be read " It
is false that 'a is true and b is false'", or loosely, "If a is true, then b is
true". But acb, like a + b, is here a relation which does not signify
"relevance" or a connection of "logical import". Suppose a = "2 + 2
= 4" and b = "Christmas is a holiday". We should hardly say "If
2 + 2 == 4, then Christmas is a holiday". Yet it is false that "2 + 2 = 4
and Christmas is not a holiday": in this example a -b = 0 is true, and
hence a c b will hold. This relation, a c b, is called "material implication ";
it is a relation of extension, whereas we most frequently interpret "implies"
as a relation of intension. But acb has one most important property in
common with our usual meaning of "a implies 6" — when a c b is true, the
case in which a is true but b is false does not occur. If a c b holds, and a is
true, then b will not be false, though it may be irrelevant. Thus "material
30 Laws of Thought, Chaps, xi-xiv.
Applications of the Boole-Schroder Algebra 215
implication" is a relation which covers more than the "implies" of ordinary
logic: a c6 holds whenever the usual "a implies 6" holds; it also holds in
some cases in which "a implies 6" does not hold.31
The application of the algebra to propositions is so simple, and so
resembles its application to classes, that a comparatively few illustrations
will suffice. We give some from the elementary logic of conditional propo
sitions, and conclude with one taken from Boole.
Example 1.
If A is B, C is D. (1)
And A is B. (2)
Let x = A is B-, y = C is D.
The two premises then are :
(1) xcy, or [4-9] -x + y == 1.
(2) x = 1, or -x = 0.
[5-7] Since -x + y = 1 and -x = 0, y = 1.
y = 1 is the conclusion " C is D ".
Example 2.
(1) If A is B, C is D.
(2) But C is not D.
Let £ = A is B- y = C is D.
(1) .re?/, or -x + y = 1.
(2) y - 0.
[5-7] Since -x + y = 1 and y = 0, -r = 1.
-x = 1 is the conclusion "A is 1? is false", or "A is not B'\
Example 3.
(1) If A isB,C'isD; and (2) if E is F, 6' is H.
(3) But either A is B or C is Z).
Let w = A isB; x = ClsD; y = E'isF; z = G is //.
(1) icc.x, or [4-9] w x = 10.
(2) ycz, or 2/3 = #•
(3) w + y - 1.
31 "Material implication" is discussed more at length in Chap, iv, Sect, i, and Chap.
v, Sect. v.
216
A Survey of Symbolic Logic
Since w + y = 1, and w x = w and y z = ?y, icx + yz = 1.
Hence [4 • 5] -w x + -w x + y z + -y z = 1 + -w x + -y z = 1.
Hence # (ir + -w) + z (y + -y) = x + z = 1.
#+3 = 1 is the conclusion ''Either C is Z) or 6? is //". This dilemma
may be diagrammed if we put our equations in the equivalent forms
(1) w-x = 0, (2) y-z = 0, (3) -10 -y = 0. In figure 25, w -z is struck
FIG. 25
out with horizontal lines, y-z with vertical, -H> -# with oblique. That
everything which remains is either x or z is evident.
Example 4.
(1) Either .1 is 7? or C is not 7).
(2) Either C is 7) or E is F.
(3) Either A is Z? or E is not /?.
Let x = A is /*; y = C is 7); 2 = E is ^.
(1) * + -2/= 1.
(2) y + s = 1, or -y-z = 0.
(3) £ + -3 = 1, or -xz = 0.
By (1), a; + -# (3 + -3) = x + -?/ z + -y -3 = 1.
Hence by (2), x + -yz = 1 = x + -y z (x + -x) = x + x -y z + -x -y z.
And by (3), -x -y z = 0. Hence x + x -y z = x = 1.
Thus these three premises give the categorical conclusion "A is 5", indi
cating the fact that the traditional modes of conditional syllogism are by
no means exhaustive.
Applications of the Boole- Schroder Algebra 217
Example 5.32
Assume the premises:
1. If matter is a necessary being, either the property of gravitation is
necessarily present, or it is necessarily absent.
2. If gravitation is necessarily absent, and the world is not subject to
any presiding intelligence, motion does not exist.
3. If gravitation is necessarily present, a vacuum is necessary.
4. If a vacuum is necessary, matter is not a necessary being.
5. If matter is a necessary being, the world is not subject to a presiding
intelligence.
Let x = Matter is a necessary being.
y = Gravitation is necessarily present.
z = The world is not subject to a presiding intelligence.
w = Motion exists.
t = Gravitation is necessarily absent.
v — A vacuum is necessary.
The premises then are :
1 (1) xc(y + f), or x-y-t = 0.
(2) t z c -w, or tzw = 0.
(3) y cv, or y -v = 0.
(4) v c -x, or v x = 0.
(5) x c z, or x -z = 0.
And since gravitation cannot be both present and absent,
(6) y t = 0.
Combining these equations :
x-y -t + tz w + ij -v + v x + x -z + y t = 0 (7)
From these premises, let it be required, first, to discover any collection
between x, "Matter is a necessary being", and y, "Gravitation is necessarily
present". For this purpose, it is sufficient to discover whether any one
of the four, x y = 0, x -y = 0, -x y = 0, or -.T -y = 0, since these are
the relations which state any implication which holds between x, or -.r,
and ?/, or -y. This can always be done by collecting the coefficients of
x y, x -y, -x y, and -x -y, in the comprehensive expression of the data,
such as equation (7), and finding which of them, if any, reduce to 1. But
32 See Boole, Laws of Thought, Chap. xiv. The premises assumed are supposed to be
borrowed from Clarke's metaphysics.
218
A Survey of Symbolic Logic
sometimes, as in the present case, this lengthy procedure is not necessary,
because the inspection of the equation representing the data readily reveals
such a relation.
From (7), [5-72] vx + -vy = 0.
Hence [1-5] v x y + -v x y = (v + -v) x y = x y = 0, or x c -y, y c -x.
If matter is a necessary being, then gravitation is not necessarily present;
if gravitation is necessarily present, matter is not a necessary being.
Next, let any connection between x and w be required. Here no such
relation is easily to be discovered by inspection. Remembering that if
a = 0, then a b = 0 and a -b = 0 ;
From (7), (-y -t + t z + y -v + v + -z + y t) w x
+ (t z + y -v + y t) w -x
+ (-y -t + y -v + v + -z + y t) -w x
+ (y-v + yt) -iv -x = 0 (8)
Here the coefficient of w x reduces to 1, for [5-85],
y -i) + D = y + v, and t z + -z = t + -z
and hence the coefficient is -y -t + y + t + v + -z + y t.
But [5-90] (-y -t + y + £) + v + -z + y t = l+v + -z + yt = 1.
Hence w x = 0, or w c -x, x c -iv.
7-
-IV
-y- --{
FIG. 26
Applications of the Boole-Schroder Algebra 219
None of the other coefficients in (8) reduces to 1. Hence the conclusion
which connects x and w is: "If motion exists, matter is not a necessary
being; if matter is a necessary being, motion does not exist".
Further conclusions, relating other terms, might be derived from the
same premises. All such conclusions are readily discoverable in the dia
gram of equation (7). In fact, the diagram is more convenient for such
problems than the transformation of equations in the algebra.
Another method for discovering the implications involved in given data
is to state the data entirely in terms of the relation c , and, remembering
that "If ac6 and bcc, then ace", as well as "acb is equivalent to
-b c-a", to seek directly any connection thus revealed between the propo
sitions which are in question. Although by this method it is possible to
overlook a connection which exists, the danger is relatively small.
IV. THE APPLICATION TO RELATIONS
The application of the algebra to relations is relatively unimportant,
because the logic of relations is immensely more complex than the Boole-
Schroder Algebra, and requires more extensive treatment in order to be of
service. We shall, consequently, confine our discussion simply to the
explanation of this interpretation of the algebra.
A relation, taken in extension, is the class of all couples, triads, or tetrads,
etc., which have the property of being so related. That is, the relation
"father of" is the class of all those couples, (x\y)> such that x is father
of y: the dyadic relation R is the class of all couples (.r; y) such that x has
the relation R to y, x R y. The extension of a relation is the class of things
which have the relation. We must distinguish between the class of couples
(x; y) and the class of couples (y- x), since not all relations are symmetrical
and .r R y commonly differs from y R x. Since the properties of relations,
so far as the laws of this algebra apply to them, are the same whether they
are dyadic, triadic, or tetradic, etc., the discussion of dyadic relations will be
sufficient.
The " product ", R x S, or R S, will represent the class of all those couples
(x; y) such that xRy and x S y are both true. The "sum ", R + S, will be
the class of all couples 0 ; y) such that at least one of the two, .r R y and
x S y, holds. The negative of R, -R, will be the class of couples 0; y) for
which x R y is false.
The null-relation, 0, will be the null-class of couples. If the class of
couples (t\ u) for which t R u is true, is a class with no members, and the
220 A Survey of Symbolic Logic
class of couples (u; w) for which v S w is true is also a class with no members,
then R and S have the same extension. It is this extension which 0 repre
sents. Thus R = 0 signifies that there are no two things, t and u, such
that t R u is true — that nothing has the relation R to anything. Similarly,
the universal-relation, 1, is the class of all couples (in the universe of dis
course).
The inclusion, RcS, represents the assertion that every couple (x; y)
for which x R y is true is also such that x S y is true; or, to put it otherwise,
that the class of couples (x; y) for which x R y is true is included in the
class of couples (u; v) for which u S v is true. Perhaps the most satisfactory
reading of R c S is " The presence of the relation R implies the presence of
the relation $". R = S, being equivalent to the pair, RcS and S c R,
signifies that R and S have the same extension— that the class of couples
(x; y) for which x Ry is true is identically the class of couples (u; v) for
which u S v is true.
It is obvious that all the postulates, and hence all the propositions, of
the Boole-Schroder Algebra hold for relations, so interpreted.
1-1 If R and S are relations (that is, if there is a class of couples (x; y)
such that x Ry is true, and a class of couples (u; v) such that u S -K is true),
then R xS is a relation (that is, there is a class of couples (w; z) such that
w R z and w S z are both true). If R and S be such that there is no couple
(w; z) for which w R z and w S z both hold, then R x $ is the null-relation, 0
— i. e., the null-class of couples.
1-2 The class of couples (x; y) for which x Ry and x R y both hold is
simply the class of couples for which x Ry holds.
1 • 3 The class of couples denoted by R x S is the same as that denoted
by SxR — namely, the class of couples (x't y) such that xRy and xSy
are both true.
1-4 The class of couples (xm, y) for which xRy, x S y, and x T y all
hold is identically the same in whatever order the relations be combined—
i. e., Rx(SxT) = (RxS) xT.
1-5 R xO = 0— i. e., the product of the class of couples for which x R y
holds and the null-class of couples is the null-class of couples.
1 • 0 For every relation, R, there is a relation -R, the class of couples
for which x Ry is false, and -R is such that:
1-61 If the relation Rx-S is null (that is, if there is no couple such
that x R y is true and x S y is false), then R xS = R (that is, the class of
couples for which x R y is true is identically the class of couples for which
x Ry and x S y are both true) ; and
Applications of the Boole-Schroder Algebra 221
1-62 If R xS - R and Rx-S = R, then R = 0— i. e., if the class of
couples for which x R y and x S y are both true is identically the class of
couples for which x R y is true, and if also the class of couples for which
x R y is true and x S y is false is identically the class of couples for which
x R y is true, then the class of couples for which x R y is true is null.
1-71= -0 — i. e., the universal class of couples is the negative of the
null-class of couples, within the universe of discourse of couples.
1-8 R+S = -(-Rx-S)-i. e., the class of couples (x\y) such that
at least one of the two, x R y and x S y, is true is the negative of the class
of couples for which x R y and x S y are both false.
1-9 RxS = Ris equivalent to R c S — i. e., if the class of couples (x ; y)
for which x R y and x S y are both true is identical with the class of couples
for which x R y is true, then the presence of R implies the presence of S',
and if the presence of R implies the presence of S, then the class of couples
(x; y) for which x R y is true is identical with the class of couples for which
x R y and x S y are both true.33
33 For a further discussion of the logic of relations, see Chap, iv, Sect. v.
CHAPTER IV
SYSTEMS BASED ON MATERIAL IMPLICATION
We are concerned, in the present chapter, with the " calculus of propo
sitions" or calculus of "material implication", and with its extension to
pro positional functions. We shall discover here two distinct modes of
procedure, and it is part of our purpose to set these two methods side by side.
The first procedure takes the Boole-Schroder Algebra as its foundation,
interprets the elements of this system as propositions, and adds to it a
postulate which holds for propositions but not for logical classes. The
result is what has been called the "Two-Valued Algebra", because the
additional postulate results in the law: For any x, if x =|= 1, then x = 0,
and if x =t= 0, then x = 1. This Two- Valued Algebra is one form of the
calculus of propositions. The extension of the Two-Valued Algebra to
propositions of the form <pxn, where xn is an individual member of a class
composed of x{, a>, a*3, etc., gives the calculus of prepositional functions.
II and 2 functions have a special significance in this system, and the relation
of "formal implication", Hx(<px c\j/x), is particularly important. In terms
of it, the logical properties of relations — including the properties treated
in the last chapter but going beyond them — can be established. This is
the type of procedure used by Peirce and Schroder.
The second method — that of Principia Mathematica — begins with the
calculus of propositions, or calculus of material implication, in a form which
is simpler and otherwise superior to the Two-Valued Algebra, then pro
ceeds from this to the calculus of propositional functions and formal impli
cation, and upon this last bases not only the treatment of relations but also
the "calculus of classes".
It is especially important for the comprehension of the whole subject
of symbolic logic that the agreement in results and the difference of method,
of these two procedures, should be understood. Too often they appear to
the student simply unrelated.
I. THE TWO-VALUED ALGEBRA1
If the elements a, b, . . . p, q, etc., represent propositions, and a x b or
a b represent the joint assertion of a and b, then the assumptions of the
1 See Schroder, Algebra der Logik: n, especially Fiinfzehnte Vorlesung. An excellent
summary is contained in Schroder's Abriss (ed. M tiller), Teil n.
222
Systems Based on Material Implication 223
Boole-Schroder Algebra will all be found to hold for propositions, as was
explained in the last chapter.2 As was there made clear, p = 0 will repre
sent "p is false", and p = 1, "p is true". Since 0 and 1 are unique, it
follows that any two propositions, p and q, such that p = 0 and q = 0,
or such that p = 1 and q = 1, are also such that p = q. p = q, in the
algebra, represents a relation of extension or "truth value", not an equiva
lence of content or meaning.
-p symbolizes the contradictory or denial of p.
The meaning of p + q is readily determined from its definition,
P + q = -(-P -q)
p + q is the denial of "p is false and q is false", or it is the proposition
"At least one of the two, p and q, is true", p + q may be read loosely,
"Either p is true or q is true". The possibility that both p and q should
be true is not excluded.
p c q is equivalent to p q = p and to p -q = 0. ;; c q is the relation of
material implication. We shall consider its properties with care later in
the section. For the present, we may note simply that p c q means exactly
"It is false that p is true and q false". It may be read "If p is true, q is
true", or "p (materially) implies q".
With the interpretations here given, all the postulates of the Boole-
Schroder Algebra are true for propositions. Hence all the theorems will
also be true for propositions. But there is an additional law which holds
for propositions:
p = (P = i)
"The proposition, p, is equivalent to ' p is true'". It follows immediately
from this that
-P = (-P = 1) = (P = 0)
"-p is equivalent to 'p is false'". It also follows that -p = -(p = 1),
and hence
-(P = 1) = (P = 0), and -(p = 0) = (p = 1)
"p = 1 is false ' is equivalent to p = 0", and "'p = 0 is false ' is equivalent
to p — 1". Thus the calculus of propositions is a two-valued algebra:
every proposition is either = 0 or == 1, either true or false. We may, then,
proceed as follows: All the propositions of the Boole-Schroder Algebra
2 However, many of the theorems, especially those concerning functions, eliminations,
and solutions, are of little or no importance in the calculus of propositions.
224 A Survey of Symbolic Logic
which were given in Chapter II may be regarded as already established in
the Two- Valued Algebra. We may, then, simply add another division of
propositions— the additional postulate of the Two-Valued Algebra and the
additional theorems which result from it. Since the last division of the
orems in Chapter II was numbered 8-, we shall number the theorems of
this section 9 • .
The additional postulate is:
9-01 For every proposition p, p — (p = 1).
And for convenience we add the convention of notation :
9-02 -(p = q) is equivalent to p =|= q>
As a consequence of 9-01, we shall have such expressions as -(p = 1) and
-(p = 0). 9-02 enables us to use the more familiar notation, p =|= 1 and
p* 0.
It follows immediately from 9-01 that the Two-Valued Algebra cannot
be viewed as a wholly abstract mathematical system. For whatever p
and 1 may be, p = 1 is a proposition. Hence the postulate asserts that
any element, p, in the system, is a proposition. But even a necessary
interpretation may be abstracted from in one important sense — no step in
proof need be allowed to depend upon this interpretation. This is the
procedure we shall follow, though it is not the usual one. It will appear
shortly that the validity of the interpretations can be demonstrated within
the system itself.
In presenting the consequences of 9-01 and 9-02, we shall indicate
previous propositions by which any step in proof is taken, by giving the
number of the proposition in square brackets. Theorems of Chapter II
may, of course, be used exactly as if they were repeated in this chapter.
9-1 -p = (p = 0).
[9-01] -p = (-p = 1). And [3-2] -p = 1 is equivalent to p = 0.
9-12 -p = (p + 1).
[9-01] p = (p = 1). Hence [3-2] -p = -(p = 1) = (p + 1).
9-13 (p* 1) = (p = 0).
[9-M2]
9-14 (p + 0) = (P = 1).
[9-13, 3-2]
9-13 and 9-14 together express the fact that the algebra is two-valued.
Every proposition is either true or false.
Systems Based on Material Implication 225
Up to this point — that is, throughout Chapter II — we have written the
logical relations "If . . . , then . . .", "Either . . . or . . .", "Both
. . . and . . .", etc., not in the symbols of the system but just as they
would be written in arithmetic or geometry or any other mathematical
system. We have had no right to do otherwise. That "... c ..."
is by interpretation "If . . . , then . . . ", and ". . . + . . ." is by inter
pretation "Either . . . or . . .", does not warrant us in identifying the
theorem "If a c 6, then -be -a" with " (a c b) c (-/; c -a) ". We have
had no more reason to identify "If . . . , then . . ."in theorems with
"... c ..." than a geometrician would have to identify the period at
the end of a theorem with a geometrical point. The framework of logical
relations in terms of which theorems are stated must be distinguished from
the content of the system, even when that content is logic.
But we can now prove that wre have a right to interchange the joint
assertion of p and q wath p xq, "If p, then <y", with pcq, etc. We can
demonstrate that if p and q are members of the class A', then p c q is a
member of K, and that "If p, then </", is equivalent to p cq. And we can
demonstrate that this is true not merely as a matter of interpretation but
by the necessary laws of the system itself. We can thus prove that writing
the logical relations involved in the theorems — "Either . . . or . . .,"
"Both ... and .. .," "If . . . , then . . ."—in terms of +, x, c,
etc., is a valid procedure.
The theorems in which these things are proved are never needed here
after, except in the sense of validating this interchange of symbols and their
interpretation. Consequently we need not give them any section number.
(1) If p is an element in A, p = 1 and /; = 0 are elements in A'.
[9-01] If p is an element in A, p = 1 is an element in A. [1-0]
If p is an element in A, -p is an element in A, and hence [9 • 1] /; = 0
is an element in K.
(2) The two, p and q, are together equivalent to p x q, or p q.
[9-01] pq = (pq = 1). [5-73] pq = 1 is equivalent to the
pair, p = 1 and q = 1, and hence [9-01] to the pair, p and q.
(3) If p and q are elements in A, then p c q is an element in A.
[4-9] p cq is equivalent to p -q = 0, and hence [9-1] to -(p-q).
But if p and q are elements in A, [1 • 6, 1 • 1] -(p -q) is an element in A.
(4) -p is equivalent to " p is false".
[9-12] -p = (p 4= 1), and [8-01] p 4= 1 is equivalent to "p = 1
is false", and hence [9-01] to "p is false".
16
226 A Survey of Symbolic Logic
(5) p c q is equivalent to "If p, then g".
[5-64] p eg gives "If p = 1, then q = 1", and hence [9-01]
"If p, then g".
And "If p, then g" gives peg, for [9-01] it gives "If p = 1, then
g = 1", and
(a) Suppose as a fact p = 1. Then, by hypothesis, g = 1, and
[2-2] peg.
(6) Suppose that p ={= 1. Then [9-14] p = 0, and [5-63] peg.
(6) If p and g are elements in K, then p = g is an element in K.
[7-1] p = g is equivalent to p-g + -pg = 0, and hence [9-1]
to-(p-g + -pg). Hence [1-6, 1-1, 3-35] Q.E.D.
(7) p = g is equivalent to "p is equivalent to g".
[2-2] p = g is equivalent to "peg and g cyj".
By (5) above, "p eg and g cp" is equivalent to "If p, then g, and
if g, then p". And this is equivalent to "p is equivalent to g".
(8) If p and g are elements in K, then p 4= g is an element in K.
[9-02] (p + g) =-(P = g).
Hence, by (6) above and 1-6, Q.E.D.
(9) p =|= 9 is equivalent to "p is not equivalent to g".
By (4) and (2) above, Q.E.D.
(10) p + g is equivalent to "At least one of the two, p and g, is true.
[1-8] p+q = -(-p-g).
By (4) and (2) above, -(-p-g) is equivalent to "It is false that
(p is false and g is false) ". And this is equivalent to "At least one
of the two, p and g, is true".
In consideration of the above theorems, we can henceforth write ". . .
" for "If . . . , then ...","... = ..." for ". . . is equivalent
to ...","... + ..." for "Either, . . or . . .", etc., for we have
proved that not only all expressions formed from elements in K and the
relations x and + are elements in K, but also that expressions which in
volve c, and =, and =j= are elements in the system of the Two-Valued
Algebra. The equivalence of "If . . . , then ..." with ". . . c . . .",
of "Both . . . and ..." with ". . . x . . .", etc., is no longer a matter
of interpretation but a consequence of 9-01, p = (p = 1). Also, we can
go back over the theorems of Chapter II and, considering them as propositions
of the Two-Valued Algebra, we can replace "If . . . , then . . .", etc.,
Systems Based on Material Implication 227
by the symbolic equivalents. Each theorem not wholly in symbols gives a
corresponding theorem which is wholly in symbols. But when we consider
the Boole-Schroder Algebra, without the additional postulate, 9-01, this
procedure is not valid. It is valid only where 9 -01 is one of the postulates —
i. e., only in the system of the Two-Valued Algebra.
Henceforth we shall write all our theorems with pcq for " If p, then 7",
p = q for "p is equivalent to r/", etc. But in the proofs we shall frequently
use "If . . . , then ..." instead of ". . . c . . . ", etc., because the
symbolism sometimes renders the proof obscure and makes hard reading.
(That this is the case is due to the fact that the Two-Valued Algebra does
not have what we shall hereafter explain as the true "logistic" form.)
9-15 0 + 1.
0 = 0. Hence [9-13]0 4= 1.
.9-16 (p4= q) = (-p = q) = (p = -?).
(1) If p = q and p = 1, then q 4= 1 and [9-13] q = 0.
And if p = 1, [3-2] -p = 0. Hence -p = q.
(2) If p 4= q and p =h 1, then [9-13] p = 0, and [3-2] -p = 1.
Hence if p =j= q, then q 4= 0, and [9- 14] q = 1 = -p.
(3) If -p = q and q = 1, then -p = 1, and [3-2] p = 0.
Hence [9-15] p 4= q.
(4) If -p = q and q 4= 1, then -p 4= 1, and [9-13] -p = 0.
Hence [3-2] p = 1, and p 4= q-
By (1) and (2), if p 4= q, then -p = q. And by (3) and (4), if
-p = q, then p =)= q- Hence p 4= q and -£> = q are equivalent.
And [3 -2] (-p = q) = (p = -q).
This theorem illustrates the meaning of the relation, =, in the calculus
of material implication. If p 4= (/. then either p = 1 and q = 0 or 7; = 0
and q = 1. But if p = 1, then -p = 0, and if p = 0, then -p = 1. Hence
the theorem. Let p represent "Caesar died", and q represent "There is
no place like home". If "Caesar died" is not equivalent to "There is
no place like home", then "Caesar did not die" is equivalent to "There
is no place like home". The equivalence is one of truth values — { = 0} or
{ = 1} — not of content or logical significance.
9-17 p = (p = 1) = (p 4= 0) = (-p = 0) = (-p 4= 1).
[9-OM3-14-16]
9-18 -p = (p = 0) = (p 4= 1) = (-P = 1) = (-P * 0).
[9-M3-14-16]
228 A Survey of Symbolic Logic
9-2 (p = l)(p = 0) = 0.
[2-4] p-p = 0. And [9-01] p = (p = 1); [9-1] -p = (p = 0).
No proposition is both true and false.
9-21 (p * l)(p * 0) = 0.
[2-4] -pp = 0. And [9-18J -p = (p * 1); [9-17] p = (p * 0).
9-22 (7; = l) + (p = 0) == 1.
[4-8] p + -p = 1. Hence [9-01-1] Q. E. D.
Every proposition is either true or false.
9-23 (p* l) + (p* 0) == 1.
[4-8, 9-01-1]
Theorems of the same sort as the above, the proofs of which are obvious,
are the following:
9-24 (p q) = (p 0 = 1) = (p g * 0) = (p = l)(g = 1) = (p 4s 0) (g =t= 0)
= (p * 0)(g == 1) - (p = l)(g 4= 0) = -(-p + -g)
= (-P + -g = 0) = [(p = 0) + (g = 0) = 0]
= [(P * 1) + (g * 1) = 0], etc., etc.
9-25 (p + g) = (p + g = 1) = (p + g 4= 0) = (p = 1) + (g = 1)
= (P * 0) + (g =(= 0) = -(-p -g) = [(p = 0)(g = 0) = 0]
= [(P * l)(g * 1) 4= 1], etc., etc.
These theorems illustrate the variety of ways in which the same logical
relation can be expressed in the Two-Valued Algebra. This is one of the
defects of the system — its redundancy of forms. In this respect, the
alternative method, to be discussed later, gives a much neater calculus of
propositions.
We turn now to the properties of the relation c . We shall include here
some theorems which do not require the additional postulate, 9-01, for the
sake of bringing together the propositions which illustrate the meaning of
"material implication".
9-3 (per/) = (-p + q) = (p -q = 0).
[4-9] (peg) = (p-g = 0) == (-p+g = 1).
[9-01] (-p + g = 1) = (-p + g).
"p materially implies g" is equivalent to "Either p is false or g is true",
and to "It is false that p is true and g false".
Since p c g has been proved to be an element in the system, " It is false
that p materially implies g" may be symbolized by -(peg).
Systems Based on Material Implication 229
9-31 -(p cry) = (-p + q = 0) = (p-q).
[3 • 4] -(-p + q) = p -q. And [9 • 3] -(p c q) = -(-p + q) .
[9.02] -(-? + </) = (-/;+</ - 0).
"p does not materially imply q" is equivalent to "It is false that either p
is false or q is true", and to "p is true and q false".
9-32 (p = 0)c(pc<7).
[5-03] Ocr/. Hence Q.E.D.
If p is false, then for any proposition ry, p materially implies q. This is
the famous — or notorious — theorem: "A false proposition implies any
proposition".
9-33 (q=- })c(pcq).
[5-01] pel. Hence Q.E.I).
This is the companion theorem: "A true proposition is implied by any
proposition".
9-34 -(pcq)c(p =• 1).
The theorem follows from 9-32 by the reductio ad absurdum,
since if -(p c<y), then [9-32] p 4= 0, and [9-14] p = 1.
If there is any proposition, q, which p does not materially imply, then p is
true. This is simply the inverse of 9 • 32. A similar consequence of 9 • 33 is:
9-35 -(pcq)c(q = 0).
If -(pcq), then [9-33] q =(= 1, and [9-13] q = 0.
If p does not materially imply q, then q is false.
9-36 -(/; c q) c (7; c -q) ; -(p c q) c (-p c q) ; -(/; c q) c (-p c -q).
[9-34-35] If -0; cry), then p = 1 and q = 0.
[3-2] If p = 1, -y; = 0, and if q = 0, then -q =-- 1.
[9-32] If -p = 0, then -p cq and -;; c-ry.
[9-331 If -q = 1, then y;c-ry.
If p does not materially imply r/, then y> materially implies the negative,
or denial, of ry, and the negative of p implies r/, and the negative of p implies
the negative of ry. If "Today is Monday" does not materially imply
"The moon is made of green cheese", then "Today is Monday" implies
"The moon is not made of green cheese", and "Today is not Monday"
implies "The moon is made of green cheese", and "Today is not Monday"
implies "The moon -is not made of green cheese".
Some of the peculiar properties of material implication are due to the
230 A Survey of Symbolic Logic
fact that the relations of the algebra were originally devised to represent
the system of logical classes. But 9-36 exhibits properties of material
implication which have no analogy amongst the relations of classes. 9-36
is a consequence of the additional postulate, p = (p = 1). For classes, c
represents "is contained in": but if a is not contained in b, it does not
follow that a is contained in not-6 — a may be partly in and partly outside
of b.
9-37 -(pcq) c(qcp).
[9 • 36] If "(p c q), then -p c -q, and hence [3 • 1] q c p.
Of any two propositions, p and q, if p does not materially imply q, then q
materially implies p.
9-38 (pq) c[(pcq)(qcp)].
[9-24] pq = (p = l)(q == 1). Hence [9-33] Q.E.D.
If p and q are both true, then each materially implies the other.
9-39 (-p-q)c[(pcq)(qcp)].
[9-24] -p-q = (-p = l)(-q = 1) = (p = 0)(g = 0).
Hence [9-32] Q.E.D.
If p and q are both false, then each materially implies the other.
For any pair of propositions, p and q, there are four possibilities:
1) p = 1, q = 1: p true, q true.
2) p = 0, q = 0: p false, q false.
3) p = 0, q = 1 : p false, q true.
4) p = 1, q = 0: p true, q false.
Now in the algebra, 0 cO, 1 c 1, and 0 c 1; but 1 cO is false. Hence in
the four cases, above, the material implications and equivalences are as
follows :
1) p c q, qcp, p = q.
2) p c q, qcp, p = q.
3) pcq, -(qcp), p 4= q.
4) ~(pcq), qcp, p ^ q.
This summarizes theorems 9-31-9-39. These relations hold regardless of
the content or meaning of p and q. Thus p cq and p = q are not the
"implication" and "equivalence" of ordinary logic, because, strictly speak
ing, p and q in the algebra are not "propositions" but simply the "truth
values" of the propositions represented. In other words, material impli-
Systems Based on Material Implication 231
cation and material equivalence are relations of the extension of proposi
tions, whereas the "implication" and "equivalence" of ordinary logic are
relations of intension or meaning. But, as has been mentioned, the material
implication, pcq, has one most important property in common with "q
can be inferred from p" in ordinary logic; if p is true and q false, pcq
does not hold. And the relation of material equivalence, p = q, never
connects a true proposition with a false one.
These theorems should make as clear as it can be made the exact
meaning and character of material implication. This is important, since
many theorems whose significance would otherwise be very puzzling follow
from the unusual character of this relation.
Two more propositions, of some importance, may be given:
9-4 (p q c r) = (q p c r) = [p c (q c r)] = [q c (p c r)}.
[1-3] pq = qp. Hence [3-2] -(p q) == -(q p), and [-(/; q) + r]
But [9 • 3] [-(p q) + r] = (p q c r), and [-(q p) + r] = (q per).
And [3 • 41] [-(p q) + r] = [(-/> + -?) + r] = [-p + (-ry + r)] = (p c (q c r)}
Similarly, [-(q p) + r] = Iqc(pcr)].
This theorem contains Peano's Principle of Exportation,
[(p q)cr]c [p c(qc r)]
"If pq implies r, then p implies that q implies r"; and his Principle of
Importation,
[y; c (q c r)] c [(p q) c r]
" If p implies that q implies r, then if p and q are both true, r is true. "
9-5 [(pq)c r] = [(p -r) c -q] = [(q -r) c -y,].
[9 -3] [(p q) c r] = [-(p q) + r] = [(-p + -q) + r] = [(-p + r) + -ry]
= [(-</ + 0 + -y;] - [-Q; -r) + -q] = [-(q -r) + -p].
[9-3] [-(p-r) + -ry] -[(p-r)c-g], and
[-(q-r)+-p] = l(q-r) c-p].
If p and q together imply r, then if p is true but r is false, q must be false,
and if q is true but r is false, p must be false. This is a principle first stated
by Aristotle, but especially important in Mrs. Ladd-Franklin's theory of
the syllogism.
We have now given a sufficient number of theorems to characterize the
Two-Valued Algebra — to illustrate the consequences of the additional
232 A Survey of Symbolic Logic
postulate p = (p = 1), and the properties of p c q. Any further theorems
of the system will be found to follow readily from the foregoing.
A convention of notation which we shall make use of hereafter is the
following: A sign =, unless enclosed in parentheses, takes precedence over
any other sign ; a sign c , unless enclosed in parentheses, takes precedence
over any + or x ; and the sign + , unless enclosed in parentheses, takes
precedence over a relation x . This saves many parentheses and brackets.
II. THE CALCULUS OF PROPOSITIONAL FUNCTIONS. FUNCTIONS OF ONE
VARIABLE
The calculus of prepositional functions is an extension of the Two-
Valued Algebra to propositions which involve the values of variables. Fol
lowing 3Ir. jlussell,3 we may distinguish propositions from prepositional
functions as follows: A proposition is any expression which is either true
or false; a propositional function is an expression, containing one or more
variables, which becomes a proposition when each of the variables is re
placed by some one of its values.
There is one meaning of " Today is Monday" for which 'today' denotes
ambiguously Jan. 1, or Jan. 2, or . . . , etc. For example, when we say
'Today is Monday' implies 'Tomorrow is Tuesday'", we mean that if
Jan. 1 is Monday, then Jan. 2 is Tuesday; if Jan. 2 is Monday, then Jan.
3 is Tuesday; if July 4 is Monday, then July 5 is Tuesday, etc. 'Today'
and 'tomorrow' are here variables, whose values are Jan. 1, Jan. 2, Jan. 3,
etc., that is, all the different actual days. When 'today' is used in this
variable sense, "Today is Monday" is sometimes true and sometimes false,
or more accurately, it is true for some values of the variable 'today', and
false for other values. "Today is Monday" is here a propositional function.
There is a quite different meaning of "Today is Monday" for which
'today' is not a variable but denotes just one thing — Jan. 22, 1916. In
this sense, if "Today is Monday" is true it is always true. It is either
simply true or simply false: its meaning and its truth or falsity cannot
change. For this meaning of 'today ', " Today is Monday " is a proposition.
'Today, ' meaning Jan. 16, 1916, is one value of the variable 'today '. When
this value is substituted for the variable, then the propositional function is
turned into a proposition.
3 See Principles of Mathematics, Chap, vii, and Principia Mathematica, i, p. 15. Mr.
Russell carries out this distinction in ways which we do not follow. But so far as is here
in question, his view is the one we adopt. Principia Mathematica is cited hereafter as
Principia.
Systems Based on Material Implication 233
We may use ^r, t(x, y), f (*, yt z)t etc., to represent prepositional
functions, in which the variable terms are x, or x and y, or x, y, and z, etc.
These propositional functions must be carefully distinguished from the
functions discussed in Chapter II. We there used /, F, and the Greek
capitals, $, *, etc., to indicate functions; here we use only Greek small
letters. Also, for any function of one variable, we here omit any parenthesis
around the variable — <px, \l/y, £x.
f(x), V(x, y), etc., in Chapter II are confined to representing such
expressions as can be formed from elements in the class A' and the relations
x and + . If x and y in *(*, y) are logical classes, then *(.r, y) is some
logical class, such as x + y or a x + b -y. Or if x in /(.r) is a proposition,
then /(.r) is some proposition such as a x or -.r + b. The propositional
functions, <px, $(xt y}, £(x, y, z), etc., are subject to no such restriction.
<px becomes a proposition when a- is replaced by one of its values, but it
does not necessarily become any such proposition as a x or -.r + b. ' x is
Monday,' 'x is a citizen of y,' 'y is between x and z'— these are typical
propositional functions. They are neither true nor false, but they become
either true or false as soon as terms denoting individual things are sub
stituted for the variables x, y, etc. All the functions in this chapter are
such propositional functions, or expressions derived from them.
A fundamental conception of the theory of propositional functions is
that of the "range of significance ". The range of significance of a function
is determined by the extent of the class, or classes, of terms which are
values of its variables. All the terms which can be substituted for x, in
<px, and 'make sense', constitute the range of ^.r. If <px be 'x is mortal',
the range of this function is the aggregate of all the individual terms for
which 'x is mortal' is either true or false. Thus the "range of significance"
is to propositional functions what the "universe of discourse" is to class
terms. Two propositional functions, <px and r/^, may be such that the
class of values of x in <px, or the range of <px, is identical with the class of
values of y in \J/y, or the range of $y. Or the two functions may have
different ranges of significance, '.r is a man ' and '.r is a poet ' will have the
same range, though the values of .r for which they are true will differ. Any .r
for which 'x is a man' is either true or false, is also such that '.r is a poet'
is either true or false. But some .r's for which '.r is a poet' is either true
or false are such that 'x precedes .r + 1' is nonsense, '.r is a poet' and
'.r precedes x + 1' have different ranges.4 It is important to note that the
4 According to Mr. Russell's "theory of types" (see Principia, i, pp. 41-42), the one
fundamental restriction of the range of a propositional function is the principle that nothing
234 A Survey of Symbolic Logic
range of <px is determined, not by x, but by <p. <px and vy are the same
function.
If we have a prepositional function of two variables, say 'x is a citizen
of y ', we must make two substitutions in order to turn it into a proposition
which is either true or false. And we conceive of two aggregates or classes—
the class of values of the first variable, x, and the class of values of the second
variable, y. These two classes may, for a given function, be identical, or
they may be different. It depends upon the function. "John Jones is a
citizen of Turkey" is either true or false; "Turkey is a citizen of John
Jones" is nonsense. But "3 precedes 5" is either true or false, as is also
"5 precedes 3". The range of x and of y in $(x, y) depends upon ^, not
upon x and y.
A convenient method of representing the values of x in <px is by x\, x2,
xz, etc. This is not to presume that the number of such values of x in <px
is finite, or even denumerable. Any sort of tag which would distinguish
these values as individual would serve all the uses which we shall make
of .TI, ,T2, £3, etc., equally well. If x±, x2, xz, etc., are individuals,5 then
<px-i, <f>x*, <p%3, etc., will be propositions; and <pxn will be a proposition.
<pxz is a proposition about a specified individual; <pxn is a proposition about
'a certain individual' which is not specified.6 Similarly, if the values of x
in \I/(x, y) be Xi, x*, .r3, etc., and the values of y be yi, yz, ys, etc., then
$(%*> 2/s), t(xz, yn), t(xm, yn), etc., are propositions.
We shall now make a new use of the operators II and S, giving them
a meaning similar to, but not identical with, the meaning which they had
in Chapter II. To emphasize this difference in use, the operators are here
set in a different style of type. We shall let 2x<px represent <pXi + <pxz + <f>Xz
+ . . . to as many terms as there are distinct values of x in <px. And Ux<px
will represent <pxi x <pxz x <pxs x . . . to as many terms as there are distinct
values of x in <px. (WTe have heretofore abbreviated ax6 to a b or a -b.
But where prepositional functions are involved, the form of expressions is
that presupposes the function, or a function of the same range, can be a value of the func
tion. It seems to us that there are other restrictions, not derived from this, upon the
range of a function. But, fortunately, it is not necessary to decide this point here.
5 "Individuals" in the sense of being distinct values of x in <px — which is the only
conception of "individual" which we require.
6 It may be urged that <pxn is not a proposition but a prepositional function. The
question is most difficult, and we cannot enter upon it. But this much may be said:
Whenever, and in whatever sense, statements about an unspecified individual can be
asserted, <pxn is a proposition. If any object to this, we shall reply "A certain gentle
man is confused". Peirce has discussed this question most acutely. (See above, pp.
93-94.)
Systems Based on Material Implication 235
likely to be complex. Consequently we shall, in this chapter, always
write "products" with the sign x .)
The fact that there might be an infinite set of values of x in <px does
not affect the theoretical adequacy of our definitions. For nothing here
depends upon the order of <pxm, ^xn, <pxp, and it is only required that the
values of x which are distinct should be identifiable or "tagable". The ob
jection that the values of x might not be even denumerable is more serious,
but the difficulty may be met by a device to be mentioned shortly.
Since <pxi, <pxz, <f>x3, etc., are propositions, #TI + <px2 + <px3 + . . . is a
proposition— the proposition, "Either <pxi or <px2 or #r3 or ... etc.". Thus
2x<px represents "For some value of x (at least one), <px is true". And
2x<px is a proposition. Similarly, ^.TJ x <px2 x ^.r3 x ... is the joint assertion
of <ATi and <pxz and <px3, etc. Thus Ht<px represents the proposition "For
all values of x, <px is true ", We may translate Zx^.r loosely by " <px is
sometimes true", and nx<px loosely by " <px is always true". This trans
lation fails of literal accuracy inasmuch as the variations of x in <px may
not be confined to differences of time.
The conception of a prepositional function, <px, and of the class of values
of the variable in this function, thus give us the new types of proposition,
<px3, (pxn, 2x<px, and Ux(px. Since the laws of the Two-Valued Algebra
hold for propositions generally, all the theorems of that system will be
true when propositions such as the above are substituted for a, b, ... p, q,
etc. (We must, of course, remember that while a, b, ... p, q, etc., in the
Two-yalued Algebra represent propositions, x in yx, etc., is not a proposi
tion but a variable whose values are individual things. In the theorems
to follow, we shall sometimes need a symbol for propositions in which no
variables are specified. To avoid any possible confusion, we shall represent
such propositions by a capital letter, P.) We may, then, assume as already
proved any theorem which can be got by replacing cr, 6, ... p, q, etc., in
any proposition of the Two-Valued Algebra, by <px3, <pxn, 2x<px, or Tix<px.
Additional theorems, which can be proved for propositions involving values
of variables, will be given below. These are to be proved by reference to
earlier theorems, in Chapter II and in Section I of this chapter. As before,
the number of the theorem by which any step in proof is taken will be given
in square brackets. Since the previous theorems are numbered up to 9-,
the additional theorems of this section will be numbered beginning with 10- .
One additional assumption, beyond those of the Two- Valued Algebra,
will be needed. The propositions which have been proved in sufficiently
236 A Survey of Symbolic Logic
general form to be used where sums and products of more than three terms
are in question all require for their demonstration the principle of mathe
matical induction. If, then, AVC wish to use those theorems in the proofs
of this section, we are confronted by the difficulty that the number of
values of x in <f>x, and hence the number of terms in 2x#r and Hx<px may
not be finite. And any use of mathematical induction, or of theorems
dependent upon that principle for proof, will then be invalid in tttis con
nection. Short of abandoning the proposed procedure, two alternatives
are open to us: we can assume that the number of values of any variable
in a propositional function is always finite; or we can assume that any
law of the algebra which holds whatever finite number of elements be involved
holds for any number of elements whatever. The first of these assumptions
would obviously be false. But the second is true, and we shall make it.
This also resolves our difficulty concerning the possibility that the
number of values of x in <px might not be even denumerable, and hence
that the notation <pxi + <pxz + <px3 + . . . and <pxi x tpx2 x <px3 x might be
inadequate. We can make the convention that if the number of values of x
in any function, <px, be not finite, <fXi+ <px2+ <pxs+ . . . , or "2x<px, and
<?Xi x <px» x (f>x3 x . . . , or Hx<px, shall be so dealt with that any theorem to
be proved will be demonstrated to hold for any finite number of values
of x in <px\ and this being proved, our assumption allows us to extend the
theorem to any case in which the values of the variable in the function are
infinite in number. This principle will be satisfactorily covered by the
convention that <pxi + <px2 + <px3 + . . . and <pxi x <px2 x <px3 x . . . shall always
be supposed to have a finite but undetermined number of terms, and any
theorem thus proved shall be presumed independent of the number of
distinct values of any variable, x, which is involved.7
This postulate, and the convention which makes it operative, will be
supposed to extend also to functions of any number of variables, and to
sums, products, and negatives of functions.
Xo further postulates are required, but the following definitions are
needed :
10-01 2p.r = 2x<px = <pxi+ <pxz+ tftf'3 + .... Def.
10-02 H(px = Hx<px = <pxi x (f>x2 x <pxz x . . .. Def.
10-03 -<p.r = -{<f>x}. Def.
7 This procedure, though not invalid, is far from ideal, as are many other details of
this general method. We shall gather the main criticisms together in the last section of
this chapter. But it is a fact that in spite of the many defects of the method, the results
which it gives are without exception valid.
Systems Based on Material Implication 237
10-031 -9xn = -{<pxn}. Def.
10-04 -Ux<px = -{nx<px}. Def.
10-05 -2x<px = -{S,0*r} Def.
The last four merely serve to abbreviate the notation.
Elementary theorems concerning propositions which involve values of
one variable are as follows:
10-1 2<px = -n-^.r.
[5-951] <f>xi+ <f>x»+ <px3+ ... = -{-^.n x-^.T2 x-^i-3 x . . . J.
10-12 U<px = -2-<px.
[5-95] ^ix^r2
10-1 states that "For some values of x, <px is true" is equivalent to the
denial of "For all values of x, <px is false". 10-12 states that "For all
values of x, <px is true" is equivalent to the denial of "For some values
of x, <px is false". These two represent the extension of De Morgan's
Theorem to propositions which involve values of variables. They might
be otherwise stated: "It is true that all x is • • " is equivalent to "It is
false that some x is not • • • "; and "It is true that some x is • • • " is equiva
lent to "It is false that all x is not
10-2 Il(px c <pxn.
[5-99] <pxi x <px-~ x <**3 x . . . c <px{
and <pxi x (px% x <pxs x . . . c <px%
and <px! x <^r2 x <px3 x . . . c ^.r3, etc., etc.
10-21 <pxnc2<f>x.
[5-991] ^c^+^roH.^*...
and <pXz c (pXi + (px^ + <px$ +
and <pXs c <pxi + pxz + <pxs + . . . , etc., etc.
By 10-2, if <px is true for all values of .r, then it is true for any given value
of .r, or "What is true of all is true of any given one". By 10-21, If <px is
true for one given value of .r, then it is true for some value of .r, or "What
is true of a certain one is true of some". It might be thought that the
implication stated by 10-21 is reversible. But we do not have 2^.r c «^.rn,
because <pxn may be <p.r2, and 2 <px c <px2 would not hold generally. For
example, let <^.r = "Today (x) is Monday". Then 2^.r will mean "Some
day is Monday", but <pxn will mean "Today (Jan. 1) is Monday", or will
mean "Today (Feb. 23) is Monday", etc. "Some day is Monday" does
238 .1 Survey of Symbolic Logic
not imply " Jan. 1 is Monday", and does not imply "Feb. 23 is Monday "-
does not imply that any one given day is Monday. xn in <pxn means " a
certain value of .r" in a sense which is not simply equivalent to "some
value of .r". Xo translation of <pxn will give its exact significance in this
respect.
10-22 II<£>.r cS^.r.
[5-1, 10-2-21]
Whatever is true of all is true of some.
10-23 H<px is equivalent to "Whatever value of x, in <px, xn may be, <pxn".
Ii<px = <pxi x <pxz x <px3 x . . . = (<pxi x <px2 x <px3 x . . . =1) [9-01]
And [5-971] <pxi x <px2 x <^r3 x . . . ==1 is equivalent to the set
<pXi = 1, <pX2 = I, pXz = 1, ....
And [9-01] <pxn = I is equivalent to <pxn.
Hence tt<px is equivalent to the set <pxlf <pxz, <p.T3, ....
This proposition is not tautological. It states the equivalence of the
product <pxi x tpxz x <f>x3 x . . with the system of separate propositions
<pxi, <f>xz, <f>xs, etc. It is by virtue of the possibility of this proposition
that the translation of tt<px as "For all values of x, ^x is true" is legitimate.
In this proof we make use of the principle, p = (p = 1) — the only case in
which it is directly required in the calculus of propositional functions.
By virtue of 10 • 23 we can pass directly from any theorem of the Two-
Valued Algebra to a corresponding theorem of the calculus of propositional
functions. If we have, for example, pc.p+q, we have also "Whatever
value of x, in <px, xn may be, <pxn c p.rn + P". And hence we have, by
10-23, Ux[<px c <px + P]. We shall later see the importance of this: it
gives us, for every theorem concerning "material implication", a cor
responding theorem concerning "formal implication".
Next, we give various forms of the principle by which any proposition
may be imported into, or exported out of, the scope of a II or S operator.
10-3 Z#c + P = S^r + P).
2 <px + P = ( <£.ri + <px<2 + <px3 +...) + P
P) + (^2 + P) + (^3 + P)+ ... [5-981]
10-31 P+Z^o- - ?X(P+ p.r).
Similar proof.
10-3 may be read; "'Either for some x, <px is true, or P is true' is equiva-
Systems Based on Material Implication 239
lent to 'For some x, either <?x is true or P is true'". And 10-31 may be
read: '"Either P is true or, for some x, <px is true' is equivalent to 'For
some x, either P is true or <px is true'".
10-32 U<f>x + P = nx(<^.r + P).
II (pX + P = (<pXiX <pXz X <pX3 X . . . ) + P
P) x... [5-941]
10-33 P+tt<px = UX(P + tpx).
Similar proof.
<( Either P is true or, for every x, <px is true" is equivalent to "For every x,
either P is true or <px is true."
10-34 SX(^ + P) = SX(P +*>*).
[4-3] 2<px + P = P + 2<px. Hence [10-3-31] Q.E.D.
10-35 ILOi' + P) = nx(P+ <px).
[10-32-33]
Exactly similar theorems hold where the relation of the two propositions
is x instead of + . The proofs are so simple that only the first need be
given.
10-36 2<px xP = S*(#E xP).
2 px x P = ( ^.TI + <f>xz + <px3 + . . . ) x P
= (<^.ri xP) + (^-2 xP) + (cp.i-3 xP) + . . . [5-94]
" <px is true for some x, and P is true", is equivalent to "For some .r, <px
and P are both true".
10-361 P xS^.i- - 2X(P x <r>.r).
10-37 n ^ x P = nj:( <px x P).
10-371 Pxn^r = nx(Px^r).
10-38 Sz(^ xP) = S,(P x ^.r).
10-381 Hx(<px xP) = nx(P x <^.r).
We should perhaps expect that a proposition, P, might be imported
into and exported out of the scope of an operator when the relation of P
to the other member of the expression is c . But here the matter is not
quite so simple.
240 A Survey of Symbolic Logic
10-4 Pc?<px = S,(Pc <px).
[9-3] PC 2(f>x = -P+2<px = -P
[5-981]
= (P c ^rO + (P c ^.r2) + (P c ^.rs) + - - . [9 • 3]
= SX(P c <px)
The relation c, in the above, is, of course, a material implication.
But it is tedious to read continually "p materially implies g". We shall,
then, translate pcq simply by "p implies 9", or by "If p, then g".
10-4 reads: "P implies that for some .r, <^.r is true" is equivalent to
"For some .r, P implies that ^>.r is true". This seems clear and obvious,
but consider the next :
10-41 Stf>.rcP = IlzO.rcP).
[9-3] S^ccP = -S#r + P = U-<px + P [10-12]
= (-^i x-^.r2 X-^.TS x . . .) + P
= (-^d + P) x (-<pxz + P) x (-^3 + P) . . . [5-941]
= (^.TiCP) X(^.T2CP) X(^.T3CP)... [9-3]
= Ux(<pxcP)
'" <px is true for some x' implies P" is equivalent to "For every x, <p'x
implies P". It is easy to see that the second of these two expressions gives
the first also: If <px always implies P, then if <px is sometimes true, P must
be true. It is not so easy to see that 2<px c P gives ttx(<px c P). But we
can put it thus: "If <px is ever true, then P is true" must mean " <px
always implies P".
10-42 PcU<px = Tlx(Pc<px).
[9-3] P c U<px = -P + Il^.r == -P + (<pxi x ^r2 x ^3 x . . .)
= (-P + <pxi) x (-P + <pxz) x (-P + #c8) x ...
[5-941]
- (P c <pxj x (P c <px2) x (P c ^cs) ... [9-3]
= Ut(P c <px)
"P implies that <px is true for every x" is equivalent to "For every xt P
implies <px".
10-43 n^TcP = 2x(<pxcP).
[9-3] n^rcp = -n^ + P - S-^.T + P [io-i]
= - <X + - .1* + - <X3 + . . . + P
Systems Based on Material Implication 241
= (-^1 + P) + (-^2 + P) + (-<pOr3 + P) + . . .
[5-981]
C P) + (<^2 C P) + (^3 C P) + . . . [9-3]
"' <px is true for every x' implies that P is true" is equivalent to "For
some x, (px implies P". At first sight this theorem seems to commit the
"fallacy of division" going one way, and the "fallacy of composition"
going the other. It suggests the ancient example about the separate hairs
and baldness. Suppose <px be "If x is a hair of Mr. Blank's, x has fallen
out". And let P be "Mr. Blank is bald". Then H<px cP will represent
"If all of Mr. Blank's hairs have fallen out, then Mr. Blank is bald".
And 2x(#rcP) will represent "There is some hair of Mr. Blank's such
that if this hair has fallen out, Mr. Blank is bald". In this example,
Ti(px cP is obviously true, but 2x(<px cP) is dubious, and their equivalence
seems likewise doubtful. The explanation of the equivalence is this: we
here deal with material implication, and <pxn c P means simply " It is
false that (<pxn is true but P is false) ". U<px cP means, in this example,
" It is false that all Mr. Blank's hairs have fallen out but Mr. Blank is not
bald"; and 2I(#rcP) means "There is some one of Mr. Blank's hairs
such that 'This hair has fallen out but Mr. Blank is not bald' is false".
No necessary connection is predicated between the falling out of any single
hair and baldness — material implication is not that type of relation.
If we compare the last four theorems, we observe that an operator in
the consequent of an implication is not changed by being extended in scope
to include the whole relation, but an operator in the antecedent is changed
from n to 2, from 2 to II. This is due to the fact that pcqis equivalent
to -<p + qf where the sign of the antecedent changes but the consequent
remains the same; and to the law -n() = 2-(), -2() = n-().
The above principles, connecting any proposition, P, with a preposi
tional function and its operator, are much used in later proofs. In fact,
all the proofs can be carried out simply by the various forms of this principle
and theorems 10-1-10-23. Since P, in the above, may be any propo
sition, #rn, 2#r, II#c, etc., can be substituted for P in these theorems.
(<px + \l/x) and (<px x^j-) are, of course, functions of x. In order that
( par + \I/x) be significant, <?x must be significant and $x must be significant,
and it is further requisite that "Either <px or $x" have meaning. Such
considerations determine the range of significance of complex functions
like (<px + \I/x) and (<px x^.r). A value of x in such a function must be at
17
242 A Survey of Symbolic Logic
once a value of x in <px and a value of x in fa: xn in <pxn and in fan, in
(<pxn + fan), denotes identically the same individual.
10-5 2 <px + 2 fa = Sx( ^ + #r) .
Since addition is associative and commutative,
" Either for some .r, ^, or for some x, fa" is equivalent to "For some x,
either <px or fa".
If it be supposed that the functions, <px and fa, may have different
ranges — i. e., that the use of the same letter for the variable is not indicative
of the range — then S^.r + 2 fa might have meaning when 2x(<p.r + fa) did
not. But in such a case the proposition which states their equivalence
will not have meaning. We shall make the convention that xn in <pxn
and xn in fan are identical, not only in (<pxn+fan) and (<f>xnxfan), but
wherever <px and fa are connected, as in S^.r + S#r. Where there is no
such presumption, it is always possible to use different letters for the
variable, as S#e + 2^y. But even without this convention, the above
theorem will always be true when it is significant — i. e., it is never false—
and a similar remark applies to the other theorems of this section.
10-51 n <px x Ufa = nz( <px x fa) .
Since x is associative and commutative, similar proof.
We might expect 2<pxx2fa = 2x(<pxxfa) to hold, but it does not.
"For some .r, x is ugly, and for some x, x is beautiful", is not equivalent to,
"For some x, x is ugly and x is beautiful". Instead of an equivalence, we
have an implication:
10-52
Sx( <px x fa) = ( (p.i-1 x fa 0
[5-21 (<pxn xfan) c <p.rn, and (<pxnxfan)cfan
Hence [5-31] 2x(^.r xfa) cS^.r, and 2,(<px xfa) c^fa
Hence [5-34] 2x(<px xfa) c2<px xZfa
Similarly, U<px + Hfa = Ux(<f>x + fa) fails to hold. "Either for every
x, x is ugly, or for every x, x is beautiful ", is not equivalent to, "For every x,
either x is ugly or .r is beautiful ". Some .T'S may be ugly and others beauti
ful. But we have:
Systems Based on Material Implication 243
10-53 II <px + Ufa cttx(<px + #r) .
[5-21] <pxnc(<pxn+\I/xn), and $xn c
Hence [5-3] H<px cHx(<px + \f/x), and Il^.r cUx((px + if/x)
Hence [5 • 33] II <px + n^x c nx( <px + $x)
In the proof of the last two theorems, we write a lemma for <pxn instead
of writing it for <pxi, for <pxz, for <pxs, etc. For example, in 10-52 we write
(<pxn x^.rn) c <pxn, instead of writing
(<pxi
<px3, etc., etc.
The proofs are somewhat more obvious with this explanation. This method
of writing such lemmas will be continued.
With two prepositional functions, <px and \f/x, we can form two impli
cation relations, Sx(#rc^) and IL^.r c i^.r). But 2x(#rc#r) states
only that there is a value of x for which either <px is false or \j/x is true:
and this relation conveys so little information that it is hardly worth while
to study its properties.
Ux(<px c^.c) is the relation of "jormal implication" "For every .r,
at least one of the two, ' <f>x is false' and ' $x is true', is a true statement"
The negative of Ux(<f>x c^x) is Sx(^.r x-^.r), so that Ux( <px c i//.r) may also
be read "It is false that there is any .r such that <px is true and $x false".
The material implication, peg, states only "At least one of the two, lp is
false' and 'q is true', is a true statement"; or, "It is false that p is true
and q false". The material implication, <pxnc\l/xn, states only "At least
one of the two, ' <p is false of xn} and '^ is true of xn', is a true statement";
or "It is false that <pxn is true and ^.rn is false". But the formal impli
cation, ns(^.rc^.r), states that however xn be chosen, it is false that <pxn
is true and $xn is false— in the whole range of <px and $x, there is not a
case in which <px is true and $x false. To put it another way, IIX(^' c ^.r)
means "Whatever has the predicate ^ has also the predicate ^"
This relation has more resemblance to the ordinary meaning of "im
plies" than material implication has. But formal implication, it should
be remembered, is simply a class or aggregate of material implications;
nx(>.r c i//.r) is simply the joint assertion of ^,r, c ^.ri, ^.r2 c ^.r«, ^.r3 c tx~,
etc., where each separate assertion is a material implication.8
s The whole question of material implication, formal implication, and the usual mean
ing of "implies", is discussed in Section v of Chap. v.
244 A Survey of Symbolic Logic
The properties of formal implication are especially important, because
upon this relation are based certain derivatives in the calculus of classes
and in the calculus of relations.
10-6 n^arc^a-) = Ux(-<px + fa) = Hx-(<px x-fa).
[9 • 3] (pXn Cfan = - <pXn + fan = -( <pXn X -fan)
Hence [10-23] Q.E.D.
10-61 ttx(<pxcfa) c(<pxncfan).
[10-2]
If (px formally implies fa, then <pxn materially implies fan-
10-611 [IIX( <px c $x) x <pxn] c fa^
[9-4, 10-61]
If <px formally implies fa and $ is true of arn, then ^ is true of xn. This is
one form of the syllogism in Barbara: for example, "If for every x, 'a; is a
man' implies 'x is a mortal', and Socrates is a man, then Socrates is a
mortal".
10 -62 nx( ??ar c #r) c Sx( <^x c fa) .
[10-22]
10 • 63 nx( <px c fa) c(Il<pxc Ufa).
[10-61] If Ux(<pxcfa), then <pxncfan
Hence [5-3] Q.E.D.
10-631 [Ux(<pxcfa) xn^cn^o*.
[9-4, 10-62]
If (px always implies fa and <px is always true, then fa is always true.
10 -64 nx( <^.r c fa) c(2<px c. 2 fa) .
[10-61, 5-31]
10-641 [nx( <f>x cfa)*2 <px] c 2 fa.
[9-4, 10-64]
If (px always implies fa and <px is sometimes true, then fa is sometimes true.
10 • 65 [nx( <px c fa) x nx(^.T c fa;)] c nx( ^.T c far).
[10-61] If nx(^.rc^.iO and Ux(fa c far), then <pxrcfan and
iA.rn c farn.
Hence [5-1] whatever value of .r, xn may be, <^.rn c farB.
Hence [10-23] nx(^arcfar)
Systems Based on Material Implication 245
This theorem states that formal implication is a transitive relation. It is
another form of the syllogism in Barbara. For example let <px = 'x is a
Greek', \f/x = 'x is a man', and £x =•• (x is a mortal'; 10-65 will then read:
"If for every x, lx is a Greek' implies 'x is a man', and for every x, 'x is a
man' implies '.r is a mortal', then for every x, '# is a Greek' implies 'x is a
mortal'".
10-65 may also be given the form:
10-651 Ux(<px c #c) c [nx(#r c £x) c Ux(<px c fr)].
[9-4, 10-65]
10 • 652 Ur(ifrx c fa;) c [Ux( <px c $x) c nx( <px c far)].
[9-4, 10-65]
10 • 66 nx( #r c i/a*) = nx(-#c c - #x) .
[3-1] (^ci/O - Htf»c -?.!•„)
Hence [2-2, 5-3] Q.E.D.
Any further theorems concerning formal implication can be derived
from the foregoing.
"Formal equivalence" is reciprocal formal implication, just as material
equivalence is reciprocal material implication. The properties of formal
equivalence follow immediately from those of formal implication.
10-67 Ux(<px = jx) = [ttx(vx c $x) xllx(tx c <px)].
Whatever value of x, xn may be, [2-2] <pxn = ^» is equivalent
to the pair, <pxn c \//xn and \f/xn c <pxn.
Hence [10-23] Q.E.D.
10-68 [Ux(<px = jx) xnx(^.r = f-r)] cnx(^.i' - T-r).
Whatever value of .r, .rn may be, if <px = ^x and £i: == far, then
<px = r«r. Hence [10-23] Q.E.D.
10-681 Ux(<px = ^.r) c[II,(^.r = far) c
[10-68, 9-4]
10-682 nz(^.r - f.c) c[Hx(^r = ^.r) c
[10-68, 9-4]
Formal equivalence, as indicated by the last three theorems, is a transitive
relation.
and
10-69 Ux(<p
nx(^c - ^')
[2-2, 10-61-62-63]
246 A Survey of Symbolic Logic
10-691 n«(^c = $x) = Ux(-<px = -#r).
[3-2, 10-23]
If we wish to investigate the propositions which can be formed from
functions of the type of (<px xfy) and (<px + ty), where the range of sig
nificance of <px may differ from that of \l/y, we find that these will involve
two operators— 2 Jl^r; ty), HtfSx(^r; ^), etc. And these are special
cases of a function of two variables. (<px x\f/y) and (<px + \l/y) are special
cases of f (.T, y). Hence we must first investigate functions of two variables
in general.
III. PROPOSITIONAL FUNCTIONS OF Two OR MORE VARIABLES
A prepositional function of two variables, <p(x, y), gives the derivative
propositions <p(xm, yn), U*<P(X, yn), S*2tfp(a:, y), 2ynxp(a:, y)-, etc. The
range of significance of <p(x, y) will comprise all the pairs (x, y) such that
<p(x, y) is either true or false. We here conceive of a class of individuals,
#1, xz, .T3, etc., and a class of individuals, ylf y,, ys, etc., such that for any
one of the x's and any one of the i/'s, <p(x, y) is either true or false.
As has already been pointed out, the function may be such that the
class of values of x is the same as the class of values of y, or the values of x
may be distinct from the values of y. If, for example, <p(x, y) be "x is
brother of #", the class of .r's for which <p(x, y) is significant consists of
identically the same members as the class of y's for which <p(x, y) is sig
nificant.9 In such a case, the range of significance of <p(x, y) is the class of
all the ordered couples which can be formed by combining any member of
the class with itself or with any other. Thus if the members of such a
class be «i, G2> as, etc., the class of couples in question will be10
(ai, ffi), (fli, a2), («i, a-s), • • .
(az, ai), («2, a,), («2, a3), ...
(a3, aO, («3, a2), (a3, a3), ...
. . . Etc., etc.
But if <p(x, y) represent "x is a citizen of ?/", or "a: is a proposition about
y", or "x is a member of the class y", the class of x's and the class of y's
for which <p(x, y) is significant will be mutually exclusive.
9 Presuming that "A is brother of A" is significant — i. e., false.
10 Schroder treats all relatives as derived from such a class of ordered couples. (See
Alg. Log., in, first three chapters.) But this is an unnecessary restriction of the logic of
relatives.
Systems Based on Material Implication 247
Although <p(x, y) represents some relation of x and y, it does not neces
sarily represent any relation of the algebra, such as x cy or x = y; and it
cannot represent relations which are not assertable.
<P(XI, y), <p(x*, y), etc., are prepositional functions of one variable, y.
Hence Hvv(xi, y), Uv<p(x±, y), ^v<p(xlt y), etc., are propositions, the meaning
and properties of which follow from preceding definitions and theorems.
And Uy<p(x,y), 2vv(x,y), Ilx<p(x,y), and 2t<t>(x, y) are propositional
functions of one variable. We can, then, define propositions involving
two variables and two operators, as follows:
11-01 UxUy<f>(x, y) == llt{ny<t>(x, y)}. Def
11-02 2xUy<p(x,y) == 2xlUy<p(x,y)}. Def.
1 1 • 03 n*Ztf <p(x, y) == nx { sy <p(x, y) } . Def.
11-04 SxStf?(.r, y) --= 2,{S^(.r, y)}. Def.
It will be seen from these definitions that our explanation of the range
of ^significance of functions of two variables was not strictly required; it
follows from the explanation for functions of one variable. The same con
vention regarding the number of values of variables and interpretation of
the propositions is also extended from the theory of functions of one
variable to the theory of functions of two.
(Where the first variable has a subscript, the comma between the two
will be omitted: <p(x_y) is p(.r2, y), etc.)
Since Uy<f>(x, y) is a propositional function of one variable, .r, the defini
tion. 10-02, gives us
IUIX*, y) == U^Il^x, y)} --= Tly<p(x,y) xlly<f>(xzy)
And the expansion of this last expression, again by 10-02, is
{ <p(xiyi) x ^Oi*/2) x <p(.ri?/s) x .
x <x2i x <>x?2) x ^(.r2z/3) x .
x { <p(x*yi) x <p(x3y«) x
x ... Etc., etc.
And similarly, by 10-01,
SJlXz, y) = Sx{n^(;r, y)} --= U
And the expansion of the last expression, by 10-02, is
x
x
x
+ . . . Etc., etc.
248 A Survey of Symbolic Logic
Or, in general, any prepositional function with two operators is expanded
into a two-dimensional array of propositions as follows:
(1) The operator nearest the function indicates the relation (+ or x)
between the constituents in each line.
(2) The subscript of the operator nearest the function indicates the
letter which varies within the lines.
(3) The operator to the left indicates the relation ( + or x ) between
each two lines.
(4) The subscript of the operator to the left indicates the letter which
varies from line to line.
Some caution must be exercised in interpreting such propositions as
2xUv<p(x, y}, etc. It is usually sufficient to read ZJIy "For some x. ami.
every y", but strictly it should be "For some x, every y is such that".
Thus 2xHy<p(x, y} should be "For some x, every y is such that <p(x, y) is
true". And Hv'2x<p(x, y) should be "For every y, some x is such that
v(x, y) is true". The two here chosen illustrate the necessity of caution,
which ma be made clear as follows:
2xlly<p(xt y) =
That is, 2xIIy<p(x, y) means "Either for x\ and every y, <p(x, y) is true,
or for #2 and every y, <p(x, y) is true, or for .T3 and every y, <p(x, y) is true,
... or for some other particular x and every y, <p(x, y) is true". On the
other hand,
Tly2x<p(x, y) = 2x<p(x, i/i) xSx^(.r, */2) xSx<^(.T, y
That is, Uy^x(p(x, y) means "For some x and yi, <p(x, y) is true, and for
some x and yz, <p{x, y) is true, and for some x and y3, <p(x, y) is true, and
. . ."; or "Given any y, there is one x (at least) such that <p(x, y) is true".
The following illustration of the difference of these two is given in Principia
Mathematical n Let <p(x, y) be the propositipnal function "If y is a proper
fraction, then x is a proper fraction greater than y". Then for all values
of y, we have 2x<p(x, ?/„), so that Iiy2x<p(x, y) is satisfied. In fact, U^x<p(x,y)
expresses the proposition: "If y is a proper fraction, then there is always
a proper fraction greater than ?/". But 2xliy<p(x, y) expresses the propo
sition: "There is a proper fraction which is greater than any proper frac
tion", which is false.
In this example, if we should read Sxny "For some x and every #";
11 See i, p. 161.
Systems Eased on Material Implication 249
IlySz "For every y and some x", we should make equivalent these two
very different propositions. But cases where this caution is required are
infrequent, as wre shall see.
Where both operators are II or both 2, the two-dimensional array of
propositions can be turned into a one-dimensional array, since every rela
tion throughout will be in the one case x , in the other + , and both of
these are associative and commutative. It follows from our discussion of
the range of significance of a function of two variables that any such func
tion, <p(x, y), may be treated as a function of the single variable, the ordered
couple, (x, y). Hence we can make the further conventions:
11-05 SsSy^Gc, y) = S(x, V)<p(x, y) == SXf y<f>(x, y).
11-06 ttxlly<p(x, y) = n(x, V)<p(x, y) == nx, y<?(x, y).
The second half of each of these serves merely to simplify notation.
1 1 • 07 If xr and ya be any values of x and y, respectively, in <p(x, y) , there
is a value of (x, y)— say, (.T, y)n— such that <f>(x, y)n == <p(xrys).
11-05 and 11-06 could be derived from 11-07, but the process is tedious,
and since our interest in such a derivation would be purely incidental,
we prefer to set down all three as assumptions.
If we wish to identify a given constituent of S*, y<p(x, y) with a con
stituent of 2x2y<p(x, y), some convention of the order of terms in Sz, y<p(x, y)
is required, because if the order of constituents in 2x2v<p(x, y) be unaltered,
this identification will be impossible unless the number of values of y is
determined— which, by our convention, need not be the case. Hence we
make, concerning the order of terms in SXf y<p(x, y), the following conven
tion: <p(xmyn) precedes <p(xrys) if m + n < r + s, and where in + n = r + *,
if n < s. Thus the order of terms in S*, y<p(x, y) will be
<p(xiyj + <p(x2yi) +
This arrangement determines an order independent of the number of values
of x, or of y, so that the equivalent of <p(x, y)n in terms of <p(xrye) can always
be specified.12 An exactly similar convention is supposed to govern the
arrangement of terms in n*, y<p(x, y) and their identification with the terms
of Uxlly<p(x, y). These conventions of order are never required in the
proof of theorems: we note them here only to obviate any theoretical
12 This arrangement turns the two-dimensional array into a one-dimensional by the
familiar device for denumerating the rationals— i. e., by proceeding along successive
agonals, beginning with the upper left-hand corner.
250 A Survey of Symbolic Logic
objection. The identification of 2x2y<p(x, y) with Sx, y<p(x, y), and of
HJIy^Gr, y) with nx, y<p(x, y), is of little consequence for the theory of
propositional functions itself, but it will be of some importance in the theory
of relations which is to be derived from the theory of functions of two or
more variables.
Having now somewhat tediously cleared the ground, we may proceed
to the proof of theorems. Since 2X, y<p(x, y) and Hx, y<p(x, y) may be re
garded as involving only one variable, (x, y), many theorems here follow
at once from those of the preceding section.
11-1 sx, y<p(x, y) = ^X^V<P(X, y} = -Hz, y-<p(x, y) = -[UxUv-<t>(x, y)}.
[11-05-06, 10-05]
11-12 HXf y<p(x, y) = Uxlly<p(x, y) = -SXf v-<p(x, y) = -{S^-pfo y)}.
[11-05-06, 10-04]
11-2 Hx, y<p(x, y) c <p(x, y)n.
[10-2]
11-21 <p(x, y)nc2Xry<p(x, y).
[10-21]
11-22 nx, y <p(x, y) c s*f y <p(x, y) .
[10-22]
11-23 nx, y<f>(x, y) is equivalent to " Whatever value of (x, y), in <p(x, y),
(x, y)n may be, <p(x, y)n".
[10-23]
11-24 UxTly<p(x, y) is equivalent to "Whatever values of x and y, in
v(x, y), x, and y, may be, <p(xry^".
[10-23] HJI^O, y) is equivalent to "Whatever value of .r, in
ILy<p(x,y), xr may be, Hy<p(xryY\ And Hy<f>(xry) is equivalent to
"Whatever value of y, in (f>(x,y), ys may be, v(xry^)". But [11-01]
the values of x in Hy<p(xry) are the values of x in <p(x, y). Hence
Q.E.D.
11-25 "Whatever value of (x,y), in <p(x,y), (x,y)n may be, <p(x, y)n"
is equivalent to "Whatever values of .r and y, in (p(x, y), xr and y, may be,
<p(xrys] "•
[11-06-23-24]
11-26 nxn, <p(x, y) c Htf <f>(xny) .
[11-01, 10-2]
Systems Based on Material Implication 251
11-27 nxn^(.r, y) = UvTLx<p(x, ?/).
Since x is associative and commutative, Q.E.D.
11-28 SxS^Gr, y) == StfSx^(a;, y).
Since + is associative and commutative, Q.E.D.
1 1 • 29 IIJI, <p(x, y) c nx ^(jr, yn) .
[11-2(3-27]
11-291 lUI^C*, y) c<p(av#,).
[2-2, 11-24]
11-3 IIJI^O, y) c SJI^Gc, y).
[11-01, 10-21]
11-31 SJI^(a;, y) cUy2x<p(x, y).
[11-03] nysx<p(;e, y) = I vfayi) + vfayd + <p(x3yi) + • • I
x { <K#i2/2) + ^22/2) + ^(^3^/2) + • • I
x { <p(xiyz) + ^(.r2z/3) + ^(.TSZ/S) + • • !
x ... Etc., etc.
Since x is distributive with reference to + , this expression is equal
to the sum of the products of each column separately, plus the sum
of all the cross-products, that is, to
A + { p(.ri2/i) x <f>(xiyz) x <p(.ri#3) x . . }
+ { <f>(xzyi) x <p(xzyz) x <f>(xzy3) x . . }
+ . . . Etc., etc.
where /I is the sum of all cross-products.
But [11-02] this is SJI^O, y) + A.
Hence SJI^Or, i/) +^4 == n^S^Gr, «/).
Hence [5-21] Sxny^(.r, y) c I^Z^Gr, y).
We have already called attention to the fact that the implication of 11-31
is not reversible— that Sxn^(.r, y) and I^S^Cr, y) are not equivalent.
11-32 nx2y<t>(x, y) c SxStf^(.r, ?/).
[11-03] n,S,
[11-04] SxS^(a-,
And [5-992] S
c 2
We have also the propositions concerning formal implication where
252 A Survey of Symbolic Logic
functions of two variables are concerned. The formal implication of
$(x,y) by <f>(x,y) may be written either UXt v[<p(xt y) c \//(xt y)] or
UsUy[<f>(xfy) Ci//(x, y)]. By 11-06, these two are equivalent. We shall
give the theorems only in the first of these forms.
11-4 ITX, y[<p(x, y) c t(x, y)] = Hx> v[-<p(x, y) + j(x, y)]
= Ux,v-[<p(x, y) x-^(z, y)].
[10-6]
11-41 nx, y[<p(x, y) c t(x, y)] c[<p(x, y)n c f(x, y)n].
[10-61]
11-411 ((nx, y[<p(x, y) c t(x, y)] x <p(x, y)n} c j(x, y)«.
[10-611]
11-42 nx, y[<p(x, y) c$(x, y)] c Sx, y[<p(x, y) cf(x, y)].
[10-62]
11-43 n*, y[<f>(x, y) c ^(.r, y)] c [n«, y^>(a;, ?/) c HXf y$(x, y)].
[10-63]
11-431 {nx, ,[^(.r, y)cf(x, y)] xHx, y^(x, ?/)} c nx, y}(x, y).
[10-631]
11-44 nr, y[<p(x, y) c ^(.T, ?/)] c [2Xt y<p(x, y) c S,f y^(.r, ?/)].
[10-64]
11-441 {nx, tf[^(a;, y) c ^(ic, y)] x Sx, ,^(.T, y)} c 2,, y$(x, y).
[10-641]
11-45 {nx, y[<p(x, y) cj(x, y)] xHIf v[^(a;, y) cf(a
[10-65]
11-451 nXftf[^(ic,2/)c^(a:,y)]
c {nx, y[^fe 2/) cr(.T, y)] cnx> J^GT, y) c{(x, y)}}.
[10-651]
11-452 nx, v[*(x,y)ct(x,y)]
c {n^, y[<^(^ y) c^(.r, ?/)] cnx, tf[^(a;, y) c rfe y)]}.
[10-652]
11-46 n,, y[<p(x, y) c^(;r, ?/)] = nx, ,[-^(.1% i/) c-^(.r, y)].
[10-66]
Similarly, we have the theorems concerning the formal equivalence of
functions of two variables.
Systems Based on Material Implication 253
11-47 n*, v[v(x, ?/) = }(x, y)] =-- (nx, v[<p(x, y)cf(x, y)]
xn,, „[*(*, y)c<p(x, y)]}.
[10-67]
11-48 {n*. y[*>(ar, y) = }(x, y)} xllx, v[t(x, y) = £(x, y)}}
cUt,y[<p(x,y) = £(x, y)}.
[10-68]
11-481 IIX, „[*>(*, ?/) = iKz, ?/)] c {IIX, ,[^0, y) = f(z, y)]
c n,f „[*>(*, y) = f(x,y)]|.
[10-681]
11-482 n«, y[^(o;, y) - rfe 2/)] c fnx, tf[^(ar, y) = t(x, y)]
cU,,v[<p(x,y) = rfe
[10-682]
11-49 Hx, J^GT, y) - t(x, y)} c [^(a;, y)n = t(x, y)n]
c[Ux>l,<p(x,y) = Ht,vt(x,y)]
c[2x, v<p(x, y) = SXf v*(xty)].
[10-69]
11-491 n«, tf[^(.r, i/) - ^fe ?/)] = nx, y[-^(ar, y) = -j(x, y)].
[10-691]
Further propositions concerning functions of two variables are simple
consequences of the above.
The method by which such functions are treated readily extends to
those of three or more variables. <p(x, y, z) may be treated as a function
of three variables, or. as a function of one variable, the ordered triad (a:, y, z) ;
just as \l/(x, y) can be treated as a function of x and y, or of the ordered
pair (x, y). Strictly, new definitions are required with each extension of
our theory to a larger number of variables, but the method of such extension
will be entirely obvious. For three variables, we should have
njiji^O, y, 2) = ux{uvuftf>(xt y, z)}
SJIyll ,¥>(&, y, z) = Sx{lI1/n2^(.T, y, z)\
Etc., etc.
It is interesting to note that the most general form for the analogues of
11-05 and 11-06 will be
n(«, „, *)*>(.r, y, z) == njl^, g)<p(xt y, z)
and S(I, „, ,)<t>(x, y, z) == SxS(y, f)^(ar, y, z)
Since njl(v, ,)<p(x, y, z) = n(j/, ,)<p(xiy, z) xn(y, g}<p(xzy, z) xn(l/, ,)^(x8y, z)
254 A Survey of Symbolic Logic
x . . ., and n(y, Z)<p(xny, 2) = Uyllz<p(xny, 2), etc., we shall be able to deduce
n(x, y, Z)<P(X, y, 2) = nji(I,, 2)^(.r, ?/, 2) = n(X| ^iwo, #, 2)
= ntfn(I> Z)<P(X, y, 2) = njiji^O, y, 2)
And similarly for S(I, „, 2). This calls our attention to the fact that <p(x,
y, 2) can be treated not only as a function of three variables or as a function
of one, but also as a function of two, x and (y, 2) or (x, y) and 2 or (x, 2)
and ?/.
In general, the conventions of notation being extended to functions of
any number of variables, in the obvious way, the analogues of preceding
theorems for functions of two will follow.
We failed to treat of such expressions as U<px xUfy, 2<px + U\f/y, etc.,
under the head of functions of one variable. The reason for this omission
was that such expressions find their significant equivalents in propositions
of the type H.xl\.y(<f>x x^z/), 2xHy(<px + ty), etc., and these are special cases
of functions of two variables. We may also remind the reader of the
difference between two such expressions as II <px + U\l/x and n tpx + U\{/y.
The ranges of the two functions, <p and \f/, need not be identical; there
may be values of .r in <px which are not values of y in $y. But in any
expression of the form <pxn x fan, xn as a value of x in #x must be identical
with xn as a value of .r in ^.r. For this reason, we have adopted the con
vention that where the same letter is used for the variable in two related
functions, these functions have the same range. Hence the case where
we have px and \[/y is the more general case, in which the functions are not
restricted to the same range. Theorems involving functions of this type
will not always be significant for every choice of v and \J/. There may even
be cases in which an implication is not significant though its hypothesis is
significant. But for whatever functions such theorems are significant,
they will be true; they will never be false for any functions, however chosen.
The meaning of an expression such as 2xUy( <px + \f/y) follows from the
definition of 2xTly<f>(x, y).
SxIIj,( <px + \j/y) = II, ( <px i + i/^) + Ily( <px2 + $y) + Uv( <pxz + ^) + . . .
= .r + z x <xi + \/z x <xi + \/3 x . . .
+ {(VP.TS + ^I) x( ^3 + ^2/2) x( ^3 + ^2/3) x . . . )
+ . . . Etc., etc.
And for any such expression with two operators we have the same type of
Systems Based on Material Implication 255
two-dimensional array as for a function of two variables in general. The
only difference is that here the function itself has a special form, <^.r + \fry
or <px x \{/y, etc.
12 • i
(1) [1-3] n^.r
(2) II <£>.r x II\l/y = ( <pX]_ x </?.r2 x ^3 x . . . ) x
= (<f>X! x n^i/) x (>.r2 x n^z/) x (<p.ra x Ityy) x . . .
[5-98]
xntf(^r8x^y)x... [10-371]
[ii-oi]
(3) By (2) and 1-3,
x tpxi) x (n^y x <^.r2)
x (H^ x (p.r3) x ...
ny(^ x <^.r2)
xlltf(^x^.r3) x ... [10- 37]
. [11-01]
(4) Similarly, ll^y xH^r = UyUx(\f^y x ^.r) = 11^(^.1- x^y).
" <px is true for every .r and \f/y is true for every z/" is equivalent to "For
every a; and every y, <px and ^z/ are both true", etc.
12-2 S^.r + S^z/ = 2^z/+2^.r = ^^^(^.r
(1) [4-3] S^i- + S^ - S^+S^r.
(2) 2<p.i- + S^ = (^'i + ^2 + ^3 +...) + ->/
[5-981]
. [10-31]
= SxZ^ + ifc/). [11-04]
(3) By (2) and 4 -3,
. [10-3]
= SxS,(^+^r). [11-04]
(4) Similarly, S^+2^.r = Stf
256 A Survey of Symbolic Logic
"Either for some x, <px, or for some y, tyy" is equivalent to "For some
x and some y, either <px or \f/y", etc.
12-3 S^r x = = * = S5 x
(1) [1-3] S^c
(2) S#e x S^y =
X S^y) + O?2 X S^y) + (^T3 X
[5-94]
) + Stf( ^2 x $y) + Sy( ^3 x ^y)
+ ... [10-361]
= S^C^cx^). [11-04]
(3) By (2) and 1-3,
x Si = S x i + S^ x
x ^cj + y x <^x2
+ Sy(^x^8)+ ..- [10-36]
= SA(^x^). [H-04]
(4) Similarly, S^y xS^o; = SyS,(^ x #r) - StfSx(^c x^).
"For some x, <^.r, and for some #, ^?/" is equivalent to "For some x and
some y, <px and ^y", etc.
12-4 n v?.r + n^ = n^ + n^ = nji^z + ^) = n*ny(^ + <?x)
= UyUx(<px + M == IlvUx(frj+ <px).
(1) [4-3] IL<f>x + TIty = U^y + U^px.
(2) H<px + n^?/ = (^aji x <^x2 x <px3 x . . . ) + n^z/
= (<pxi + ttty) x (^2 + n^i/) x (^T3 + n^y) x ...
[5-941]
+ M x... [10-371]
= nsntf(^ + ^). [ii-oi]
(3) By (2) and 4 -3,
+ n^?/ = (n^y + ^o x (n^ + ^2)
x n +
.T3) x... [10-37]
(4) Similarly, Uty + H<f>x = UvUx(^y + <px) = Uyttx(<px + -
"Either for every x, <px, or for every y, \f/y" is equivalent to "For every x
Systems Based on Material Implication 257
and every y, either <px or \f/y", etc. At first glance this theorem may seem
invalid. One may say: "Suppose <px be 'If a- is a number, it is odd',
and $y be 'If y is a number, it is even'. Then U<px + U^y will be 'Either
every number is odd or every number is even', but Uxllv(<px + \f/y) will be
'Every number is either odd or even'". The mistake of this supposed
illustration lies in misreading TLxUv(<px + \^y). It is legitimate to choose,
as in this case, <px and \f/y such that their range is identical: but it is not
legitimate to read UxUy(<px + $y) as if each given value of x were connected
with a corresponding value of y. To put it another way: nzllx(#c + #c),
as a special case of Uxlly(<px + ty), would not be "For every value of x,
either <px or \l/x", but would be "For any two values of x, or for any value
of x and itself, either <px or $x" . Thus HJIj/^T + 1/^) in the supposed
illustration would not be as above, but is in fact "For any pair of numbers,
or for any number and itself, either one is odd or the other is even" — so
that U<px + TLfy and Ilxlly(<px + $y) would here both be false, and are equiva
lent.
A somewhat similar caution applies to the interpretation of the next
two theorems. The analogues of these, in <p(x, y), do not hold.
12-5 2<f>x + Wy = n^y + 2 <px = 2XIIV( <f>x + ty) == ^x
(1) [4-3]
(2) By proof similar to (2) in 12-2, 2<px + Ufy = Sxnv(^r + ty)
And by proof similar to (3) in 12 • 2, S <px + Ufy = S,ntf(^ + <px)
(3) By proof similar to (2) in 12-4, H^y + S
And by proof similar to (3) in 12-4, Uifry + S^r == Uy^x
"Either for some or, ^r, or for every y, ^" is equivalent to "For some x
and every y, either <?x or ^?/", etc.
12-6 s<^.r
(1) [1-3] S^r
(2) By proof similar to (2) in 12-3, S^c xltyy == S*ntf(^c x
And by proof similar to (3) in 12-3, 2<px xH^z/ == S,Hy(^ x
(3) By proof similar to (2) in 12-1, Ityy xS^c « ntf2
And by proof similar to (3) in 12-1, Ityy xS^r == Hy2
"For some x, <px, and for every y, fy" is equivalent to "For some x and
every y, <px and ^?/", etc.
18
258 A Survey of Symbolic Logic
We may generalize theorems 12-1-12-6 by saying that for functions
of the type (<px + \f/y) and ( (px x \j/y) the order of operators and of members
in the function is indifferent; and for propositions of the type
nl
2J
the operators may be combined, and the functions combined in the relation
between the propositions.
It will be unnecessary to give here the numerous theorems which follow
from 10-5-12-6 by the principles pqcp, pcp + q, and Il^xc'Spx, etc.
For example, 10-51,
IlipX
gives at once
(i) nx(<px
(2)
(3)
(4)
(5)
(6)
Etc., etc.
And 12-2, 2<px + 2\j/y = 2x2v(<f>x + \fry), etc., gives
(1) S,
(2)
(3)
(4)
(5) 2<px
(6) 2^z
Etc., etc.
Another large group of theorems, only a little less obvious, follow from
the combination of H<px cH.<px, or S#ccS#r, with Ii\f/y c S^y, giving
by 5 -3,
(3) 2 <px + Tl\{/y c 2 (px + ^^y
(4) 2 <px x TL\f/y c 2 <px x 2^?/
Etc., etc.
Each of these has a whole set of derivatives in which n (px + U\I/y is replaced
Systems Based on Material Implication 259
by H.xlly(<px + 1/'?/), etc., H<px xZi/'Z/ by nxSJ/(^x^), etc. We give, in
summary form, the derivatives of (2), by way of illustration:
Any one of c any one of
xU\l/y
x <px)
x <px) Stfnx(^ x
UyUx( 9x x ^) Svnz( <px x
x n ipx zty x n <?x
, etc., etc.
, etc., etc.
^), etc., etc.
This table summarizes one hundred fifty-six theorems, and these are only a
portion of those to be got by such procedures.
Functions of the type of (<f>xx$y) and ((px + M give four different
kinds of implication relation: (1) njl^rc^); (2) nxSv(^c^);
(3) 2xny(<pxcty)', and (4) SxStf(^c c $y). With the exception of the
first, these relations are unfamiliar as "implications", though all of them
could be illustrated from the field of mathematics. Xor are they par
ticularly useful: the results to be obtained by their use can always be got
by means of material implications or formal implications. Perhaps
UJly(<px c $y) is of sufficient interest for us to give its elementary properties.
y^ x
i c Ufy) x (^2 c
12-7 nxntf(#cc
[11-01]
... [10-42]
And this last expression is equivalent to the set
<pxi c Uifry, <pxz c H^y, <px3 c nty, etc.
12-71 {IIJIJ/Ozci/'?/) x^rB}
[9-4, 12-7]
260 A Survey of Symbolic Logic
If for every x and every y, <px implies \l/y, and for some given x, <px is true
then $y is true for every y.
12-72 IltTlv( <px c }y) = S ^.r c Ilty = II - #c
(1) If IIJI^Oo; c^y), then [12-7] <p
Hence [5-991] S^cltyi/.
And if S^ccltyy, then [10-42] tty(2 <px c ty) , and hence [10-41]
(2) [9-3] S^c
HxHy(<px c $y) is equivalent to "If there is some x for which px is true,
then $y is true for every y".
12-73 {njiw(#rc^) xnyn,(^cf2)} cnxn,(^c^).
[12-72] If IlJIyC^rc^) and nvn«(^i/cfz), then
and S^y c Hfz.
But [10-21] Il^yc^y. Hence [5-1] S«*ccII£3, and [12: 72]
This implication relation is here demonstrated to be transitive. In fact,
it is, so to speak, more than transitive, as the next theorem shows.
12-74 {(S^ccS^) xntfn,(^cfz)} cIIJI.C^c cfz).
[12-72] nvna(^crz) = s^z/cn^.
And [5-1] if S^c cZfy and S^y c Hfz, then S^.T c Hfz, and [12-72]
12-75 (nx
[12-72] ns
And [5-1] if 2<pxcttty and H^y c Hfz, then S^c cHfz, and [12-72]
IV. DERIVATION OF THE LOGIC OF CLASSES FROM THE CALCULUS OF
PROPOSITIONAL FUNCTIONS
The logic of classes and the logic of relations can both be derived from
the logic of propositional functions. In the present chapter, we have
begun with a calculus of propositions, the Two-Valued Algebra, which
includes all the theorems of the Boole-Schroder Algebra, giving these
theorems the propositional interpretation. We have proved that, con
sidered as belonging to the calculus of propositions, these theorems can
validly be given the completely symbolic form: "If . . . , then . . ."
Systems Based on Material Implication 261
being replaced by "... c ...","... is equivalent to ..." by ". . .
= . . .", etc. The Two- Valued Algebra does not presuppose the Boole-
Schroder Algebra; it simply includes it.
Suppose, then, we make the calculus of propositions — the Two-Valued
Algebra — our fundamental branch of symbolic logic. We derive from it
the calculus of prepositional functions by the methods of the last two
sections. We may then further derive the calculus of logical classes, and a
calculus of relations, by methods which are to be outlined in this section-
and the next.
The present section will not develop the logic of classes, but will present
the method of this development, and prove the possibility and adequacy
of it. At the same time, certain differences will be pointed out between
the calculus of classes as derived from that of propositional functions and
the Boole-Schroder Algebra considered as a logic of classes. In order to
distinguish class-symbols from the variables, x, y, z, in propositional func
tions, we shall here represent classes by a, 0, y, etc.
For the derivation of the logic of classes from that of propositional
functions, a given class is conceived as the aggregate of individuals for
which some propositional function is true. If <pxn represent "a: is a man",
then the aggregate of x's for which <px is true will constitute the class of
men. If, then, z(<?z) represent the aggregate of individuals for which the
propositional function <pz is true, z(<pz) will be "the class determined by
the function <pz", or "the class determined by the possession of the char
acter <£>".13 We can use a, 0, 7, as an abbreviation for z(<pz), 2(^2), z(&),
etc. a = z(tpz) will mean "a is the class determined by the function <pz",
(In this connection, we should remember that <px and <pz are the same
function.)
The relation of an individual member of a class to the class itself will
be symbolized by e. xn e a represents " xn is a member of a "-—or briefly
"xn is an a". This relation can be defined.
13 We here borrow the notation of Principia. The corresponding notation of Peirce
and Schroder involves the use of S, which is most confusing, because this S has a meaning
entirely different from the S which is an operator of a propositional function. But in
Principia, z(<pz) does not represent an aggregate of individuals; it represents " : such
that <pz". And z(tpz) is not a primitive idea but a notation supported by an elaborate
theory. Our procedure above is inelegant and theoretically objectionable: we adopt it
because our purpose here is expository only, and the working out of an elaborate technique
would impede the exposition and very likely confuse the reader. As a fact, a more satis
factory theory on this point makes no important difference.
262 A Survey of Symbolic Logic
13-01 xn€z(<pz) = <pxn Def.
"xn is a member of the class determined by <pz" is equivalent to " <pxn is
true".
(For convenience of reference, we continue to give each definition and
theorem a number.)
The relation "a is contained in ft" is the relation of the class a to the
class ft when every member of a is a member of (3 also. We shall symbolize
"a is contained in ft" by a c ft. The sign c between a and ft, or between
z(vz) and 2(^2), will be "is contained in"; c between propositions will
be "implies ", as before. xn e a is, of course, a proposition; x e a, a prepo
sitional function.
13-02 acft = Ux(xeacxeft) Def.
a c ft is equivalent to "For every x, cx is an a' implies (x is a ft'".
nx(> e o: c x e (3) is a formal implication. It will appear, as we proceed
that the logic of classes is the logic of the formal implications and formal
equivalences which obtain between the propositional functions which deter
mine the classes.
13-03 (a = ft) = Ux(x e a = x e (3) Def.
a = ft is equivalent to "For every x, {x is a member of a' is equivalent
to 'x is a member of /3'". a = ft thus represents the fact that a and ft
have the same extension — i. e., consist of identical members.
xne a, a c ft, and a = ft are assertable relations — propositions. But
the logical product of two classes, and the logical sum, are not assertable
relations. They are, consequently, defined not by means of propositions
but by means of functions.
13-04 axft = x[(xea) x(xe ft)} Def.
The product of two classes, a and ft, is the class of x's determined by the
propositional function "x is an a and x is a ft". The class of the x's for
which this is true constitute a* ft, the class of those things which are
both as and /3's.
The relation x between a and ft is, of course, a different relation from x
between propositions or between propositional functions. A similar remark
applies to the use of + , which will represent the logical sum of two classes,
as well as of two propositions or propositional functions. This double
use of symbols will cause no confusion if it be remembered that a and ft,
£(<pz) and 2(^2), etc., are classes, while x e a is a propositional function,
and xn e a, a c ft, and a = ft are propositions.
Systems Based on Material Implication 263
13-05 a + /3 = x{(xea) + (x6(3)} Def.
The sum of two classes, a and /?, is the class of z's such that at least one
of the two, 'x is an «' and fx is a 0', is true, or loosely, the class of z's
such that either Z is an a or x is a /3.
The negative of a class can be similarly defined:
13-06 -a = x-(xta) Def.
The negative of a is the class of or's for which 'x is an a' is false.
The "universe of discourse", 1, may be defined by the device of selecting
some prepositional function which is true for all values of the variable.
Such a function is (f.r cfz), whatever prepositional function £x may be.
13-07 1 = x({xc£x) Def.
1 is the class of xs for which £x implies £x.u Since this is always true, 1 is
the class of all x's. The "null-class", 0, will be the negative of 1.
13-08 0 = -1 Def.
That is, by 13-06,, 0 = x-(txc£x), and since -(fac^) is false for all
values of x, the class of such x's will be a class with no members.
Suppose that a = 2(^2) and /3 = 2(^2). Then, by 13-01, xn * a == <pxn.
Hence a c ft will be Ux(<f>x c$x), and a = 0 will be ttx(<px == #r). This
establishes at once the connection between the assertable relations of
classes and formal implication and equivalence. To illustrate the way in
which this connection enables us to derive the logic of classes from that of
propositional functions, we shall prove a number of typical theorems.
It will be convenient to assume for the whole set of theorems:
a = 2O), ]8 = 2(*z), 7 = &(&)
13-1 0 = x-({xc{x).
0 = -1. Hence [13-06] 0 = x -(x * 1).
[13 • 01 • 07] x e 1 = f.r c £c. Hence 0 = x -(s*.r c f .r) .
13-2 nx(xel).
[13-01-06] xnel == (£xnc£xn).
Hence Ux(x e 1) = Ux(£xc{x).
But [2 - 2] frn c r-rn. Hence [10 • 23] n,(^ c £x) .
Every individual thing is a member of the "universe of discourse".
14 This defines, not the universe of discourse, but "universe of discourse",— the range
of significance of the chosen function, f . With 1 so defined, propositions which invol
the classes £(**), «(*«), etc., and 1, will be significant whenever <p, t, etc., a
the same range, and true if significant.
264 A Survey of Symbolic Logic
13-3 IIx-(.£eO).
[13 -01 -06 -07] a-neO = -(f.rn cf.rn).
Hence [3-2] ~(xn e 0) = (r«n c far») .
But $xn c f.r«. Hence [10-23] nx(fx c fr), and Hx -(a; e 0).
For every x, it is false that x e 0 — no individual is a member of the null-
class.
13-4 «cl.
[13-01-06] Xntl = frrnCf.rn).
Since a = %(<pz), [13-01] xn e a = ipxn-
[9-33] On cf.rn) c [«*cn c (fzn c £&„)].
Hence since f.rn c f.rn, <p.Tn c ($xn c £rn).
Hence [10-23] ttx[<px c ({x c{x)], and [13-2] acl.
Any class, a, is contained in the universe of discourse. It will be noted
(13-2 and 13-4) that individuals are members of 1, classes are contained in 1.
In the proof of 13-4, we make use of 9-33, "A true proposition is implied
by any proposition". $xn c $xn is true. Hence it is implied by <pxn. And
since this holds, whatever value of x, xn may be, therefore,
But (px is the function which determines the class «; fa: cfa:, the function
which determines 1. Hence <pxn is xn e a, and £xn c£xn is xn e 1. Conse
quently we have 11^ (a:caca:cl). And by the definition of the relation
"is contained in", this is a c 1.
13-5 Oca.
[9-1] -On e 0) is equivalent to (xn e 0) = 0.
Hence [13-3] (xn e 0) - 0, and [9-32] (xn e 0) c <pxn.
Hence [13-01] xneOcxne a, and [10-23] Ux(xeQcxe a).
Hence [13-02] 0 c a.
The null-class is contained in every class, a. In this proof, we use 9-32,
"A false proposition implies any proposition". -(fa:ncfa:n) is false, and
hence implies <pxn. But -(fxcfx) is the function which determines 0;
and <px, the function which determines a. Hence 0 c a.
The proofs of the five theorems just given are fairly typical of those
which involve 0 and 1. But the great body of propositions make more
direct use of the connection between the relations of classes and formal
implications or equivalences. This connection may be illustrated by the
following:
Systems Based on Material Implication 265
13-6 z( (pz) c z(^z) = nx( (px c \j/x) .
[13-02] z((pz) cz(^z) = Ux[xe z((pz) c.ce
[13-01] xn e z((pz} = (pxn, and xn e z(\f/z) = $
Hence [2-1] Ut[x e z(<pz) ex e £(^z)] = Ux(<px c
"The class determined by (pz is contained in the class determined by \f/z"
is equivalent to "For every x, (px implies \frx".
[13-03] [z(<pz) = z(^z)] = Hx[xez(<pz) = a-ez(^z)].
[13-01] xn e z((pz) == <pxn, and xn e z(^z) = \l/xn.
Hence Hx[x e z(<pz) = x e z(^z)] = Hx(<px = fa)-
"The class determined by <pz is equivalent to the class determined by ^z"
is equivalent to "For every x, <px is equivalent to $x".
[10-66] Ilx[x e z(<pz) ex e z(tz)] == Ex(-[.r e z(^z)] c-[.r e z(^z)]).
Hence IIx(a: e a ex e /3) = nx[-(.i- e /3) c-(x e a)].
[13-01-06] -(a; e a) = a; e -a, and -(x e 0) = .T e -/3.
Hence [13-02] («c/3) - (-/3c-«).
13-9 [(aC|8) x(/3cT)] c(aC7).
[13-6] (a c j8) = Ux(<px c ^tr), (|8 c 7) = Hx(tx c ^.i-), and (a c 7)
And [10-65] [Ux(<f>x c \f/x) x nx(^.r c £.r)] c nx(^a; c ^).
The relation "is contained in" is transitive. 13-9 is the first form of the
syllogism in Barbara. The second form is:
13-91 [(aC]8) x(.Tnea)] c(arBc j8).
[13-6] (aC]8) = nx(^c^.r).
[13-01] (xne a) == <pxn, and (xne 0) = ^n.
And [10-611] [nx(^.r c ^r) x ^.rn] c\j/xn.
If the class a is contained in the class & and .rn is a member of a, then .rn
is a member of /3.
13-92 [(a = ]8) x(/3 == 7)] c (a =: 7)-
[lo-/J \<2 Py ~ -iix^cp.c ~ Y /> \fj
And [10-68] [Hx(^,r = ^.r) xHx(^.r = ^.r)] cHx(^ = ^r)-
The last three theorems illustrate particularly well the direct connection
266 A Survey of Symbolic Logic
between formal implications and the relations of classes. 13-6 and 13-7
are alternative definitions of a c /3 and a = 0. Similar alternative defini
tions of the other relations would be : l5
-[z(<pz*)] = z(-<pz)
z(<pz)
We may give one theorem especially to exemplify the way in which
every proposition of the Two-Valued Algebra, since it gives, by 10-23, a
formal implication or equivalence, gives a corresponding proposition con
cerning classes. We choose for this example the Law of Absorption.
13-92 [a+(aX]8)] = a.
[13-04-05] [a+(aX|8)] = x{(xea) + [(xea) x(.re/3)]}.
Hence [13-01] {xn e [a + (a x/3)]}
= {(xnea) + [(xnea)x(xne(3)}}. (1)
But [13-03] {[«+(« x/3)] = a}
= n,[{(.T e a) + [(x e a) X (x e 0)]} == (x e «)J. (2)
But [13-01] (xn e a) = <pxn, (xn e 0) = $xn, and by (2),
{[a+(a*{3)] = a} = Ux{[<px + (<px xjx)] = <px}
But [5-4] [<pXn+(<pXnX\l/Xn)] = V%n>
Hence [10-23] TLx{[<px + (<?x xifrx)] = <px}.
All but the last two lines of this proof are concerned with establishing
the connection between [a + (a x /3)] = a and the formal equivalence
na{[#c+(#ex#r)] = <px]
Once this connection is made, we take that theorem of the Two- Valued
Algebra which corresponds to [a + (a x /3)] = «, namely 5-4, (p + .p q) = p,
substitute in it <pxn for p and if/xn for g, and then generalize, by 10-23, to
the formal equivalence which gives the proof. An exactly similar pro
cedure will give, for most theorems of the Two-Valued Algebra, a corre
sponding theorem of the calculus of classes. The exceptions are such
propositions as p = (p = 1), which unite an element p with an implication
or an equivalence. In other words, every theorem concerning classes can
be derived from its analogue in the Two- Valued Algebra.
We may conclude our discussion of the derivation of the logic of classes
15 As a fact, these definitions would be much more convenient for us, but we have
chosen to give them in a form exactly analogous to the corresponding definitions of Prin-
cipia (see i, p. 217).
Systems Based on Material Implication 267
from the logic of prepositional functions by deriving the set of postulates
for the Boole-Schroder Algebra given in Chapter II. This will prove that,
beginning with the Two- Valued Algebra, as a calculus of propositions, the
calculus of classes may be derived. This procedure may have the appear
ance of circularity, since in Section I of this chapter we presumed the
propositions of the Boole-Schroder Algebra without repeating them. But
the circularity is apparent only, since the Two- Valued Algebra is a distinct
system.
The postulates of Chapter II, in a form consonant with our present
notation, can be proved so far as these postulates express symbolic laws.
The postulates of the existence, in the system, of -a when a exists, of a x 0
when a and 0 exist, and of the class 0, must be supposed satisfied by the
fact that wre have exhibited, in their definitions, the logical functions which
determine a x (3, -a, and O.16
14-2 (a Xa) = a.
[13-01] xn e a = <fXn.
Hence [13-04] xn e(a*a) = [(xn e a) x (xn e a)] = (<pxn x «^.rn).
Hence [13-03] [(aXa) = a] = Hx{[.re (a x a)] = x e a]
= Ht[(<px x <px) == <&].
But [1-2] (<f>xn x <pxn) = <pxn.
Hence [10-23] Ut[(<px x <px) == <px].
14-3 (ax0) = (/3 x a).
[13-03] [(ax0) = (0 x a)] = Ux{[x e (a x 0)] = [x e (ft x a)]}
= IIx{[(.r6a)x(.r60)] = [(.r e /3) x(.r € a)]}.
[13-01-04]
Hence [13-01] [(ax/3) = (0 x a)] = nj(#rx#r) = (*x x <px)}.
But [1-3] (<pxnxtxn) = (#r» x ^rn).
Hence [10-23] Ux[(<f>x x^.r) = (^ x «^.r)].
14-4 (aX0) XT == ax(0x7).
[13-03] [(«X0)X7 = «X(0XT)] =
= nx[{.T6[(aX/3)x7]l == {a?e[ax(0x7)]}J
= nx[{[(.T e a) x (x e (3)] x (.r e 7) J == { (.r e a) x [(.r € 0) x (.r e 7)] }J.
[13-01-04]
Hence [13-01] [(«x0) x7 - ax(0x7)]
16 A more satisfactory derivation of these existence postulates is possible when the
theory of propositional functions is treated in greater detail. See Principia, I, pp. 217-18.
268 A Survey of Symbolic Logic
But [1-04] (<pxnxtxn) x£xn = <pxnx(\f/xnx^xn).
Hence [10-23] Hx{[(<px x^x) x &] = [<px x(+x x &)]}.
14-5 axO = 0.
[13-1-01] .TncO = -(fr»c£rB).
[13-03-04] [axO = 0] = Uxl[(xnea)x(xneQ)] = (xneO)}
= n.u^x-o-zc^)] = -(fzcMOj. [13-01]
'But [2-2, 9-01] (r.r»cfarB) = 1, and [3-2] -({xnc{xn) = 0.
Hence [1-5] [<pxn x-(£xn c £xn)] = 0 = -(fr» c£c»).
Hence [10-23] ns{[^r x-frr c^)] = -(fa; cfr)}.
0, in the fourth and fifth lines of the above proof, is the 0 of the Two-Valued
Algebra, not the 0 of the calculus of classes. Since the general method of
these proofs will now be clear, the remaining demonstrations can be some
what abbreviated.
14-61 [(«x-|8) = 0] c[(ax/3) = a].
[13 -01 -02 -04 -06] The theorem is equivalent to
But [13-3] Ilx-(xeO), and hence [9-1] ILx[(x e 0) - 0].
Hence the theorem is equivalent to
IT* { [Or x -i/a*) = (xeQ)]c[(<pxxifrx) = <px]}
But [13-3] Hx-(xeQ), and hence [9-1] TLx[(x e 0) = 0].
Hence the theorem is equivalent to
Ux{[(<px x-fr) = 0] c[(^ x^) = <px]}
But [1-61] [(<f>Xn X-\j/Xn) = 0] C[(<pxn X$Xn) = <pXn}.
Hence [10-23] Q.E.D.
14-62 {[(«Xj8) = a] x[(«x-/3) = a]} c (a = 0).
The theorem is equivalent to
n,[{[(#rx£r) == ^]x[(^x-^) = <px]} c[<px = (x e 0)]]
But [13-3, 9-1] nx[(.r€0) = 0].
Hence the theorem is equivalent to,
n,[{[(#cx^.T) == ^i']x[(^x-^.r) = ^r]} c(<px = 0)]
But [1-62] {[(?.r«x£rn) == <pxn] x[(^xnx-^xn} == <pxn}} c(^.rn = 0).
Hence [10-23] Q.E.D.
The definition, 1 = -0, follows readily from the definition given of 0 in
this section. The other two definitions of Chapter II are derived as follows:
14-8 O+/3) - -(-ax-j8).
The theorem is equivalent to Ilx[(<px + \f/x) = -(-$x x-\f/x)].
Systems Based on Material Implication 269
But [1-8] (<pxn+tXn) = -(-<?xnx-\l/xn).
Hence [10-23] Q.E.D.
14-9 (aCjS) = [(ax/3) = a].
The theorem is equivalent to IIjOz c V/.r) = Uz[(<px x$x) == <px].
But [1-9] (<f>xnc^xn) is equivalent to [(^c»x^ictt) == v*cn].
Hence [10-23-69] Q.E.D.
Since the postulates and definitions of the calculus of classes can be
deduced from the theorems of the calculus of prepositional functions, it
follows that the whole system of the logic of classes can be so deduced.
The important differences between the calculus of classes so derived and
the Boole-Schroder Algebra, as a logic of classes, are two: (1) The Boole-
Schroder Algebra lacks the e-relation, and is thus defective in application,
since it cannot distinguish the relation of an individual to the class of
which it is a member from the relation of two classes one of which is con
tained in the other; (2) The theorems of the Boole-Schroder Algebra
cannot validly be given the completely symbolic form, while those of the
calculus of classes derived from the calculus of prepositional functions can
be given this form.17
V. THE LOGIC OF RELATIONS
The logic of relations is derived from the theory of propositional func
tions of two or more variables, just as the logic of classes may be based
upon the theory of propositional functions of one variable.
A relation, R, is determined in extension when we logically exhibit the
class of all the couples (x, y) such that x has the relation R to y. If <p(x, y)
represent "x is parent of y", then x $[<p(x, y)} is the relation "parent of "
This defines the relation in extension: just as the extension of "red" is
the class of all those things which have the property of being red, so the
extension of the relation "parent of" is the class of all the parent-child
couples in the universe. A relation is a property that is common to all the
couples (or triads, etc.) of a certain class; the extension of the relation is,
thus, the class of couples itself. The calculus of relations, like the calculus
of propositions, and of classes, is a calculus of extensions.
"Oftentimes, as in Schroder, Alg. Log., i, the relations of propositions in the algebra
of classes have been represented in the symbols of the propositional calculus befo
calculus has been treated otherwise than as an interpretation of the Book-Schroder Algebra.
But in such a case, if these symbols are regarded as belonging to the system, tl
is invalid.
270 A Survey of Symbolic Logic
We assume, then, the idea of relation : the relation R meaning the class
of couples (x, y) such that x has the relation R to y.
R = x y(x Ry), S = w z(w S z), etc.
This notation is simpler and more suggestive than R = x y[<p(x, y)],
S = w z[\[/(iv, z)], but it means exactly the same thing. A triadic relation,
T, will be such that
T = x y z[T(x} y} z)}
or T is the class of triads (x, y, z) for which the prepositional function
T(x, y, z) is true. But all relations can be defined as dyadic relations. A
triadic relation can be interpreted as a relation of a dyad to an individual —
that is to say, any function of three variables, T(x, y, z), can be treated as a
function of two variables, the couple (x, y) and z, or x and the couple (y, z) .
This follows from the considerations presented in concluding discussion of
the theorems numbered 11-, in section III.18 Similarly, a tetradic relation
can be treated as a dyadic relation of dyads, and so on. Hence the theory
of dyadic relations is a perfectly general theory.
Definitions exactly analogous to those for classes can be given.
15 -01 (x, y)n e z w[R(z, ic)] = R(x, y)n. Def.
It is exactly at this point that our theoretical considerations of the equiva
lence of <p(x, y)n and <p(xr ya) becomes important. For this allows us to
treat R(x, y), or (x Ry), as a function of one or of two variables, at will;
and by 11-07, we can give our definition the alternative form:
15-01 (xm yn)ezw(zRw) = xm R yn. Def.
"The couple (xm yn) belongs to the field, or extension, of the relation deter
mined by (z R w) " means that xm R yn is true.
15-02 RcS = Ux,v[(xRy) c(xSy)]. Def.
This definition is strictly parallel to 13-02,
(ac/3) = Hx (xe acxe /3)
because, by 15-01, (x R y) is (x, y) e R and (x S y) is (x, y) e S. A similar
remark applies to the remaining definitions.
15-03 (R = S) = nx, y[(x Ry) = (xS y)]. Def.
R and S are equivalent in extension when, for every x and every y, (x R y)
and (x S y) are equivalent assertions.
18 See above, pp. 253 ff.
Systems Based on Material Implication 271
15-04 R xS = x y [(x Ry)* (x S y)]. Def.
The logical product of two relations, R and S, is the class of couples (x, y.
such that x has the relation R to y and x has the relation S to y. If R is
"friend of", and S is "colleague of", R x S will be "friend and colleague of")
15-05 R + S = xy[(xRy) + (xSy)]. Def.
The logical sum of twyo relations, R and S, is the class of couples (x, y) such
that either x has the relation R to y or x has the relation S to y. R + S
will be " Either fl of or S of".
15-06 -R = xy-(xRy). Def.
-R is the relation of x to y when x does not have the relation R to y.
It is important to note that R x S, R + S} and -# are relations: x(R x *$)#,
z(# + S)y, and # -E y are significant assertions.
The "universal-relation" and the "null-relation" are also definable
after the analogy to classes.
15-07 1 = x y [?(x, y)c{(x, y)]. Def.
x has the universal-relation to y in case there is a function, f, such that
$(x, y) c£(x, y), i. e., in case x and y have any relation.
15-08 0 = -1. Def.
Of course, 0,1, + and x have different meanings for relations from their
meanings for classes or for propositions. But these different meanings ot
0, + , etc., are strictly analogous.
As was pointed out in Section III of this chapter, for every theorem
involving functions of one variable, there is a similar theorem involving
functions of two variables, due to the fact that a function <p(xt y) may be
regarded as a function of the single variable (x, y). Consequently, for
each theorem of the calculus of classes, there is an exactly corresponding
theorem in the calculus of relations. We may, then, cite as illustrations of
this calculus the analogues of the theorems demonstrated to hold for
classes; and no proofs will here be necessary. These proofs follow from the
theorems of Section III, numbered 11-, exactly as the proofs for classes
are given by the corresponding theorems in Section II, numbered 10 • .
15-1 0 = x$-[t(x,y)c£(x, y)].
The null-relation is the relation of x to y when it is false that $(x, y) implies
£(x, y), i. e., when x has no relation to #.19 Of course, there is no such
(x, y) couple which can significantly be called a couple.
19 As in the case of the 1 and 0 of the class calculus, the 1 and 0 of relations, defined as
272 A Surrey of Symbolic Logic
15-2 nx, y[(x, y) e 1].
Every couple is a member of the universe of couples, or has the universal
(dyadic) relation.
15-3 nx,y-[(x, 2/)eO].
No couple has the null-relation.
15-4 jRcl.
15-5 Ocfl.
Every relation, R, is implied by the null-relation and implies the universal
relation; or, whatever couple (.r, y) has the null-relation has also the
relation 72, and whatever couple has any relation, R, has also the universal-
relation.
15-6 (RcS) = ttx,y[(xRy)c(xSy)}.
For relations, RcS is more naturally read " R implies S" than " R is con
tained in S". By 15-6, " R implies S" means "For every x and every y,
if x has the relation R to y, then x has the relation S to y". Or " R implies
S" means "Every (x, y) couple related by R are also related by S".
15-7 (R = S) = Ux, y[(x Ry) = (x S y)].
Two relations, R and S, are equivalent when the couples related by JR. are
also related by S, and vice versa (remembering that = is always a reciprocal
relation c).
15-8 (RcS) = (-Sc-R).
If the relation R implies the relation S, then when S is absent R also will
be absent.
15-9 [(RcS) *(ScT)]c(RcT).
The implication of one relation by another is a transitive relation.
15-91 [(RcS)x(xmRyn)]c(xmSyn).
If R implies S and a given couple are related by R, then this couple are
related also by S.
15-92 [(R = S) x(S == T)]c(R = T).
The equivalence of relations is transitive.
If it be supposed that the postulates concerning the existence of rela
tions are satisfied by exhibiting the functions which determine them, then,
we have defined them, are such that propositions involving them are true whenever sig
nificant, and significant whenever the prepositional functions determining the functions in
question have the same range.
Systems Based on Material Implication 273
as in the case of classes, we can derive the postulates (or remaining postu
lates) for a calculus of relations from the theorems of the calculus of prepo
sitional functions. The demonstrations would be simply the analogues of
those already given for classes, and may be omitted.
16-2 (RxR) = R.
16-3 (RxS) = (S xR).
16-4 (RxS) xT = Rx(SxT).
16-5 RxO = 0.
16-61 [(Rx-S) = 0}c[(RxS) = R].
16-62 H(RxS) ---- R] x[(Rx-S) = R}} c(R = 0).
16-8 (R + S) = -(-Rx-S).
16-9 (RcS) = [(RxS) = R].
These theorems may also be taken as confirmation of the fact that
the Boole-Schroder Algebra holds for relations. In fact, "calculus of
relations" most frequently means just that — the Boole-Schroder Algebra
with the elements, a, b, c, etc., interpreted as relations taken in extension.
So far, the logic of relations is a simple analogue of the logic of classes.
But there are many properties of relations for which classes present no
analogies, and these peculiar properties are mosl important, h. Fact,
the logistic development of mathematics, worked out by Peirce, Schroder,
Frege, Peano and his collaborators, and Whitehead and Russell, has de
pended very largely upon a further study of the logic of relations. While
we can do no more, within reasonable limits, than to suggest the manner
of this development, it seems best that the most important of these proper
ties of relations should be given in outline. But even this outline cannot
be complete, because the theoretical basis provided by our previous dis
cussion is not sufficient for completeness.
Every relation, R, has a converse, *R, which can be defined as follows:
17-01 *R = y x (x Rij). Def.
If x has the relation R to y, then y has the converse relation, »R, to x.
It follows at once from the definition of (xmyn) e R that
xm R yn = yn *R xm
because (xm R yn) = (xmyn) * R = (ynxm) e "R = Vn "R xm>
The converse of the converse of R is R.
w(wfl) = R
19
274 A Survey of Symbolic Logic
since »(*>R) = x y (y »R x) = x y (x R y) = R. (This is not a proof:
proof would require that we demonstrate
n,( „[(*, y) e ~(~R) = (x, y) e R]
But it is obvious that such a demonstration may be given. In general,
we shall not pause for proofs here, but merely indicate the method of proof.)
The properties of symmetrical relations follow from the theorems con
cerning converses. For any symmetrical relation T, T = *->T. The uni
versal relation, 1, and the null-relation, 0, are both symmetrical:
(x 1 y) = [rO, y) cf(s, y)] == 1 == [f(y, x) c{(y, .r)] - (y I x)
(The "1" in the middle of this 'proof is obviously that of the calculus of
propositions. Similarly for 0 in the next.)
(X 0 y) = -[r(.r, y) c $•(*, y)] = 0 = -[f (y, x) c {(y, x)] = (y 0 a-)
It is obvious that if two relations are equivalent, their converses will be
equivalent:
(# = £) = (»R = „$)
Not quite so obvious is the equivalent of (RcS), in terms of *R and ^S.
We might expect that (R c S) would give («/S c^). Instead we have
(RcS) = (*R c «S)
for (fl c ,S) = HXf y[(.i- E y) c (x S y)] = Ux, y[(y "R x) c (y ^S x)]
= (vRc^S)
"'Parent of implies 'ancestor of" is equivalent to '"Child of implies
'descendant of".
The converses of compound relations is as follows:
«(/JxS) = vRx^S
for x »(R x S)y = y(R x S)x = (y R x) x(yS x) = (x »R y) x (x ^S y)
= x(*Rx"S)y
If x is employer and exploiter of y, the relation of y to .r is "employee of
and exploited by". Similarly
If x is either employer or benefactor of y, the relation of y to x is "either
employee of or benefitted by".
Other important properties of relations concern "relative sums" and
Systems Based on Material Implication 275
"relative products". These must be distinguished from the non-relative
sum and product of relations, symbolized by + and x . The non-relative
product of "friend of" and "colleague of" is "friend and colleague of":
their relative product is "friend of a colleague of". Their non-relative
sum is "either friend of or colleague of": their relative sum is "friend of
every non-colleague of". We shall denote the relative product of R and S
by R \ S, their relative sum by R t S.
17-02 R S = xz{2y[(xRy) x(ySz)]}. Def.
R\S is the relation of the couple (x, z) when for some y, x has the relation
R to y and y has the relation S to z. x is friend of a colleague of z when,
for some y, x is friend of y and y is colleague of z.
17-03 R-tS = xz{ny[(xRy) + (ySz)}}. Def.
R t S is the relation of x to z when, for every y, either x has the relation
R to y or y has the relation S to z. x is friend of all non-colleagues of z
when, for every y, either x is friend of y or y is colleague of z.
It is noteworthy that neither relative products nor relative sums are
commutative. "Friend of a colleague of" is not "colleague of a friend of".
Nor is "friend of all non-colleagues of" the same as "colleague of all non-
friends of". But both relations are associative.
R\(S\T) = (R\S)\T
for 2t{(wR x) x [X(S T)z] } --= 2X { (w R x) x zv[(x Sy)x(yTz)]}
= 2y2x{(wRx)x[(xSy)x(yTz)]}
= 2v2,{[(wRx)x(xSy)]x(yTz)}
= 2v(2x[(wRx)x(xSz)]x(yTz)}
= 2v{[w(RS)y]x(yTz)}
"Friend of a (colleague of a neighbor of)" is "(friend of a colleague) of a
neighbor of".
Similarly, R t (S t T) = (R t S) t T
"Friend of all (non-colleagues of all non-neighbors of)" is "(friend of all
non-colleagues) of all non-neighbors of".
De Morgan's Theorem holds for the negation of relative sums and prod
ucts.
-(R\S) = -flt-S
for -{ 2t[(x Ry)*(yS z)] } --= ntf -[(* R y) x (y S z)}
276 A Survey of Symbolic Logic
The negative of "friend of a colleague of" is "non-friend of all colleagues
(non-non-colleagues) of".
Similarly, -(R t S) = -R -S
The negative of "friend of all non-colleagues of" is "non-friend of a non-
colleague of".
Converses of relative sums and products are as follows:
for x *(R | S)z = z(R \ S)x = ?y[(z Ry)*(yS x)]
= 2v((y S x) x(zR y)]
If x is employer of a benefactor of 'z, then the relation of z to x is " bene-
fitted by an employee of".
Similarly, ^(R t /S) = ^S t ^R
If x is hater of all non-helpers of z, the relation of z to a; is "helped by all
who are not hated by".
The relation of relative product is distributive with reference to non-
relative addition.
R\(S+T) = (RS) + (R T)
for x[R \ (S + T)]z = ?y{(x Ry)x [y(S + T)z] }
— X f (T 7? ii\ x F (11 »Si ?'^ + T?y y7 2^1 1
•^^2/ 1 v**' •tl/ c// L\c/ / \" / J )
= 2y{[(x Ry)*(yS z)] + [(x Ry)x(y T z)}}
Similarly, (R + S) \ T = (R \ T) + (S \ T)
"Either friend or colleague of a teacher of" is the same as "either friend
of a teacher of or colleague of a teacher of".
A somewhat curious formula is the following:
R\(S*T)c(R\S)x(R\T)
It holds since x[R \ (S x T)]z = ?y{(x Ry)x [y(S x T)z] }
= Zy{(xRy)x[(ySz)x(yTz)}}
and since a x (b xc) = (a x6) x (a xc),
= zy{[(x Ry)*(yS z)] x [(x Ry)x(yT z)] }
c 2,[(.T Ry)*(yS z)] x ?y[(x Ry)*(yT z)]
And this last expression is [x(R\S)z] x[x(R\ T)z].
Systems Based on Material Implication 277
If x is student of a friend and colleague of z, then x is student of a friend and
student of a colleague of z. The converse implication does not hold, be
cause "student of a friend and colleague" requires that the friend and the
colleague be identical, while "student of a friend and student of a col
league" does not. (Note the last step in the 'proof ', where Sy is repeated,
and observe that this step carries exactly that significance.)
Similarly, (flxS) T c (R\ T) x (S \ T)
The corresponding formulae with t instead of are more complicated
and seldom useful; they are omitted.
The relative sum is of no particular importance, but the relative product
is a very useful concept. In terms of this idea, "powers" of a relation are
definable :
#2 = R | JR, #3 = R2 | R, etc.
A transitive relation, S, is distinguished by the fact that S2 c 8, and hence
SncS. The predecessors of predecessors of predecessors ... of x are
predecessors of x. This conception of the powers of a relation plays a
prominent part in the analysis of serial order, and of the fundamental proper
ties of the number series. By use of this and certain other concepts, the
method of "mathematical induction" can be demonstrated to be com
pletely deductive.20
In the work of De Morgan and Peirce, "relative terms" were not given
separate treatment. The letters by which relations were symbolized were
also interpreted as relative terms by a sort of systematic ambiguity. Any
relation symbol also stood for the class of entities which have that relation
to something. But in the logistic development of mathematics, since that
time, notably in Principia Mathematical relative terms are given the
separate treatment which they -really require. The "domain" of a given
relation, R— that is, the class of entities which have the relation R to some
thing or other-may be symbolized by D'fl, which can be defined as follows:
17-04 D'R = &[2y(xRy)]. Def.
The domain of R is the class of x's determined by the function "For some y,
x has the relation R to y". If R be "employer of ", D'R will be the class
of employers.
The "converse domain" of fl— that is, the class of things
20 See Principia, i, Bk. n, Sect. E.
21 See i, *33. The notation we use for domains and converse domains i
cipia.
278 A Survey of Symbolic Logic
which something or other has the relation R — may be symbolized by CL'R
and similarly defined:
17-05 (I'R = y[2,(xRy)]. Def.
The converse domain of R is the class of y's determined by the function
"For some x, a- has the relation R to y". If R be "employer of", d'R
will be the class of employees.
The domain and converse domain of a relation, R, together constitute
the "field "of R, C'R.
17-06 C'R = xi2y[(xRy) + (yRx)]}.
The field of R will be the class of all terms which stand in either place in
the relation. If R be "employer of", C'R is the class of all those who are
either employers or employees.
The elementary properties of such "relative terms" are all obvious:
xneC'R = 2y[(xnRy) + (yRxn)}
C'R = D'R + a'R
However, for the logistic development of mathematics, these properties
are of the highest importance. We quote from Prirwipia Mathematical 22
" Let us ... suppose that R is the sort of relation that generates a series,
say the relation of less to greater among integers. Then D'R = all integers
that are less than some other integer = all integers, Q'R = all integers
that are greater than some other integer = all integers except 0. In this
case, C'R = all integers that are either greater or less than some other
integer = all integers .... Thus when R generates a series, C'R becomes
important. ..."
We have now surveyed the most fundamental and important characters
of the logic of relations, and we could not well proceed further without
elaboration of a kind which is here inadmissible. But the reader is warned
that we have no more than scratched the surface of this important topic.
About 1890, Schroder could write "What a pity! To have a highly
developed instrument and nothing to do with it". And he proceeded to
make a beginning in the bettering of this situation by applying the logic
of relatives to the logistic development of certain portions of Dedekind's
theory of number. Since that time, the significance of symbolic logic has
been completely demonstrated in the development of Peano's Formulaire
22 1, p. 261.
Systems Based on Material Implication 279
and of Principia Mathematica. And the very head and front of this develop
ment is a theory of relations far more extended and complete than any
previously given. We can here adapt the prophetic words which Leibniz
puts into the mouth of Philalethes: "I begin to get a very different opinion
of logic from that which I formerly had. I had regarded it as a scholar's
diversion, but I now see that, in the way you understand it, it is a kind of
universal mathematics."
VI. THE LOGIC OF Principia Mathematica
We have now presented the extensions of the Boole-Schroder Algebra—
the Two-Valued Algebra, prepositional functions and the propositions
derived from them, and the application to these of the laws of the Two-
Valued Algebra, giving the calculus of propositional functions. Beyond
this, we have shown in outline how it is possible, beginning with the Two-
Valued Algebra as a calculus of propositions, to derive the logic of classes
in a form somewhat more satisfactory than the Boole-Schroder Algebra,
and the logic of relations and relative terms. In so doing, we have presented
as much of that development which begins with Boole and passes through
the work of Peirce to Schroder as is likely to be permanently significant.
But, our purpose here being expository rather than historical, we have not
followed the exact forms which that development took Instead, we have
considerably modified it in the light of what symbolic logicians have learned
since the publication of the work of Peirce and Schroder.
Those who are interested to note in detail our divergence from the
historical development will be able to do so by reference to Sections VII
and VIII of Chapter I. But it seems best here to point out briefly what
these alterations are that we have made. In the first place, we have
interpreted S^.r, II ^r, S^(.r, y), etc., explicitly as sums or products of
propositions of the form ^rn, iK-rmyn), etc. Peirce and Schroder avoided
this, in consideration of the serious theoretical difficulties. But while
they did not treat II <px as an actual product, S#c as an actual sum, still
the laws which they give for propositions of this type are those which result
from such a treatment. There is no slightest doubt that the method by
which Peirce discovered and formulated these laws is substantially the one
which we have exhibited. And this explicit use of n^.r as the symbol for a
product, S#r as the symbol of a sum, makes demonstration possible where
otherwise a large number of assumptions must be made and, for further
principles, a much more difficult and less obvious style of proof resorted to.
280 A Survey of Symbolic Logic
In this part of their work, Peirce and Schroder can hardly be said to have
formulated the assumptions or given the proofs.
In the second place, the Boole-Schroder Algebra — the general outline
of which is already present in Peirce 's work — probably seemed to Peirce
and Schroder an adequate calculus of classes (though there are indications
in the paper of 1880 that Peirce felt its defects). With this system before
them, they neglected the possibility of a better procedure, by beginning
with the calculus of propositions and deriving the logic of classes from the
laws which govern propositional functions. And although the principles
which they formulate for propositional functions are as applicable to func
tions of one as of two variables, and are given for one as well as for two,
their interest was almost entirely in functions of two and the calculus of
relatives which may be derived from such functions. The logic of classes
which we have outlined is, then, something which they laid the foundation
for, but did not develop.
The main purposes of our exposition thus far in the chapter have been
two: first, to make clear the relation of this earlier treatment of symbolic
logic with the later and better treatment to be discussed in this section;
and second, to present the logic of propositional functions and their deriva
tives in a form somewhat simpler and more easily intelligible than it might
otherwise be. The theoretically sounder and more adequate logic of Prin-
cipia Mathematica is given a form which — so far as propositional functions
and their derivatives is concerned — seems to us to obscure, by its notation,
the obvious and helpful mathematical analogies, and requires a style of
proof which is much less obvious. With regard to this second purpose, we
disclaim any idea that the development we have given is theoretically
adequate ; its chief value should be that of an introductory study, prepara
tory to the more complex and difficult treatment which obviates the the
oretical shortcomings.
Incidentally, the exposition which has been given will serve to indicate
how much we are indebted, for the recent development of our subject, to
the earlier work of Peirce and Schroder.
The Peirce-Schroder symbolic logic is closely related to the logic of
Peano's Formulaire de Mathematiques and of Principia Mathematica. This
connection is easily overlooked by the student, with the result that the sub
ject of his first studies — the Boole-Schroder Algebra and its applications — is
likely to seem quite unrelated to the topic which later interests him — the
logistic development of mathematics. Both the connections of these two
Systems Based on Material Implication 281
and their differences are important. We shall attempt to point out both.
And because, for one reason, clearness requires that we stick to a single
illustration, our comparison will be between the content of preceding
sections of this chapter and the mathematical logic of Book I, Principia
Mathematical
The Two- Valued Algebra is a calculus produced by adding to and re
interpreting an algebra intended primarily to deal with the relations of
classes. And it has several defects which reflect this origin. In the first
place, the same logical relation is expressed, in this system, in two different
ways. We have, for example, the proposition "If peg and q c r, then
per", where p, q, and r are propositions. But "if . . . , then ..."
is supposed to be the same relation which is expressed by c in p c q, q c r,
and per. Also, "and " in " p c q and q c r " is the relation which is other
wise expressed by x — and so on, for the other logical relations. The
system involves the use of "if . . . , then . . . ", "... and . . .",
"either . . . or . . .", "... is equivalent to . . .", and "... is not
equivalent to . . . ", just as any mathematical system may; yet these are
exactly the relations c , x , + , = , and =H whose properties are supposed to
be investigated in the system. Thus the system takes the laics of the logical
relations of propositions for granted in order to prove them. Xor is this
paradox removed by the fact that we can demonstrate the interchange-
ability of "if . . . , then ..." and c, of ". . . and ..." and x, etc.
For the very demonstration of this interchangeability takes for granted
the logic of propositions; and furthermore, in the system as developed,
it is impossible in most cases to give a law the completely symbolic form
until it has first been proved in the form which involves the non-symbolic
expression of relations. So that there is no way in which the circularity
in the demonstration of the laws of propositions can be removed in this
system.
Another defect of the Two- Valued Algebra is the redundance of forms.
The proposition p or "p is true" is symbolized by p, by p = 1, by p 4= ^»
23 Logically, as well as historically, the method of Peano's Formulaire is a sort of
intermediary between the Peirce-Schroder mode of procedure and Principia. The general
method of analysis and much of the notation follows that of the Formulaire. But the
Formulaire is somewhat less concerned with the extreme of logical rigor, and somewhat
more concerned with the detail of the various branches of mathematics. Perhaps for this
reason, it lacks that detailed examination and analysis of fundamentals which is the dis
tinguishing characteristic of Principia. For example, the Formulaire retains the ambiguity
of the relation D (in our notation, c ): p?q may be either ''the class p is contained in
the class q", or "the proposition p implies the proposition q". In consequence, the Formu
laire contains no specific theory of propositions.
282 A Survey of Symbolic Logic
etc., the negation of p or "p is false" by -p, p = 0, -p = 1, p =}= 1, etc.
These various forms may, it is true, be reduced in number; p and -p may
be made to do service for all their various equivalents. But these equivalents
cannot be banished, for in the proofs it is necessary to make use of the fact
that p = (p = 1) = (p 4= 0), -p = (p = 0) = (-p = 1), etc., in order to
demonstrate the theorems. Hence this redundance is not altogether
avoidable.
Both these defects are removed by the procedure adopted for the
calculus of propositions in Principia Mathematical Here p = 1, p = 0,
etc., are not used; instead we have simply p and its negative, symbolized
by ~p. And, impossible as it may seem, the logic of propositions which
every mathematical system has always taken for granted is not presumed.
The primitive ideas are: (1) elementary propositions, (2) elementary
propositional functions, (3) assertion, (4) assertion of a prepositional func
tion, (5) negation, (6) disjunction, or the logical sum; and finally, the
idea of "equivalent by definition", which does not belong in the system
but is merely a notation to indicate that one symbol or complex of symbols
may be replaced by another. An elementary proposition is one which
does not involve any variables, and an elementary propositional function
is such as "not-p" where p is an undetermined elementary proposition.
The idea of assertion is just what would be supposed — a proposition may
be asserted or merely considered. The sign h prefaces all propositions
which are asserted. An asserted propositional function is such as "A is A "
where A is undetermined. The disjunction of p and q is symbolized by
p v q, instead of p + q. pv q means " At least one of the two propositions,
p and q, is true".
The postulates and definitions are as follows:
#1-01 poq. = m~pvq. Df.
"p (materially) implies q" is the defined equivalent of "At least one of
the two, (p is false' and 'q is true', is a true proposition". (The explana
tion of propositions here is ours.) p^q is the same relation which we
have symbolized by p c q, not its converse.
(The propositions quoted will be given the number which they have in
Principia. The asterisk which precedes the number will distinguish
them from our propositions in earlier chapters or earlier sections of this
chapter.)
The logical product of p and q is symbolized by p q, or p • q>
24 See Bk. i, Sect. A.
Systems Based on Material Implication 283
*3-01 p.q. = - ~(~p v~ry). I)f.
"p is true and g is true" is the defined equivalent of "It is false that at
least one of the two, p and </, is false". This is, of course, a form of De
Morgan's Theorem — in our notation, (p q) = -(-/> + -(/).
The (material) equivalence of p and q is symbolized by p = q or p . = . r/.
#4-01 p = q . = ,p?q . q^p. Df .
"p is (materially) equivalent to q" is the defined equivalent of "p (ma
terially) implies q and g (materially) implies p". In our notation, this
would be (p = q) = (pcq)(qcp). Note that ... == ... and . . .
Df are different relations in Principia.
The dots in these definitions serve as punctuation in place of parentheses
and brackets. Two dots, :, takes precedence over one, as a bracket over a
parenthesis, three over two, etc. In #4-01 we have only one dot after =,
because the dot between p D q and q D p indicates a product : a dot, or two
dots, indicating a product is always inferior to a stop indicated by the
same number of dots but not indicating a product.
The postulates of the system in question are as follows:
*!•! Anything implied by a true elementary proposition is true. Pp.
("Pp." stands for "Primitive proposition".)
*1-11 When <px can be asserted, where x is a real variable, and <pxo\f/x
can be asserted, where x is a real variable, then #r can be asserted, where x
is a real variable. Pp.
A "real variable" is such as p in -p.
#1-2 \-:pvp.?.p. Pp.
In our notation, (p + p} c p.
*l-3 \-lq*-D.pvq. Pp.
In our notation, q c (p + q).
*l-4 \-mpvqmOmqvp. Pp.
In our notation, (p + q) c(q + p}.
*l-5 \-:p v(q vr) .3. q v (p vr). Pp.
In our notation, [p + (q + r)] c [q + (p + r)].
#1-6 h! • <7 ^ r . ^ : p v q . D . p v r. Pp.
In our notation, (</ c r) c [Q; + (/) c Q; + r)].
Note that the sign of assertion in each of the above is followed by a
284 A Survey of Symbolic Logic
sufficient number of dots to indicate that the whole of what follows is
asserted.
$K-l-7 If p is an elementary proposition, ~p is an elementary proposition.
PP.
*1-71 If p and q are elementary propositions, pvq is an elementary
proposition. Pp.
#1-72 If <pp and \I/p are elementary prepositional functions which take
elementary propositions as arguments, <pp v \f/p is an elementary proposi-
tional function. Pp.
This completes the list of assumptions. The last three have to do
directly with the method by which the system is developed. By *l-7,
any proposition which is assumed or proved for p may also be asserted to
hold for ~p, that is to say, ~p may be substituted for p or q or r, etc., in
any proposition of the system. By #1-71, p v q may be substituted for p
or q or r, etc. And by *1 • 72, if any two complexes of the foregoing symbols
which make sense as " statements" can be treated in a certain way in the
system, their disjunction can be similarly treated. By the use of all three
of these, any combination such as p v q, p • q, p 3 q, p = q, p i> q . q D p,
~p . v . p v q, ~p v ~ry, etc., etc., may be substituted for p or q or r in any
assumed proposition or any theorem. Such substitution, for which no
postulates would ordinarily be stated, is one of the fundamental operations
by which the system is developed.
Another kind of substitution which is fundamental is the substitution
for any complex of symbols of its defined equivalent, where such exists.
This operation is covered by the meaning assigned to "... = ... Df ".
Only one other operation is used in the development of this calculus
of elementary propositions — the operation for which *!•! and ^1-11 are
assumed. If by such substitutions as have just been explained there
results a complex of symbols in which the main, or asserted, relation is o ,
and if that part of the expression which precedes this sign is identical with a
postulate or previous theorem, then that part of the expression which
follows this sign may be asserted as a lemma or new theorem. In other
words, a main, or asserted, sign D has, by *!•! and *1-11, the significant
property of "If . . . , then . . .". This property is explicitly assumed
in the postulates. The main thing to be noted about this operation of
inference is that it is not so much a piece of reasoning as a mechanical, or
strictly mathematical, operation for which a rule has been given. No
Systems Based on Material Implication 285
"mental" operation is involved except that required to recognize a previous
proposition followed by the main implication sign, and to set off what
follows that sign as a new assertion. The use of this operation does not,
then, mean that the processes and principles of ordinary logic are tacitly
presupposed as warrant for the operations which give proof.
What is the significance of this assumption of the obvious in # 1 • 1, # 1 - 1 1,
*l-7, #1-71, and *l-72? Precisely this: these postulates explicitly
assume so much of the logical operations as is necessary to develop the
system, and beyond this the logic of propositions simply is not assumed.
To illustrate this fact, it will be well to consider carefully an exemplary
proof or two.
#2-01 \-ipD~p.o.~p
Dem. Taut - M-: ~p v ~p . D . ~p (1)
01)] Vip?~p.?.~p
"Taut" is the abbreviation for the Principle of Tautology, #1-2 above.
~plp indicates that ~p is substituted in this postulate for p} giving (1).
This operation is valid by #1-7. Then by the definition #1-01, above,
p D ~/j is substituted for its defined equivalent, ~p v ~p, and the proof is
complete.
*2-05 \-mmqDrm3ipoqm3mp3r
Dem. Sum - M-: • q => r . D : ~p v q . D . ~p v r (1)
[(1) . O1-01)] \-: . qor .D: pDq .? . p?r
Here "Sum" refers to *l-6, above. And (1) is what *!•(> becomes when
~p is substituted for p. Then, by #1-01, p^q and pir are substituted
for their defined equivalents, ~pvq and ~pvr, in (1), and the resulting
expression is the theorem to be proved.
The next proof illustrates the use of *1 • 1 and #1-11.
*2 • 06 \-l ."3 m"3l -zr ."3 •'Dr
Dem.
p,
Dr.D:pDg.D.pDr:.3:«p3g'.3Sg'Dr«D.p3r (1)
[#2 -05] hs • £ 3 r . D : p 3 # . s . p a r (2)
[(1) . (2) . *1-11] h: - />3r/.D: q?r .0. por
286 A Survey of Symbolic Logic
"Comm " is *2-04, previously proved, which is p . D . q D r : D : q D . p D r.
When, in this theorem, q D r is substituted for p, p D q for q, and p D r for r,
it becomes the long expression (1). Such substitutions are valid by *l-7,
#1-71, and the definition ^1-01: if p is a proposition, ~p is a proposition;
if ~p and q are propositions, ~p v q is a proposition; and p D ry is the defined
equivalent of ~p v q. Thus p D q can be substituted for p. If we replace
the dots by parentheses, etc., (1) becomes
h { (q D r) 3 [(p D </) D (/> D r)] } 3 { (p D ry) D [(ry D r) D (p D r)] }
But, as (2) states, what here precedes the main implication sign is identical
with a previous theorem, *2-05. Hence, by #1-11, what follows this
main implication sign — the theorem to be proved — can be asserted.
Further proofs would, naturally, be more complicated, but they involve
no principle not exemplified in the above. These three operations — sub
stitutions according to *l-7, *1-71, and *l-72; substitution of defined
equivalents; and "inference" according to #1-1 and #1-11 — are the only
processes which ever enter into any demonstration in the logic of Principia.
The result is that this development avoids the paradox of taking the logic
of propositions for granted in order to prove it. Nothing of the sort is
assumed except these explicitly stated postulates whose use we have ob
served. And it results from this mode of development that the system is
completely symbolic, except for a few postulates, #1 • 1, #1-7, etc., involving
no further use of "if . . . , then . . .", "either ... or ...","... and
" etc
• • • j \^ M .
We have now seen that the calculus of propositions in Principia Mathe-
matica avoids both the defects of the Two-Valued Algebra. The further
comparison of the two systems can be made in a sentence : Except for the
absence, in the logic of Principia, of the redundance of forms, p, p = 1,
p =|= 0, etc., etc., and the absence of the entities 0 and 1, the two systems
are identical. Any theorem of this part of Principia can be translated
into a valid theorem of the Two-Valued Algebra, and any theorem of the
Two-Valued Algebra not involving 0 and 1 otherwise than as {=0} or
{ = 1 J can be translated into a valid theorem of Principia. In fact, the
qualification is not particularly significant, because any use of 0 and 1 in
the Two-Valued Algebra reduces to their use as { = 0 } and { = 1 ) . For 0
as a term of a sum, and 1 as a factor, immediately disappear, while the
presence of 0 as a factor and the presence of 1 in a sum can always be other
wise expressed. But p = 0 is -p, and p = 1 is p. Hence the two systems
Systems Based on Material Implication 287
are simply identical so far as the logical significance of the propositions
they contain is concerned.25
The comparison of our treatment of propositional functions with the
same topic in Principia is not quite so simple.28
In the first place, there is, in Principia, the "theory of types," which
concerns the range of significance of functions. But we shall omit con
sideration of this. Then, there are the differences of notation. Where
we write U<px, or Hx(px, Principia has (or) . <px\ and where we write S#r,
or 2x<px, Principia has (Kx) . <px. A further and more important difference
may be made clear by citing the assumptions of Principia.
*9-01 -{(.?) .?&}. = . (3.r) . ~#r. Df.
*9-02 ~{(3a:) .?&}. = . fa) . ~<px. Df.
*9-03 (x) . <px .vp : = . (x) . <px vp. Df.
25 This may be proved by noting that, properly translated, the postulates of each system
are contained amongst the propositions of the other. Of the postulates in Principia,
rendered in our notation:
^1-01 is (peg) = (-p + q), which is contained in our theorem 9-3.
^•1 -2 is (p + p) cp, which is a consequence of our theorems 2-2 and 5-33.
•sfcl -3 is, p c (p + q), which is our theorem 5-21.
^•1 -4 is (p + q) c(q + p), which follows from our theorem 4-3, by 2-2.
^•1-5 is, p + (</ + r) cq + (p + ?•), which is a consequence of our theorems 4-3 and 4-4,
by 2-2.
^el-6 is (q cr) c [(p + q) c(p + r)], which is a consequence of our theorem 5-31, by 2-2.
The remaining (non-symbolic) postulates are tacitly assumed in our system.
Of our postulates, 1-1-1-9 in Chap, n and 9-01 in Chap, iv:
•1 is a consequence of ^1-7 and ^1-71 in Principia.
•2 is -^4 -24 in Principia.
•3 is ^-4-3 in Principia.
•4 is ^2-3 in Principia.
•5 is equivalent to "If x = 0, then a x = 0", hence to -x c -(a x), which is a consequence
of *3-27 in Principia, by *2-16.
1-61, in the form -(x -a) c (x a = x), is a consequence of -#-4-71 and ^-4-61 in Principia,
by ^4-01 and *3-26.
1-62, in the form [(y a = y)(y -a = y)]c-y, is a consequence of #4-71, *5-16, and
^•2-21 in Principia.
1-7 is equivalent to [(a; = 1)Q/ = 0)] c (x = -?/), hence to (x -y) c (x =•• -y), which is an
immediate consequence of ^-5-1 in Principia.
1-8 is ^K-4-57 in Principia.
1-9 is ^4-71 in Principia.
9-01 is equivalent to (q = 1) c [p = (p = q)}, hence to q c [p = (p = q)], which is an
immediate consequence of -^5-501 in Principia.
26 See Principia, i, 15-21.
288 A Survey of Symbolic Logic
In this last, note the difference in the scope of the "quantifier" (x) on the
two sides. If the dots be replaced by parentheses, *9 • 03 will be
{[(x) . ^r] vp] = {(x) . [tpxyp]}
A similar difference in the scope of (x) or (3.r) on the two sides characterizes
each of the further definitions.27
*9-04 p . v . (x) . <px : = . (x) . p v <px. Df.
*9 • 05 (3.r) . <px . v . p : = . (3.r) . <px v p. Df .
#9 -OG p *** ("3.x) . <px : = . (3z) . p v <px. Df.
*9-07 (.r) . <px mvm (3#) . ^ : = : (#) : (3#) . ^ v ^. Df.
*9 • OS (3y) . ^ . v . (.I-) . ^r : = : (.T) : (3y) . $y v «^.r . Df .
Besides these definitions, there are four postulates (in addition to those
which underlie the calculus of elementary propositions).
*9 • 1 h: <?x . D . (3 z) . <pz. Pp.
*9 • 1 1 hs <f>x v «^?/ . D . (3 2) . <pZ. Pp.
*9- 12 What is implied by a true premiss is true. Pp.
#9-13 In any assertion containing a real variable, this real variable may
be turned into an apparent variable for which all possible values are asserted
to satisfy the function in question. Pp.
By our method, every one of these assumptions, except ^9-12, is a
proved proposition. In our notation,
#9-01 is -H<px = S -<px, which is our theorem 10-1, with -<px substituted
for <px.
*9-02 is -2^.i- = n-^.r, which is our theorem 10-12, with -<?x substi
tuted for <px.
*9-03 is tt<px + P = nx(^c + P), which is our theorem 10-32.
*9-04 is P + n^r = IIX(P+ <px), which is 10-33.
*9-05 is S^ + P = 2x(^-c + P), which is 10-3.
*9-06 is P+ S^r - 2X(P+ <px), which is 10-31.
*9-07 is U<px + Z^y = Ux2y(<px + $y), which is contained in 12-5,
*9-08 is 2ty+tt<px = Ux2y(ty + «^.r), which is also contained in 12-5.
The postulates require explanation. The authors of Principia use
<py, <pz, etc., to represent values of the function <px. In other words, where
we have written <pxn they simply change the letter. This is a valid con-
"Ibid., i, 135-38.
Systems Based on Material Implication 289
vention (though it often renders proofs confusing) because the range of <px
is determined by <p, not by x, and x is— conventions aside— indifferent.
z in <pz, where we should write <pxn, is called a "real variable", x in (x) . <px
and (3or) . <px, an "apparent variable". With this explanation, it is clear
that:
*9-l is <pxnc2<px, which is 10-21.
#9-11 is <pxm + <pxnc2<px, which is an immediate consequence of 10-21,
by 5-33.
*9-13 is "If whatever value of x, in <px, xn may be, <pxn, then n^.r," and
this implication is contained in the equivalence stated by 10-23.
These principles which are assumed in Principia Mathematica are suf
ficient to give all further propositions concerning functions of one variable,
without assuming (x) . <*r to be the product of <px^ <px2, etc. (or <py, <pz,
etc.), (3a-) . <px to be the sum of <pxlf <f>x2, etc. These are simply assumed
as new primitive ideas, (x) . <?x meaning " <px for all values of x", (3or) . <px
meaning " <px for some values of a:". This procedure obviates all questions
about the number of values of x in <px — which troubled us — and secures
the universality of theorems involving prepositional functions without any
discussion or convention covering the cases in which the values of the vari
able are infinite in number. The proofs in Principia reflect this difference
of method. They are, in general, what ours might have been if we had
based all further proofs directly upon 10-23 and the propositions con
necting 2<f>x + P with 2x(<px + P), etc., not making any use, after 10-23,
of the properties of TLtpx as a product, or of 2<^.r as a sum.
The theory of functions of two variables, in Principia Mathematica,
requires two further assumptions:
*11 -01 (x, y) . <p(x, y) . = : (x) : (y) . <p(x, y). Df.
*11 -03 (3z, y) . <p(x, y) . = : (3.x) : (30) . <p(x, y). Df.
These are identically our assumptions:
11-06 nx, y<p(x, y) == Hxllv<f>(x, y), and
11-05 2XlV<p(x,y) = SxS^fey).
The difference between the treatment of propositional functions which
we have given and the treatment in Principia is not necessarily correlated
with the difference between our treatment of propositions and theirs. The
method by which we have developed the theory of propositional functions
20
290 A Survey of Symbolic Logic
might exactly as well have been based upon the calculus of elementary
propositions in Principia as upon the Two-Valued Algebra. A few minor
alterations would be sufficient for this change. The different procedure
for propositional functions, in the twro cases, is a difference to be adjudged
independently, without necessary reference to the defects of the Two-
Valued Algebra which have been pointed out.
Beyond the important differences wrhich have been mentioned, there
are minor and trivial divergences between the two systems, due to the
different use of notation. Neglecting these, we may say that the two
methods give the same results, with the following exceptions:
1. There are certain complexities in Principia due to the theory of
types.
2. In Principia the conditions of significance are explicitly investigated.
3. Principia contains a theory of "descriptions", account of which is
here omitted.
But none of these exceptions is a necessary difference. They are due to
the more elementary character of our presentation of the subject. We
may, then, say loosely that the two methods give identical results.
The calculus of classes and of relations which we have outlined in the
preceding sections bear a similar relation to the logic of classes and of
relations in Principia; that is to say, there is much more detail and com
plexity of theory in Principia, but so far as our exposition goes, the two are
roughly the same. And here there is no important difference of method.
It should now be clear how the logic of Principia is related to the logic
we have presented, following in the main the methods of Peirce and Schroder.
There is much difference of method, and, especially in the case of the cal
culus of propositions, this difference is in favor of Principia. And in
Principia there is much more of theoretical rigor and consequent complexity:
also there are important extensions, especially in the theory of "descrip
tions" and the logic of relatives. But so far as the logic which we have
expounded goes, the two methods give roughly identical results- When
we remember the date of the work of Peirce and Schroder, it becomes clear
what is our debt to them for the better developments which have since
been made.
CHAPTER V
THE SYSTEM OF STRICT IMPLICATION1
The systems discussed in the last chapter were all based upon material
implication, p cq meaning exactly "The statement, * p is true and q false/
is a false statement". We have already called attention to the fact that
this is not the usual meaning of "implies". Its divergence from the
"implies" of ordinary inference is exhibited in such theorems as "A false
proposition implies any proposition", and "A true proposition is implied
by any proposition".2
The present chapter intends to present, in outline, a calculus of propo
sitions which is based upon an entirely different meaning of " implies "-
one more in accord with the customary uses of that relation in inference
and proof. We shall call it the system of Strict Implication. And we shall
refer to Material Implication, meaning either the Two-Valued Algebra or
the calculus of propositions as it appears in Principia Mathematica, since
the logical import of these two systems is identical. It will appear that
Strict Implication is neither a calculus of extensions, like Material Impli
cation and the Boole-Schroder Algebra, nor a calculus of intensions, like
the unsuccessful systems of Lambert and Castillon. It includes relations
of both types, but distinguishes them and shows their connections. Strict
Implication contains Material Implication, as it appears in Principia
Mathematica, as a partial-system, and it contains also a supplementary
partial-system the relations of which are those of intension.
The numerous questions concerning the exact significance of implication,
and the ordinary or "proper" meaning of "implies", will be discussed in
Section V.
It will be indicated how Strict Implication, by an extension to proposi-
tional functions, gives a calculus of classes and class-concepts which exhibits
their relations both in extension and in intension. In this, it provides the
1 Various studies toward this system have appeared in Mind and the Journal of Phi
losophy (see Bibliography). But the complete system has not previously been printed.
We here correct, also, certain errors of these earlier papers, most notably with reference to
triadic "strict" relations.
2 For further illustrations, see Chap, n, Sect, i, and Lewis, "Interesting Theorems in
Symbolic Logic," Jour. Philos., Psych., etc. x (1913), p. 239.
291
292 A Survey of Symbolic Logic
calculus of intensions, so often attempted before, so far as such a calculus
is possible at all.
I. PRIMITIVE IDEAS, PRIMITIVE PROPOSITIONS, AND IMMEDIATE CONSE
QUENCES
The fundamental ideas of the system are similar to those of MacColl's
Symbolic Logic and its Applications. They are as follows:
1. Propositions: p, q, r, etc.
2. Negation: -p, meaning "p is false".
3. Impossibility: ~p, meaning "p is impossible", or "It is impossible
that p be true".3
4. The logical product: p xg or p q, meaning "p and q both", or "p is
true and q is true".
5. Equivalence: p = q, the defining relation.
Systems previously developed, except MacColl's, have only two truth-
values, "true" and "false". The addition of the idea of impossibility
gives us five truth- values, all of which are familiar logical ideas:
(1) p, "pis true".
(2) -p, "pis false".
(3) ~p, "p is impossible".
(4) — p, "It is false that p is impossible" — i. e., "p is possible".
(5) — p, "It is impossible that p be false" — i. e., "p is necessarily
true".
Strictly, the last two should be written -(~p) and -(-p): the parentheses
are regularly omitted for typographical reasons.
The reader need be at no pains to grasp -~p and — p as simple ideas:
it is sufficient to understand -p and ~p, and to remember that each such
prefix affects the letter as already modified by those nearer it. It should
be noted that there are also more complex truth-values. — p is equivalent
to p, as will be shown, but - ~ -p, p, p, etc., are irreducible.
We shall have occasion to make use of only one of these, p, " It is
false that it is impossible that p be true" — i. e., "p is possibly false".4 ,
Each one of these complex truth-values is a distinct and recognizable idea,
though they are seldom needed in logic or in mathematics.
3 We here use a symbol, ~, which appears in Prindpia Mathematica with a different
meaning. The excuse for this is its typographical convenience.
4 MacColl uses a single symbol for -~p, "p is possibly true" and p, "p is possibly
false".
The System of Strict Implication 293
The dyadic relations of propositions can be defined in terms of these
truth- values and the logical product, p q?
1-01 Consistency. poq = -~(pq}. Def.
~(pq), "It is impossible that p and q both be true" would be "p and q
are inconsistent". Hence -~(pq), "It is possible that p and q both be
true", represents "p and q are consistent".
1-02 Strict Implication. p-*q = ~(p-q)- Def.
1-03 Material Implication. pcq = -(p-q). Def.
1-04 Strict Logical Sum. p *q = ~(-p-r/). Def.
1-05 Material Logical Sum. p + q = -(-p -q)- Def.
1-06 Strict Equivalence, (p = q) = (p •* q) (q -* p) . Def.
We here define the defining relation itself, because by this procedure we
establish the connection between strict equivalence and strict implication.
Also, this definition makes it possible to deduce expressions of the type,
p = q — something which could not otherwise be done.5 But p = q re
mains a primitive idea as the idea that one set of symbols may be replaced
by another.
1 • 07 Material Equivalence, (p = q) = (p c q) (q c p). Def.
These eight relations — the seven defined above and the primitive rela
tion, p q— divide into two sets, p q, p c q, p + q, and p = q are the relations
which figure in any calculus of Material Implication. We shall refer to
them as the "material relations", p o q, p -i q, p*q, and p = q involve
the idea of impossibility, and do not belong to systems of Material Impli
cation. These may be called the " strict relations ". We may anticipate a
little and exhibit the analogy of these two sets, which results from the
theorem
~(pq) = -(p°ti
shortly to be proved.
Strict relations : Material relations :
piq =-(pO-q) pcq = -(p -q)
pAq = _(_po-<7) P+<1 = -(-P-0)
(p = q) = -(po-q) x-(qo-p) (p = q) = ~(p -tf) *-(cl~P">
• The "circularity" here belongs inevitably to logic. No mathematician hesitates to
prove the equivalence of two propositions by showing that " If theorem A, then theorem B,
and if theorem B, then theorem A". But to do this he must already know that a reciprc
"if . . . then . . ." relation is equivalent to an equivalence. And the italicized
to" represents a relation which must be assumed.
294 A Survey of Symbolic Logic
The reader will, very likely, have some difficulty in distinguishing in meaning
p -J q from p c q, p A q from p + q. The above comparison may be of assist
ance in this connection, since it translates these relations in terms of p o q
and p q. We shall be in no danger of confusing p o q, "p is consistent with
g," with p q, "p and q are both true".
Both p A q and p + q would be read "Either p or q". But p A q denotes
a necessary connection; p + q a merely factual one. Let p represent " To
day is Monday", and q, "2 + 2 = 4:". Then p+q is true but p A q is
false. In point of fact, at least one of the two propositions, "Today is
Monday" and "2 + 2 = 4", is true; but there is no necessary connection
between them. "Either . . . or . . ." is ambiguous in this respect. Ask
the members of any company whether the proposition "Either today is
Monday or 2 + 2 = 4" is true, and they will disagree. Some will confine
"Either ... or ..." to the p*q meaning, others will make it include
the p + q meaning; few, or none, will make the necessary distinction.
Similarly, the difference between p = q and p = q is that p = q denotes
an equivalence of logical import or meaning, while p = q denotes simply
an equivalence of truth-value. As was shown in Chapter II, p = q may be
accurately rendered "p and q are both true or both false". Here again,
the strict relation, p = q, symbolizes a necessary connection; the material
relation, p = g, a merely factual one.
The postulates of the system are as follows :
1-1 p q-i q p
If p and q are both true, then q and p are both true.
1-2 q p -I p
If q and p are both true, then p is true.
1- 3 p -*p p
If p is true, then p is true and p is true.
1-4 p(qr)4q(pr)
If p is true and q and r are both true, then q is true and p and r are both
true.
1-5 p-i-(-p)
If p is true, then it is false that p is false.
1-6 (p-tq)(q-lr)-l(p-lr)
If p strictly implies q and q strictly implies r, then p strictly implies r.
The System of Strict Implication 295
1 • 7 ~p-i-p
If it is impossible that p be true, then p is false.
1-8 p -J q = ~q -J ~p
"p strictly implies q" is equivalent to "'q is impossible' strictly im
plies 'p is impossible'".
The first six of these present no novelty except the relation -J . They
do not, so far, distinguish this system from Material Implication. But,
as we shall see shortly, the postulates 1 • 7 and 1-8 are principles of trans
formation; they operate upon the other postulates, and on themselves,
and thus introduce the distinguishing characteristics of the system. Postu
late 1 -7 is obvious enough. Postulate 1 -8 is equivalent to the pair,
(p-iq) -i (- ~p -J - ~q) If p implies q, then ' p is possible ' implies
' q is possible'.
(~p -J ~q) -J (-p -J -q) If 'p is impossible' implies 'q is impossible',
then (p is false' implies 'q is false'.
These two propositions are more "self-evident" than the postulate, but
they express exactly the same relations.
(To eliminate parentheses, as far as possible, we make the convention
that the sign =, unless in parentheses, takes precedence over any other
relation ; that -1 and c take precedence over A , + , o , and x ; that A
and + take precedence over o and x ; and that -J takes precedence over
c . Thus
pq + -p -q -ipcq is [(p q) + (-p -</)] -* (p c q)
and p c q r = (p c q)(p c r) is [pc.(q r)] = [(p c q)(p c r)]
However, where there is a possibility of confusion, we shall put in the
parentheses.)
The operations by which theorems are to be derived from the postulates
are three:
1. Substitution.— Any proposition may be substituted for p or q or r,
etc. If p is a proposition, -p and ~p are propositions. If p and q are
propositions, p q is a proposition. Also, of any pair of expressions related
by = , either may be substituted for the other.
2. Inference. — If p is asserted and p -J q is asserted, then q may be
asserted. (Note that this operation is not assumed for material impli
cation, p c q.)
3. Production— It p and q are separately asserted, p q may be asserted.
These are the only operations made use of in proof.
296 A Survey of Symbolic Logic
In order to make clearer the nature of the strict relations, and particu
larly strict implication, we shall wish to derive from the postulates their
correlates in terms of strict relations. This can be done by the use of
postulate 1 • 8 and its consequences, for by 1-8 a relation of two material
relations can be transformed into a relation of the corresponding strict
relations. But as a preliminary to exhibiting this analogy, we must prove a
number of simple but fundamental theorems. These working principles
will constitute the remainder of this section.
The first theorem will be proved in full and the proof explained. The
conventions exemplified in this proof are used throughout.
2-1 pq-tp
1-6 [pq/p; qp/q; p/r}: 1-1 xl-2-i (p q*p)
This proof may be read: "Proposition 1-6, when p q is substituted for p,
qp for q, and p for r, states that propositions 1-1 and 1-2 together imply
(P q •* P) "• The number of the proposition which states any line of proof
is given at the beginning of the line. Next, in braces, is indication of any
substitutions to be made, "p q/p" indicates that p q is to be substituted,
in the proposition cited, for p; "p+ q/r" would indicate that p+ q was to
be substituted for r, etc. Suppose we take proposition 1 • 6, which is
and make the substitutions indicated by {p q/p; qp/p; p/r}. We then
get
(pq-lqp)(qp-lp) H (pqlp).
This is the expression which follows the brace in the above proof. But
since p q -l q p is 1-1, and q p -J p is 1-2, we write 1-1 x 1 • 2 instead of
(p q-iqp)(qplp). This calls attention to the fact that what precedes
the main implication sign is the product of two previous propositions.
Since 1-1 and 1-2 are separately asserted, their product may be asserted;
and since this product may be asserted, what it implies — the theorem to be
proved — may be asserted. The advantage of this way of writing the proofs
is its extreme brevity. Yet anyone who wishes to reconstruct the demon
stration finds here everything essential.
2-11 (p = q)4(p4q)
2-1 {p-lq/p; q-ip/q}: (p * q) (q •* p) 4 (p •* q)
1-06: (p = q) = (p
The System of Strict Implication 297
2-12 (p = (/HfeHp)
Similar proof, 1-2 instead of 2-1.
2-2 (p-i q) -J (~g -J ~p)
1-06: 1-8 = [(p4q)4(~q4~p)}[(~q*~P}*(P*<l)} (l)
2-1: (1) H Q.E.D.
In this last proof, we introduce further abbreviations of proof as follows:
(1), or (2), etc., is placed after a lemma which has been established, and
thereafter in the same proof we write (1), or (2), etc., instead of that lemma.
Also, we shall frequently write "Q.E.D." in the last line of proof instead
of repeating the theorem to be proved. In the first line of this proof, the
substitutions which it is necessary to make in order to get
1 -8 = [(p •* q) * (~q -* ~p)][(~<7 H ~p) -J (P •< </)]
are not indicated because they are obvious. And in the second line, state
ment of the required substitutions is omitted for the same reason. Such
abbreviations will be used frequently in later proofs.
Theorem 2-2 is one of the implications contained in postulate 1-S.
By the definition, 1-6, any strict equivalence may be replaced by a pair of
strict implications. By postulate 1-2 and theorem 2-1, either of these
implications may be taken separately.
2-21 (~q*~p)4(p4q)
1 • 2 : [(1) in proof of 2 • 2] -J Q.E.D.
This is the other implication contained in postulate 1-8.
2-3 (-P * q) * (~q * P)
1-1 {-q/p', -p/q}: -q-p-*-p-q
2.2 {-g -p/p; -p -q/q} : (1) -J [~(-/> -<?) -* ~H "rfl
1-02: (2) = Q.E.D.
2-4 pip
1-2 {p/q}: pp-lp
1-G: l-3x(l)-{ Q.E.D.
2-5 -(-p)*p
2--i {-p/p}: -p*-p
2-3 {-p/q}: (1) -I Q.E.D.
2-51 -(-p} =p
1-06: 2-5x1-5 = Q.E.D.
298 A Survey of Symbolic Logic
2-6 (-p -i -g) -J (g -J p)
2-3 {-q/q}: (-p -J -g) -i [-(-q) H p] (1)
2-51: (1) = Q.E.D.
2-61 (p -j -g) -i (fy -j -p)
2-6 {-pip}: K-p) -* -gM (g •< -P) (1)
2-51: (1) = Q.E.D.
2-62 (p •« q) -I (-g •* -p)
2-61 {-q/q}: [p -i -(-g)] -J (-g H -p) (1)
2-51: (1) = Q.E.D.
2-6J (png) = H-t-p)
1-06: 2-62x2-6 = Q.E.D.
2-64 (p*-q) = (q*-p)
1-06: 2-61x2-3 = Q.E.D.
Theorems 2-3, 2-6, 2-61, and 2-62 are the four forms of the familiar
principle that an implication is converted by changing the sign of both
terms.
2-7 (~p4~q}4(-p4-q)
2-21 [plq; q/p}: (~p -J ~g) -j (q * p) (1)
2-62 {p/q; q/p}: (q -J p) H (-p -t -q) (2)
1-6: (1) x (2) H Q.E.D.
2-71 (-p -J -q) -i (~p -J ~q)
2.6: (-p -i -?) H (g •« p) (1)
2-2 {^/p; p/gj: (g S p) -i (~jM ~g) (2)
1-6: (l)x(2)^Q.E.D.
(-p-t-q) = (~p*~q)
1-06: 2-71 x2-7 = Q.E.D.
2-7 {-p/p; -g/g} : (~ -p -J ~ -g) -J [-(-p) -J -(-g)] (1)
2-51: (1) = Q.E.D.
2-73 (p4q)-i(~-p-i~-q)
2-71 {-p/p', -q/q} : [-(-p) -i -(-g)] 4 (- -p s - -g) (1)
2-51: (1) = Q.E.D.
2-7J1 (p-ig) = (~-p^~-g)
1-06: 2-73x2-72 = Q.E.D.
2-74 (p^g)H(-^p^-^)
2-62 {-g/p; ~p/g} : (~g ^ -p) H (- ^p -i - ^g) (1)
1-6: 2-2x(l)^ Q.E.D.
The System of Strict Implication 299
2-75 (- ~p -i - ~g) -i (p H ry)
2-6 {~p/p; ~q/q] : (- ~p -J - ~q) -J (~g -I ~p) (1)
1-6: (1) x2-21-jQ.E.D.
2-76 (p-Jg) = (-T •!--</)
1-06: 2-74x2-75 - Q.E.D.
2-77 (p -i g) = (- ~p -i - ~g) - (~ -p -i ~ -g) = (-g -{ -p) = (~g -I ~p)
2-76 x2-731 x2-G3 x2-712 = Q.E.D.
"p implies g" is equivalent to "' p is possible' implies 'q is possible'" is
equivalent to "'p is necessary' implies 'q is necessary'" is equivalent to
"q is false' implies 'p is false'" is equivalent to "'g is impossible' implies
'p is impossible'".
2-6-2-77 are various principles for transforming a strict implication.
These are all summed up in 2 • 77. The importance of this theorem will be
illustrated shortly.
2-8 pq = qp
1-1 {g/p; pfq}: q p -I p q (1)
1-06: 1-1 x(l) - Q.E.D.
2-81 p = pp
1-2 {plq}: pplp (1)
1-06: 1-3 x(l) = Q.E.D.
2-9 p(qr) = q(p r)
1-4 {g/p; plq}: q(p r) -i p(q r) (1)
1-06: 1-4 x(l) = Q.E.D.
2-91 p(qr) = (p q)r
2-8: p(qr) = p(r q)
2-9: p(rq) = r(p q)
2-8: r(pq) = (p q)r
The above theorems constitute a preliminary set, sufficient to give
briefly most further proofs.
II. STRICT RELATIONS AND MATERIAL RELATIONS
We proceed now to exhibit a certain analogy between strict relations
and material relations; between truths and falsities on the one hand and
necessities, possibilities, and impossibilities on the other. This analogy
runs all through the system: it is exemplified by 2-77.
300 A Survey of Symbolic Logic
l-l pq-iqp 3-llpOq-iqOp
If p and q are both true, then q If p and q are consistent, then q
and p are both true. and p are consistent.
1-2 q pip 3-12 qop*-~p
If q and p are both true, then p It q and p are consistent, then
is true. it is possible that p be true.
1-3 p-tpp 3-13 -~p 4 pop
If p is true, then p and p are If it is possible that p be true,
both true. then p is consistent with itself.
1-4 p(qr)4q(pr) 3-14 p o (q r) 4 q o (p r)
The correspondence exhibited in the last line seems incomplete. But
we should note with care that while
p(q r) = q(p r) = (p q)r
and any one of these may be read " p, q, and r are all true", po(qor)
is not "p, q, and r are all consistent", p o (q or) means "p is consistent
with the proposition 'q is consistent with r'". Let p = "Today is Tues
day"; q = "Today is Thursday"; r = "Tomorrow is Friday". Then
qor is true. And it happens to be Tuesday, so p is true. Since p and
q or are both true in this case, they must be consistent: p o (q or) is true.
But "p, q, and r are all consistent" is false. "Today is Tuesday" is incon
sistent with "Today is Thursday" and with "Tomorrow is Friday".
Suppose we represent "p, q, and r are all consistent" by p o q o r. Then
as a fact, p o q o r will not be equivalent to p o (q or). Instead, we shall
have
p o q or = p o (qr) = q o (p r) = (p q) or
"p, q, and r are all consistent" is equivalent to " p is consistent with the
proposition 'q and r are both true'", etc. We may, then, add two new
definitions:
3-01 pqr = p(qr). Def.
3 - 02 poqor = po(qr). Def.
3-02 is typical of triadic, or polyadic, strict relations: when parentheses
are introduced into them, the relation inside the parentheses degenerates
into the corresponding material relation. In terms of the new notation of
3-01 and 3-02, the last line of the above table would be
p q r -i q p r poqorlqopor
which exhibits the analogy more clearly.
The System of Strict Implication 301
We must now prove the theorems in the right-hand column of the table
3-11 poq-iqop
2-74 Iqp/p', qp/q}: I • 1 -I [- ~(P q) -I - ~(q p}} (1)
1-01: (1) = Q.E.D.
3-12 qop-l-~p
2-74 (qp/p; p/q}: l-2*[- ~(qp) *- ~p] (1)
1-01: (1) = Q.E.D.
3-13 — p H p op
2-74 {pp/p}: l-3-J[-~iM-~(Pp)] (1)
1.01: (1) = Q.E.D.
3-14 p o(qr) -lqo(pr)
2-74 [p(qr)/p-, q(pr)/q}: 1 -4 -I - ~[p(r/ r)] -J - ~[?(p r)] (1)
1.01: (1) = Q.E.D.
(In the above proof, the whole of what 1 • 4 is stated to imply should
be enclosed in a brace. But in such cases, since no confusion will be oc
casioned thereby, we shall hereafter omit the brace.)
3-15 po (q r) = (pq) or = qo (p r)
2-76: 2-9H-~[p(<7r)] = - ~[q(P r)] (1)
2-76: 2-91-J-~[p(r/r)] - - ~[(p q)r] (2)
1-01: (2)x(l) = Q.E.D.
An exactly similar analogy holds between the material logical sum,
p + q, and the strict logical sum, p A q.
3-21 p + q-iq + p 3-31 p*(j-lqAp
"At least one of the two, p and "Necessarily either p or </" im-
q, is true" implies "At least one plies "Necessarily either q or p".
of the two, q and p, is true".
3-22 p*p + q 3-32 - -p -J p A q
If p is true, then at least one of If p is necessarily true, then
the two, p and q, is true. necessarily either p or q is true.
3-23 p + p + p 3-33 p*p4~-p
If at least one of the two, p and If necessarily either p is true or
p, is true, then p is true. p is true, then p is necessarily true.
3-24 p+(q + r) 4q+(p + r) 3-34 p A (0 + r) -J ? A (p + r)
As before, the analogy in the last line seems incomplete, and as before,
it really is complete. And the explanation is similar, p + (7 + r) and
302 A Survey of Symbolic Logic
q + (p + r) both mean "At least one of the three, p, q, and r, is true". But
p A (q A r) would not mean " One of the three, p, q, and r, is of necessity
true". Instead, it would mean "One of the two propositions, p and
'necessarily either q or r\ is necessarily true". To distinguish p*(q + r)
from p A (q A r) is rather difficult, and an illustration just now, before we
have discussed the case of implication, would probably confuse the reader.
We shall be content to appeal to his ' intuition' to confirm the fact that
"Necessarily one of the three, p, q, and r, is true" is equivalent to "Neces
sarily either p is true or one of the two, q and r, is true" — and this last is
p A (q + r). If we chose to make definitions here, similar to 3-01 and 3-02,
they would be
p + q + r = p + (q + r)
and p^qAr = p A (q + r)
Proof of the theorems in the above table is as follows :
3-21 p + q-iq + p
1-1 {-q/p; -p/q}: -q-pl-p-q (1)
2-62: (l)*-(-p-q)i-(-q-p) (2)
1-05: (2) = Q.E.D.
3 - 22 p-tp + q
Similar proof, using 1 -2 in place of 1-1.
3 - 2 3 p + p-ip
Similar proof, using 1 • 3.
3 - 24 p + (q + r) -i q + (p + r)
1-4 {-q/p; -p/q; -r/r}: -q(-p -r) -J -p(-q -r) (1)
2-62: (1) i-[-p(-q-r)] -j -[-g(-p -r)] (2)
2-51: (2) = -{-p-[-(-q-r)]} * -{-q -[-(-p -r)]} (3)
1 - 05 : (3) = p + -(-q -r) -J q + -(-p -r) (4)
1-05: (4) = Q.E.D.
3 - 25 p + (q + r) = (p + q) + r = q + (p + r)
Similar proof, using 2-9 and 2-91, and 1-06.
3-31 p A g -J q A p
1-1 {-q/p; -p/q}: -q-pi-p-q (1)
2-2: (l)4~(-p-q)4~(-q-p) (2)
1-04: (2) = Q.E.D.
3 - 32 ~ -p H p A q
Similar proof, using 1-2 in place of 1-1.
The System of Strict Implication
303
3 • 3 3 p A p -j — p
Similar proof, using 1-3.
3 - 34 p A (q + r) -J q A (p + ?•)
1-4 {-q/p; -p/q; -r/r}: -q(-p -r) -J -p(-q -r) (1)
2 - 2 : (1) -I ~[-p(-q -r)} -J ~[-q(-p -r)} (2)
2-51: (2) - H-p-hH-r)]} H-H+(-p-r)ll (3)
1 - 04 : (3) = p A -(-fy -r) ., q A -(-p _r) (4)
1-05: (4) = Q.E.D.
3 - 35 p A (q + r) = (p + q) A r = ry A (p + /•)
Similar proof, using 2-9 and 2-91, and 1-06.
Again, an exactly similar analogy holds between material implication,
peg, and strict implication, p H q.
3-41 (pcq)-t(-qc-p)
If p materially implies g, then ' q
is false' materially implies (p is
false'.
3-42 -p-*(pcq)
If p is false, then p materially
2-62 (p -I q) -J (-q -I -p)
If 2? strictly implies g, then * q is
false' strictly implies 'p is false'.
implies any proposition, q.
3-43 (pc-p)-i-p
If p materially implies its own
negation, then p is false.
3-44 ccr]^[c(cr)]
3-52 ~p-l(p-l0)
If p is impossible (not self-con
sistent, absurd), then p strictly im
plies any proposition, q.
3-53 (p-*-p)4~p
If p strictly implies its own nega
tion, then p is impossible (not self-
consistent, absurd).
3-54 [-t(cr)]4[*(pcr)}
The comparison of the last line presents peculiarities similar to those
noted in previous tables. The significance of 3-54 is a matter which can
be better discussed when we have derived other equivalents of p -J (q cr).
The matter will be taken up in detail further on.
The theorems of this last table, like those in previous tables, are got
by transforming the postulates 1-1, 1-2, 1-3, and 1-4. In consideration
of the importance of this comparison of the two kinds of implication, we
may add certain further theorems which are consequences of the above.
3-45 p-i(qcp) 3-55 ~-p-l(</-»p)
If p is true, then every proposi- If p is necessarily true, then p is
tion, q, materially implies p. strictly implied by any proposi
tion, q.
304 A Survey of Symbolic Logic
3-46 (-pep) -ip 3-56 (-pipH~-p
If p is materially implied by its If p is strictly implied by its own
own denial, then p is true. denial, then p is necessarily true.
3-47 -(pcq)-ip 3-57 -(p-ig)-i-~p
If p does not materially imply If p does not strictly imply any
any proposition, q, then p is true. proposition, q, then p is possible
(self-consistent) .
3-48 -(pcq)4-q 3-58 -(p •* q) -* q
If p does not materially imply q, If p does not strictly imply g,
then q is false. then p is possibly false (not neces
sarily true) .
Note that the main or asserted implication, which we have translated
" If . . . , then . . . ", is always a strict implication, in both columns.
3-42 and 3-45-3-48 are among the best known of the "peculiar" the
orems in the system of Material Implication. For this reason, their ana
logues in which the implication is strict deserve special attention. Let us
first note that ~ -p -J (-p -J p) is a special case of 3 • 55. This and 3 • 56 give
us at once
~ -P = (-P -J p)
This defines the idea of "necessity". A necessarily true proposition—
e. g., "I am", as conceived by Descartes — is one whose denial strictly
implies it. Similarly, p-l(-pcp) is a special case of 3-45. And this,
with 3-46, gives
p = (-p c p)
A true proposition is one which is materially implied by its own denial.
This point of comparison throws some light upon the two relations.
The negative of a necessary proposition is impossible or absurd.
~p -i (p -i -p) is a special case of 3-52. This, with 3-53, gives
-P = (P -* -P)
And p -i -p is equivalent to -(pop). Thus an impossible or absurd propo
sition is one which strictly implies its own denial and is not consistent
with itself. Correspondingly, we get from 3-42 and 3-43
-P = (P c -P)
A false proposition is one which materially implies its own negation.
It is obvious that material implication, as exhibited in these theorems,
The System of Strict Implication 305
is not the relation usually intended by "implies", but it may be debated
whether the corresponding properties of strict implication are altogether
acceptable. We shall revert to this question later. At least, these propo
sitions serve to define more sharply the nature of the two relations.
Proof of the above theorems is as follows:
3-41 (pcq) -i (-r/c-p)
1-1 [p/q; -q/p}: -q p 4 p -q (1)
2- 62: (1) 4 -(p -q) -J -(-q p) (2)
2-51: (2) =-(p-q)*-(-q-(-p)} (3)
1-03: (3) = Q.E.D.
3 • 42 -p 4 (p c q)
1-2 {p/q; -q/p}: p -q 4 p (1)
2-62: (1) 4 -p 4 -(p -q) (2)
1-03: (2) - Q.E.D.
3 • 43 (p c. -p) 4 -p
Similar proof, using 1 -3.
3 • 44 p c (q c r) 4 q c (p c r)
1-4 {q/p', p/q-, -r/r}: q(p -r) 4 p(q -r) (1)
2-62: (l)^-[p(g-r)]H-[g(p-r)] (2)
2-51: (2) = -{p-[-(g-r)]} 4-{g.[-(p-f)]} (3)
1 • 03 : (3) - 2J c -(ry -r) -I ry c -(/; -r) (4)
1-03: (4) - Q.E.D.
3 • 45 p4(qc.p)
3-42 {-p/p; -q/q] : -(-/>) -I (-/> c-f/) (1)
3.41: (_2;c_ry)H[-(-r/)c -(-;;)] (2)
2-51: (2) - (-pc-r/)-i(7Cp) (3)
1-6: (l)x(3)H-(-y;)-i(r/c^ (4)
2-51: (4) = Q.ED.
3 • 46 (-p c p) -j p
3-43 {-p/2;}: [-pc-(-p)H-C-p) (1^
2-51: (1) = Q.E.D.
3-47 ~(pcq)4p
2-62 {-p/p; pcq/q}: 3-42 4 -(p c q) 4 -(-p)
2-51: (1) = Q.E.D.
3-48 -(pcq)4-q
3-45 {^/p; p/^}: q4(pcq) (D
2-62: (1) -i Q.E.D.
21
306 A Survey of Symbolic Logic
3-52 ~p -J (p -J g)
2-1 i-q/q}: p-q*p (1)
1.8: (1) =~p-i~(p-g) (2)
1-02: (2) = Q.E.D.
3-53 (p-J-p)-J~p
Similar proof, using 1-3.
3-54 [p-*(gcr)H[g-J(pcr)]
1-4 [q/p; p/q; -r/r}: q(p -r) * p(q -r) (1)
1-8: (1) = ~[p(g-r)H~[g(p-r)] (2)
2-51: (2) - ~{p-[-(g-r)]H~{<H-(p-r)]) (3)
1 -02: (3) = [p S -(<? -r)] •* [g H -(p -r)] (4)
1-03: (4) = Q.E.D.
3 • 55 ~ -p •* (g -J p)
3-52 !-p/p; -g/g) : ~ -p -J (-p -^ -g) (1)
2-6: (-p -l -g) -l (g -l p) (2)
1-6: (1) x (2) -i Q.E.D.
3-56 (-p-lp)4~-p
3-53 {-p/p}: [-p-j-(-p)H~-:p (1)
2-51: (1) = Q.E.D.
3-57 -(p-ig)-*-~p
2-62: 3-52 -^ Q.E.D.
J-58 -(p-J g) -J g
2-62: 3-55 -^ Q.E.D.
The presence of this extended analogy between material relations and
strict relations in the system enables us to present the total character of
the system with reference to the principles of transformation, 1 • 7 and 1 • 8,
in brief and systematic form. This will be the topic of the next section.
III. THE TRANSFORMATION {-/~<
We have not, so far, considered any consequences of postulate 1 • 7,
-p -} -p, "If p is impossible, then p is false". They are rather obvious.
4-1 ~-p*p
1-7 {-pip}'. --P-I-(-P) (1)
2-51: (1) = Q.E.D.
If p is necessary, then p is true.
4-12 p-i-~p
2-61 {~p/p; p/q}: 1- 7 * Q.E.D.
If p is true, then p is possible.
The System of Strict Implication 307
4-13 ~-p-J-~p
1-6: 4-1 x4-12-iQ.E.D.
If p is necessary, then p is possible.
4-14 pq-ipoq
4-12 lpq/p}: pq*-~(pq) (1)
1-01: (1) = Q.E.D.
4-15 (p*q)4(pcq)
1-7 (p-q/p}: ~(p-q)*~(p-q) 0)
1-02: (1) = (piq)-i-(p-q) (2)
1-03: (2) = Q.E.D.
4-16 p*q-ip + q
1-7 {-p-q/p}: -(-p-gH-Hp-tf) (1)
1.Q4: (1) = p A </ -I -(-p -g) (2)
1-05: (2) - Q.E.D.
4-17 -(poq)4-(pq)
2-62: 4-14 = Q.E.D.
By virtue of theorem 4-15, any strict implication which is asserted—
i. e., is the main relation in the proposition — may be replaced by a material
implication. And by 4 -1C, any strict logical sum, A, which should be
asserted, may be reduced to the corresponding material relation, + . The
case of the strict relation " consistent with", o , is a little different. It
follows from 4-17 that for every theorem in the main relation o is denied,
that is, -(. . . o . . .), there is an exactly similar theorem in which the main
relation is that of the logical product, that is, -(. . . x . . .).
It is our immediate object to show that for every strict relation which is
assertable in the system, the corresponding material relation is also assert-
able. It is, then, important to know how these various relations are present
in the system. The only relations so far asserted in any proposition are
-J and = . Since == is expressible in terms of -i , we may take -J as the
fundamental relation and compare the others with it.
4-21 p -J q = -p A q
1-02: p*q = ~(p-g) C1)
2-51: (1) =p*q = ~[-(-p)~q] (2)
1-04: (2) = Q.E.D.
4-22 p A g = -p -J q
4-21 {-p/p}: -p-i? = -Hp) A?
2-51: (1) =* Q.E.D.
308 A Survey of Symbolic Logic
For every postulate and theorem in which the asserted relation is -J ,
there is a corresponding theorem in which the asserted relation is A , and
vice versa.
Consider the analogous relations, c and + .
4 • 23 p c q = -p + q
1-03: pcq = -(p-q) 0)
2-51: (1) = pcq = -[-(-p)-q] (2)
1-01: (2) = Q.E.D.
4-24 p + q = -pcq
4-23 {-p/p} : -pcq = -(-p) + q (1)
2-51: (1) = Q.E.D.
For every theorem in which the asserted relation is c , there is a corre
sponding theorem in which the asserted relation is + , and vice versa.
The exact parallelism between 4-21 and 4-23, 4-22 and 4-24, corrob
orates what 4-16 tells us: that wherever the relation A is asserted, the
corresponding material relation, + , may be asserted.
4-25 p-iq = -(po-q)
1-02: p*q = ~(p-q) (1)
2-51: (l)=p*q = -[--(?-?)] (2)
1-01: (2) - Q.E.D.
4 • 26 p o q = -(p -i -q)
4-25 {-q/q}: p*-q = -[po -(-?)] (1)
2-51:- (1) = p*-q = -(poq) (2)
2-11: (2)-l(2M-0H-(pog) (3)
2-12: (2)4-(poq)*(p*-q) (4)
2-3: (3)-ipo0-i-(2>-*-0) (5)
2-61: (4) -* -(p -l -ry) H p o r/ (6)
1-06: (5) x(6) = Q.E.D.
For every postulate and theorem in which the relation -* is asserted,
there is a corresponding theorem in which the main relation is o but this
relation is denied: and for every possible theorem in which the relation o
is asserted, there will be a corresponding theorem in which the main relation
is -J but that relation is denied. o and -i are connected by negation.
An exactly similar relation holds between p q and p c q.
1-03 pcq = -(p-q)
4-27 pq = -(pc-q)
Proof similar to that of 4-26, using 1-03 in place of 1 -02.
The System of Strict Implication 309
The parallelism here corroborates 4-17: for every possible theorem in
which the relation o is denied, there is a theorem in which the corre
sponding logical product is denied. But the implications of 4-1-4-17 are
not reversible, and a theorem in which the relation o is asserted does not
give a theorem in which any material relation is asserted. To put it another
way: of — p, p, and - ~p, the weakest is — p and it cannot be further
reduced. But the truth- value of any consistency is [ — ] — p o q = — (p q).
The reduction of = to the corresponding material relation, =, is obvious.
4-28 Hypothesis: p = q. To prove: p = q.
2.11: Hyp. H (p -J q) (1)
2-12: Hyp. •< (<M p) (2)
4-15: (IHCpcg) (3)
4-15: (2)-i(?cp) (4)
1-07: (3)x(4) == (p = q)
For every theorem in which the relation = is asserted, there is a cor
responding theorem in which the relation = is asserted.
We have now shown at length that, confining attention to the main
relations in theorems, there are two sets of strict relations which appear
in the system: (1) relations =, -i , and A which are asserted, and relations
o which are denied; (2) relations o which are asserted, and relations
= , -i , and A which are denied. Wherever a relation of the first described
set appears, it may be replaced by the corresponding material relation.
Any relation of the second set will be equivalent to some relation o which
is asserted — its truth- value will be [- ~]. Such relations cannot be further
reduced; they do not give a corresponding material relation. But under
what circumstances will relations of this second sort appear? Examination
of the postulates will show that they can occur as the main relation in the
orems only through some use of 1-7 and its consequences, for example,
pq-ipoq, p-l-~p, and - -p -J - ~p. In other words, they can occur only
where the corresponding material relation is already present in the system.
Hence for every theorem in the system in which a relation of the type
p o q is asserted, there is a theorem in which the corresponding material
relation, p q, is asserted.
Consequently, for every theorem in the system in which the main relation
is strict there is an exactly similar theorem in which the main relation is the
corresponding material relation.
WTherever strict relations appear as subordinate, or unasserted, relations
in theorems, the situation is quite similar. These are reducible to the
310 A Survey of Symbolic Logic
corresponding material relations through some use of 1-8 and its conse
quences. Note particularly theorem 2 • 77,
(P * q) = (~ ~P -* - ~?) = (~ -P -* ~ -9) = (-? •* -P) = (~? -* ~P)
The truth-value of any strict relation will, by its definition, be [~] or [~ -]
or [ — •]. And where two such are connected by -J or any equivalent rela
tion, they may be replaced by the corresponding relation whose truth-value
is simply positive or is [-] — and this is always a material relation.
We may now illustrate this reduction of subordinate strict relations:
43 [(p •* q) -J (r -J *)] -* [(p c g) -I (res)]
2-7 {p -q/p; r -s/q] : [~(p -q) H -(r -s)} -i [-(p -g) -I -(r -$)] (1)
1-02: (1) = [(p -J q) -I (r -I *)] •* [-(p -q) •* -(r -s)] (2)
1-03: (2) = Q.E.D.
4- 31 [(p A g)s (r A s)] •* [(p + q) -J (r + *)]
Similar proof, (-p -q) in place of (p -q), etc.
4-32 [(poq)-l(ros)]*[(pq)-i(rs)]
2-75 {pq/p; rs/q}: [- ~(p ?) -1 - ~(r s)H [(p ?) •* (r *)] (1)
1-01: (1) = Q.E.D.
4- 33 [(p -i q) H (r -J *)] S [(p c g) c (r c 5)]
4-15: [(pcg)-l(rc5)H[(pcg)c(rc*)] (1)
1-6: 4-3 x(l)-i Q.E.D.
4-34 [(p A q) H (r A*)] -^ [(p + g) c (r + s)]
Similar proof, using 4-31 in place of 4-3.
4-35 [(pog)H(ro*)H[(pg)c(r*)]
Similar proof, using 4-32.
Note that as a subordinate relation, p o q reduces directly.
In theorems 4 • 3-4 • 32, postulate 1 • 8 only has been used, and the reduc
tion of strict relations to material relations is incomplete. In theorems
4 • 33-4 • 35, postulates 1 • 8 and 1 • 7 have both been used, and the reduction
is complete. In these theorems, dyads of dyads are dealt with. The
reduction extends to dyads of dyads of dyads, and so on. We may illustrate
this by a single example which is typical.
Hypothesis: [(p 4 q) -J (-p A q)] 4 [(p o -q) -I -(p -J q)]
To prove: [(p c q) c (-p + q)] c [(p -q) c -(p c q)}
(The hypothesis is true, though it has not been proved.)
2-71 lp -q/p; p -q/q} : [-(p -q) H -(p -q)] -I [~(p -g) -I -(p -q)] (1)
1-02, 1-03, 1-04, 1-05, and 2-51:
(1) = [(p c q) H (-p + q)] -I [(p -i g) 4 (-p A g)] (2)
The System of Strict Implication 311
1-6: (2) xllyp. -I [(per/) -i (-p + r/)] -i [(p o-g) -t -(/; -J g)l (3)
2-72 {p -g/p; p -g/g} : [- -(? -g) -i - ~(p -r/)] -I [(p -g) •< (p -r/)] (4)
1-01, 1-02, 1-03, and 2-51:
(4) = ((p o -g) -i -(p -i g)] « [(p -r/) -t -(p c g)] (5)
1-6: (3) x (5) -I [(p c g) S (-p + g)] -J [(p -r/) -* -(p c g)] (6)
4-33: (6HQ.E.D.
In any theorem in which ~p is related to ~g, or — p to - ~g, or ~ -p to
~ -g, ~ may be replaced by -. This follows immediately from 2-77. We
illustrate briefly the reduction in those cases in which ~r, or — r, or — r,
is related to p o g, or p A g, or p -J g.
4-36 (pog-J-~r) -1 (p g H r)
2-75 {p g/p; r/g} : [- -(p g) H -r] -J (p q H r) (1)
1-01: (1) = (poq*-~r)-l(pq4r) (2)
4-15: (pg-ir) -* (pgcr) (3)
1-6: (2) x (3) -i Q.E.D.
4-37 (pAg-i~r) ^(p + gc-r)
2-7 {-p -g/p; r/g} : [-(-p -q) H -r] -J [-(-p -g) -J -r] (1)
4 . 15 : [-(-p -q) -* -r] ^ [-(-p -g) c -r] (2)
1-6: (l)x(2HK-p-gH~rH[-(-p-g)c-r] (3)
1-04 and 1-05: (3) = Q.E.D.
A dyad of triadic strict relations, e. g., p o (g r) -J g o (p r), reduces
just like a dyad of dyads, because a triadic strict relation is a dyadic strict
relation— with a dyadic material relation for one member. But a triad
of dyadic strict relations behaves quite differently. Such is postulate 1 • 6,
This does not look like a strict triad, but it is, being equivalent to
(p -l g) -l [(g -J r) c (p -{ r)]
which obviously has the character of strict triads generally. The sub
ordinate relations in such a triad cannot be reduced by any direct use of
1-8 and its consequences. However, all such relations can be reduced.
The method will be illustrated shortly by deriving
(pcg)(gcr) c(pcr)
from the above.
What strict relations, then, cannot be reduced to the corresponding
material relations? The case of asserted relations has already been dis-
312 A Survey of Symbolic Logic
cussed. For subordinate relations, the question admits of a surprisingly
simple answer. All the relations of the system can be expressed in terms
of some product and the various truth values — the truth values of ~ -p, p,
- ~p, -p, and ~p. Let us remind ourselves :
p o q --= - ~(p q) pq = - -(p q)
pi q = ~(p -g) p c q = -(p -q)
p A q = ~(-p -q) p+q = -(-p -q)
The difference between the truth-value of p and that of -p, between ~p
and — p, between — p and — -p, does not affect reduction, because — p
can be regarded as - -(p) or as ~(-p); p as (p) or as - ~(-p);
and p is also -(-p). Hence we may group the various types of expression
which can appear in the system under three heads, according to truth-value :
Hor[~-] [ ] or [-] [--] or [---]
p-lq ptq
p = q p = q
p*q p+q
pq poq
-(poq) -(pq)
-(p+q) ~(p*q)
-(p = q) -(p = q)
-(pcq) ~(p-lq)
-p -p p
~-p p -~p
In this table, the letters are quite indifferent: replacing either letter by
any other letter, or by a negative, or by any relation, throughout the table,
gives a valid result. The blank spaces in the table could also be filled;
for example, the first line in the third column would be (p -q). But,
as the example indicates, the missing expressions are more complex than
any which are given, and possess little interest. The significance of the
table is this: //, in any theorem, two expressions which belong in the same
column of this table are connected, then these expressions may be reduced by
postulate 1-8 and its consequences. For, by 2-77, a relation of any two in
the same column gives the corresponding relation of the corresponding two
in either of the other columns. But any theorem which relates expressions
which belong in different columns of this table is not thus reducible, since
any such difference of truth- value is ineradicable. This table also sum-
The System of Strict Implication 313
marizes the consequences of postulate 1-7: any expression in the table
gives the expression on the same line with it and in the next column to the
right. It follows that expressions in the column to the left also give the
expressions on the same line in the column to the right, since -J is transitive.
Just as postulate 1 • 7 is the only source of asserted strict relations
which are not replaceable by the corresponding material relations, so also
the only theorems containing irreducible subordinate relations are con
sequences of 1-7. For this postulate is the only one in which different
truth-values are related, and is the only assumed principle by which an
asserted (or denied) truth- value can be altered. But if we simply substi
tute - for ~ in 1 • 7, it becomes the truism, -p -* -p. As a consequence, for
every proposition in the system which contains strict relations or the truth-
values, [~], [ — ], [ — ], or [- — ], in any form, in such wise that these truth-
values cannot be reduced to the simple negative, [-], or the simple positive
(the truth- value of p), by the use 1-8, the theorem which results if we
simply substitute - for ~ in that proposition is a valid theorem. Or, to
put it more clearly, if less accurately; if any theorem involve [~], explicitly
or implicitly, in such wise that it cannot be reduced to [-] by the use of 1-8,
still the result of substituting ~ for - is valid. For example, 4 • 13, ~ -p ^ — ;;,
cannot be reduced by 1-8; ~ -p and - ~p are irreducibly different truth-
values. But substituting - for ~, we have -(-p) -»-(-;;), and hence -(-p)
c-(-p), or p^p- Propositions such as the pair ~-p4-~p and -(-p)
c-(-p) may be called "pseudo-analogues". If we reduce completely, so
far as possible, all the propositions which involve [~] or strict relations, by
the use of 1-7 and 1-8 and their consequences, and then take the pseudo-
analogues of the remaining propositions, we shall find such pseudo-analogues
redundant. They will all of them already be present as true analogues of
propositions which are completely reducible. This transformation by
means of postulates 1-7 and 1-8, by which strict relations give the cor
responding material relations, may be represented by the substitution
scheme
p c q, p = q, -(p q), p + q, -p, p, ~(p)
P*q, p = </, -(p°</)» PM» ~l>* ~-P* -(-~P)
We put -(p o q) and -(p q), -(- ~p) and -(p), because p o q as a main rela
tion in theorems is reducible only when it is denied, and - ~p is reducible
only through its negative. As we have now shown (except for triads of
dyads, the reduction of which is still to be illustrated), propositions involving
314
A Survey of Symbolic Logic
expressions below the line are still valid when the corresponding expressions
above the line are substituted.
The transformation by {-/~} of all the assumptions and theorems of the
system of Strict Implication which can be thus completely reduced, and the
rejection of remaining propositions which involve expressions below the line
(or the substitution for them of their pseudo-analogues), gives precisely the
system of Material Implication.
All the postulates and theorems of Material Implication can be derived
from the postulates and definitions of Strict Implication: the system of
Strict Implication contains the system of Material Implication. We may
further illustrate this fact by deriving from previous propositions the
postulates and definitions of the calculus of elementary propositions as
it appears in Principia Mathematical
(Principia, ^1-01)
pcq = -p + q
is theorem 4-23.
4-41
4 - 42
4 - 4 3
4-44
4-45
pq = --p + -q
1-05 {-p/p', -q/q}: -p + -q = -[-(-p)-(-
2-51: (1) =-p + -q = -(p q)
2-63: (2) = -(-p + -q) = -[-(p q)]
2-51: (3) = Q.E.D.
(p = q) = (pcq)(qcp)
is the definition, 1-07.
p + p c p
3-23: p + p-lp
4-15: (1) -i Q.E.D.
q c p + q
1-2 {-q/p; -p/q}: -p-q-i-q
2-61: (1) -J q ^ -(-p -q)
1-05: (2) = q-lp + q
4-15: (3) -i Q.E.D.
p + q cq + p
3-21: p + q-iq + p
4-15: (!)H Q.E.D.
p+ (q + r) cq + (p + r)
3-24: p + (q + r) -J q + (p + r)
4-15: (1)-^ Q.E.D.
6 Pp. 98-101, 114, 120.
(Principia, ^3-01)
(i)
(2)
(3)
(Principia, *4-01)
(Principia,
(Principia,
(Principia,
(Principia,
(1)
CD
(2)
(3)
CD
CD
The System of Strict Implication 315
For the proof of the last postulate in the set in Principia Mathematica
certain lemmas are needed which are of interest on their own account.
4-51 p q c r = p c (q c r) = qc(pcr)
1-03 {pq/p-, r/q}: pqcr = -[(p q) -r] (1)
2-91 and 2-9: (1) = pry cr = -[p(q -r)] = -[q(p -r)] (2)
2-51: (2) = p0cr = -{p-[-(9-r)]} = -|<H-(p-r)]} (3)
1-03: (3) = Q.E.D.
4-52 p q-lr = p-l (qcr) = ql (per)
1-02 {p q/pi r/q} : pq -J r = ~[(p q) -r]
Remainder of proof, similar to the above.
4-53 [(pcq)p]-lq
2-4 {pcq/p}: (p c 0) -1 (p c g) (1)
4-52 {pcq/p; p/q-, q/r}: (I) = Q.E.D.
It is an immediate consequence of 4-53 that " If p is asserted and p c q
is asserted, then q may be asserted", for, by our assumptions, if p is asserted
and p c q is asserted, then [(p c q)p] may be asserted. And if this is asserted,
then by 4-53 and our operation of " inference", q can be asserted. But
note that the relation which validates the assertion of q is the relation s in
the theorem. This principle, deduced from 4-53, is required in the system
of Material Implication (see Principia, #1-1 and #1-11).
4-54 (~pc~q) *(qcp)
4-3: 2-21-^Q.E.D.
4-55 (p-*q)l(prcqr)
1-6 {-r/r}: (p-lg)(^ -r) -i (p l -r)
4-52: (1) = (p-Jg)HKg-l-r)c(p-l-r)] (2)
1-02 and 2-51: (2) = (p -I g) -* Kg r) c ~(p r)] (3)
4-54: -(gr)c-(pr)-l(2;rcgr) (4)
1-6: (3) x (4) •< Q.E.D.
4-56 (pcq) c(prcgr)
4-55 {(pcq)p/p}: 4-53 H [(p c?)p]r eg r
2-91: (1) = [(pc9)(pr)]c9r (2)
4-51: (2) = Q.E.D.
4-57 p cq = -qc-p
4-3: 2-62H(pCtfH(-?c-p) (1)
4-3: 2-6-l(-gc-p)-j(pC(/) (2)
1-06: (l)x(2) = Q.E.D.
316 A Survey of Symbolic Logic
4-58 (pcq)(qcr)c(pcr)
4-56 {-r/r}: (p c q) c (p -r c q -r) (1)
4-57: (1) - (pc</)c[-(g-r)c-(p-r)] (2)
1-03: (2) = (pcq)c[(qcr)c(pcr)] (3)
4-51: (3) = Q.E.D.
4 • 58 is the analogue, in terms of material relations, of 1-6. The method
by which we pass from 1-6 to 4-58 illustrates the reduction of triads of
strict dyads in general. This reduction begins in the first line of the proof
of 4-55. Here 1 -6 is put in the form
(p •*?)•*[(?•* r) c (p •* r)]
12 345
The relation numbered 4 is already a material relation. This is character
istic of strict triads. Relations 3 and 5 are reduced together by some
consequence of 1-8, in a form in which the asserted relation is material.
Then, as in 4 • 56, relations 1 and 2 are reduced together by the use of 4 • 53
as a premise. This use of 4-53 is quite puzzling at first, but will become
clearer if we remember its consequence, " If p is asserted and p c q is asserted,
then q may be asserted". This method, or some obvious modification of it,
applies to the reduction of any triad of strict dyads which the system gives.
We can now prove the last postulate for Material Implication.
4 - 59 (q c r) c[(p + q) c(p + r)] (Principia, #1-6)
4-58 {-pip} : (-pcq)(qcr) c (-per) (1)
4-51: (1) = (gcr)c[(-pcg)c(-pcr)] (2)
4-24: (2) - Q.E.D.
These are a sufficient set of symbolic postulates for Material Impli
cation, as the development of that system from them, in Principia Mathc-
matica, demonstrates. However, in the system of Strict Implication, those
theorems which belong also to Material Implication are not necessarily
derived from the above set of postulates. They can be so derived, but the
transformation {-/~} produces them, more simply and directly, from their
analogues in terms of strict relations.
IV. EXTENSIONS OF STRICT IMPLICATION. THE CALCULUS OF CONSIST
ENCIES AND THE CALCULUS OF ORDINARY INFERENCE
From the symmetrical character of postulate 1-8, and from the fact
that postulate 1-7 is converted by negating both members, i. e., p -i- ~p,
The System of Strict Implication 317
it follows that, since the transformation j-/~) is possible, an opposite
transformation, {~/-}, is possible. And since implications are reversed
by negating both members, those expressions which are transformed directly
by j-/~) will be transformed through their negatives by |~/-), while those
expressions which are transformed through their negatives by {-/~j will
be transformed directly by }~/-J. Hence we have
-(peg), -(/; = g), pq, -(p + (
This substitution scheme may be verified by reference to the table on
page 312. The transformation }-/~} represents the fact that expressions
in the column to the left, in this table, give expressions in the middle column:
{~/-J represents the fact that expressions in the middle column give ex
pressions in the column to the right. {-/-} eliminated strict relations:
j~/-j eliminates material relations. As in the previous case, so here, a
dyad of dyadic relations, or a relation connected with p, -p, ~p, etc., can
be transformed by 1-8 and its consequences when and only when the
connected expressions appear in the same column of that table. Thus the
transformation J~/-j is subject to the same sort of limitation as is (-/-}.
The transformation {~/-J, eliminating material relations, has already
been illustrated in those tables in Section II, in which theorems in terms of
strict relations were compared with analogous propositions in terms of
material relations. Theorems in the right-hand column of those tables
result from those in the left-hand column by the transformation {-/-).
The proofs of 3-11, 3-12, 3-13, 3-14, 3-31, 3-32, 3-33, 3-34, 3-52, 3
and 3-55 indicate the method of this transformation. Theorem 3-54 indi
cates a limitation of it. As we have noted, triadic strict relations are not
expressible in terms of strict dyads alone. Consequently, in the case of
triadic relations, the transformation {-/-} cannot be completely carried
out. This is an important limitation, since postulate 1 • (>, which is necessary
for any generality of proof, is a triadic strict relation. It means that any
system of logic in which there are no material relations cannot symbolize
its own operations. Since strict relations are the relations of intension,
this is an important observation about calculuses of intension in general.
The vertical line in the substitution scheme is to indicate that the
transformation {-/-} is arbitrarily considered to be complete when no
material relations remain in the expression, p and -p will be transformed
when connected with a material relation which is transformed; when not
318 A Survey of Symbolic Logic
so connected, p and -p remain. They could be transformed in all cases,
but the result is needlessly complex and not instructive.
The system, or partial-system, which results from the transformation
{~/-} may be called the Calculus of Consistencies. It can be generated
independently by the following assumptions :
Let the primitive ideas be: (1) propositions, p, q, r, etc., (2) -p, (3) ~p,
(4) p o q, (5) p = q.
Let the other strict relations be defined :
II. p -J q = -(p O -q)
For postulates assume :
III. poq-iqop
IV. q o p -J — p
V. - ~p -J p o p
VI. = -
Assume the operations of " Substitution " and "Inference" as before,
but in place of "Production" put the following: If p -J q is asserted and q -J r
is asserted, then p -J r may be asserted. By this principle, proof is possible
without the introduction into the postulates of triadic relations.
The' system generated by these assumptions is purely a calculus of
intensions. It is the same which would result from performing the trans
formation {~/-j upon all the propositions of Strict Implication which
admit of it, and rejecting any which still contain expressions, other than p
and -p, below the line. It contains, amongst others, all those theorems
concerning strict relations (except the triadic ones) which were exhibited
in Section II in comparison with analogous propositions concerning material
relations.
More interest attaches to another partial-system contained in Strict
Implication. If our aim be to create a workable calculus of deductive
inference, we shall need to retain the relation of the logical product, p q,
but material implication, p eg, and probably also the material sum, p + q,
may be rejected as not sufficiently useful to be worth complicating the
system with. The ideas of possibility and impossibility also are unnecessary
complications. Such a system may be called the Calculus of Ordinary
Inference. The following assumptions are sufficient for it.
The System of Strict Implication 319
Primitive Ideas: (1) Propositions, p, q, r, etc., (2) -p, (3) p H q, (4) p g,
(5) p - ?.
Definitions:
A. p A g = -p -J g
B. pOq = -(p*-q)
C. (p = q) = (p-ig)(g-ip)
[D. p + g = -(-p-g)] Optional.
Postulates:
E. (-p -J g) •< (-g -l p)
F. p g -J p
G. p-«pp
H. p(g r) -J g(p r)
I. p H -(-p)
J. (p-!g)(g-ir)-l(p-Jr)
K. p q -J p o g
L. (p g •* r s) = (p o g H r o 5)
All of these assumptions are propositions of the system of Strict Impli
cation. A. is 4-22, B. is 4-26, C. is 1-06, and D. is 1-05; E. is 2-3, F. is
2-1, G. is 1-3, H. is 1-4, I. is 1-5, J. is 1-06, K. is 4-26, and L. is an im
mediate consequence of 4 • 32 and 4 • 35. The Calculus of Ordinary Inference
is, then, contained in the system of Strict Implication. It consists of all
those propositions of Strict Implication, which do not involve the relation
of material implication, peg [or the material logical sum, p + g]. But
where, in Strict Implication, we have -p, we shall have, in the Calculus of
Ordinary Inference, -(pop) or p -J -p. Similarly ~ -p will be replaced
by -(-p o -p) or -p -J p, and - ~p by p o p or -(p -J -p). In other words,
for 'p is impossible' we shall have {p is not self-consistent' or ' p implies
its own negation'; for 'p is necessary' we shall have 'the negation of p is
not self-consistent' or 'the negation of p implies p'; and for ' p is possible'
we shall have 'p is self -consistent ' or (p does not imply its own negation'.
The Calculus of Ordinary Inference contains the analogues, in terms of
p g, p A q, and p -J r/, of all those theorems of Material Implication which
are applicable to deductive inference. It does not contain the useless and
doubtful theorems such as "A false proposition implies any proposition",
and "A true proposition is implied by any proposition". As a working
320 A Survey of Symbolic Logic
system of symbolic logic, it is superior to Material Implication in this
respect, and also in that it contains the useful relation of consistency, p o q.
On the other hand, it avoids that complexity which may be considered an
objectionable feature of Strict Implication.
The system of Strict Implication admits of extension to prepositional
functions by methods such as those exhibited in the last chapter. For
the working out of this extension, several modifications of this method are
desirable, but, for the sake of brevity, we shall adhere to the procedure
which is already familiar so far as possible. In view of this, the outline
to be given here should be taken as indicative of the general method and
results and not as a theoretically adequate account. Since, as we have
demonstrated, the system of Material Implication is contained in Strict
Implication, It follows that, with suitable definitions of U<px and S#E,
the whole theory of prepositional functions, as previously developed, may
be derived from Strict Implication. H<px will here be interpreted more
explicitly than before, " px is true in all (actual) cases/' or " <px is true of
every x which '.exists'". And 'Znpx will mean " tpx is true in some (actual)
case", or "There 'exists' at least one x for which <px is true". The novelty
of the calculus of prepositional functions, as derived from Strict Implica
tion, will come from the presence, in that system, of ~p, — p, ~ -p, and the
strict relations. We might expect that if <px is a prepositional function,
~$x would be a prepositional function. But such is not the case: ~(px is a
proposition. For example, "It is impossible that 'x is a man but not
mortal'" is a proposition although it contains a variable. So is "Nothing
can be both A and not-.4", which predicates the impossibility of ".i1 is A
and x is not- A ". It would be an error to suppose that all the propositions
which contain variables are such because they involve the idea of impossi
bility, or necessity, but the most notable examples, the laws of mathe
matics, are propositions, and not prepositional functions, for precisely this
reason. When stated in the accurate hypothetical form — i. e., as the
implications of certain assumptions — they are necessary truths.
Since ~ <px is a proposition, — <px, — <px, and all the strict relations of
propositional functions will be propositions. If ipx and \f/x are prepositional
functions, then
<px o \f/x is the proposition - ~( <px x \f/x) ;
<px A \[/x is the proposition ~(-<px x-^r);
<px •* \f/x is the proposition ~($x x-\I/x)
The System of Strict Implication 321
We shall have the law, ~<px4U-<px, "If <px is impossible, then it is
false in all cases". Hence also, 2 <px -J - - #c, "If <px is sometimes true,
then <px is possible". The first of these gives us one most important con
sequence,
( <px -i fa) -1 IIX( <px c fa)
"If it is impossible that <px be true and fa false, then in no (actual) case
is <px true but fa false", or "If <px strictly implies \l/x, then <px formally
implies fa". This connects the novel theorems of this theory of prepo
sitional functions with the better known propositions which result from
the extension of Material Implication. Similarly we shall have
( <px A fa) -j nx( <px + fa)
and S, ( <px x fa) -j ( <px o fa)
If we use z(<pz) to denote the class determined by <pz, that is, the class
of all x's such that <px is true, then we derive the logic of classes from this
calculus of propositional functions, by the same general type of procedure
as that exhibited in Section III of Chapter IV. If we let a =
(3 = &(fa), the definitions of this calculus will be as follows:
On e a) = (pxn
"x is a member of the class a, determined by the function <pz'' means
" <pxn is true".
(ttC/3) = U*(<pxcfa)
(a = /3) = (<px = fa)
(a = j8) = IIX(#C s ^.T)
-a = x(-<f>x), or -a = ^ -0 e a)
(a x |3) = x(<px x ^.r), or (a x ]8) = x[(x e a) x (x e /3)]
(a + j8) - .T(^T + ^), or (a + j3) = ^[(a; e a) + (a; 6 /3)]
1 == £(Sx -J fa:)
0 = -1
a c /3 is the relation "All members of a are also members of 0"— a relation
of extension. It is defined by "In every (actual) case, either <px is false
or fa is true"; or "There is no (actual) case in which ?x is true and fa
false "-nx(^c^r) == U,(-<px + fa) - II, -(<p* x-**). a -I 0 is the cor
responding relation of intension: it is defined by "Necessarily either <px is
false or fa is true", or "It is impossible that <f>x be true and fa false",
22
322 A Survey of Symbolic Logic
that is, (vx-*$x) = (-<pxA\f/x) = ~(<px x-\f/x). a -J ft may be correctly
interpreted " The class-concept of a, that is, <p, contains or implies the
class-concept of ft, that is, ^". That this should be true may not be at
once clear to the reader, but it will become so if he study the properties of
a -J ft, and of <?x 4 \f/xt in this system.
Since we have (<px -* \f/x) -J Ux( #x c \[/x)
we shall have also (a -i ft) -J (« c 0)
If the class-concept of a implies the class-concept of ft, then every member
of a will be also a member of (3. The intensional relation, -i , implies the
extensional relation, c. But the reverse does not hold. The old "law" of
formal logic, that if a is contained in ft in extension, then ft is contained in a
in intension, and vice versa, is false. The connection between extension
and intension is by no means so simple as that.
This discrepancy between relations in extension and relations in inten
sion is particularly evident in cases where one of the classes in question is
the null-class, 0, or the universe of discourse, 1. As was pointed out in
Chapter IV, we shall have for every "individual", x,
x e 1, and -(x e 0)
Also, for any class, a, we shall have
a c 1, and 0 c a
These last two will hold because, since fa: •* fa: is always true when significant,
-(fa: H fa:) always false, we shall have, for any function, <px,
and ILKfa: -J fa:) c <px]
We shall have these because "A false proposition materially implies any
proposition", and UA true proposition is materially implied by any propo
sition." But since it does not hold that "A false proposition strictly implies
any proposition", or that "A true proposition is strictly implied by any
proposition", we shall not have
<px -J (fa: -{ far)
or -(fa: -J fa:) -i <px
And consequently we shall not have
a -* 1, or 0 -J a
If 7 is a null class, we shall have " All members of 7 are also members of ft,
The System of Strict Implication 323
whatever class 0 may be". But we shall not have "The class-concept of y
implies the class concept of 0, whatever class 0 may be". The implications
of a class-concept are not affected by the fact that the class has no members.
The relation, a = 0, is material or extensional equivalence, "The
classes a and 0 consist of identical members"; a = ft is strict or intensional
equivalence, "The class-concept of a is equivalent to the class-concept of 0".
It is obvious that
(a = 0) -I (a - 0)
but that 2/z0 reverse does not hold. The relation between intensions and
extensions is unsymmetrical, not symmetrical as the medieval logicians
would have it. And, from the point of view of deduction, relations of
intension are more powerful than relations of extension.
a+0 and a x (3 are relations of extension — the familiar "logical sum"
and "logical product" of two classes. What about the corresponding
relations of intension? This most important thing about them — there are
none. Consider the equivalences,
(a A 0) = x(<px A #r) = x[(x e a) A (x e 0)]
and (a o 0) = x(<px o #r) - x[(x e a) o (x € (3)}
(px o i/a* is a proposition — the proposition — (<px x^.r), "It is possible that
<px and \l/x both be true". And being a proposition, either it is true of
every x or it is true of none. So that a o 0, so defined, would be either
1 or 0. Similarly <^.r A \f/x is a proposition, either true of every .r or false
of every or; and a A (3 would be either 1 or 0. Consequently, a A 0 and
a o (3 are not relations of a and 0 at all. The product and sum of classes
are relations of extension, for which no analogous relations of intension
exist. This is the clue to the failure of the continental successors of Leibniz.
They sought a calculus of classes in intension: there is no such calculus, unless
it be confined to the relations a -J 0 and a = (3. Holland really came in
sight of this fact when he pointed out to Lambert the difficulties of logical
"multiplication" and logical "division".7
The presentation of the calculus of propositional functions and calculus
of classes here outlined,— and of the similar calculus of relations,— would
involve many subtle and vexatious problems. But we have thought it
worth while to indicate the general results which are possible, without dis
cussing the problems. But there is one important problem which involves
the whole question of strict implication, material implication, and formal
7 See above, p. 35.
324 A Survey of Symbolic Logic
implication, which must be discussed — the meaning of "implies". This
is the topic of the next section.
V. THE MEANING OF "IMPLIES"
It is impossible to escape the assumption that there is some definite
and "proper" meaning of "implies". The word denotes that relation
which is present when we "validly" pass from one assertion, or set of
assertions, to another assertion, without any reference to additional "evi
dence". If a system of symbolic logic is to be applied to such valid infer
ence, the meaning of "implies" which figures in it must be such a "proper"
meaning. We should not hastily assume that there is only one such
meaning, but we necessarily assert that there is at least one. This is no
more than to say: there are certain ways of reasoning that are correct or
valid, as opposed to certain other ways which are incorrect or invalid.
Current pragmaticism in science, and the passing of "self-evident
axioms" in mathematics tend to confuse us about this necessity. Pure
mathematics is no longer concerned about the truth either of postulates or
of theorems, and definitions are always arbitrary. Why, then, may not
symbolic logic have this same abstractness? What does it matter whether
the meaning of "implies" which figures in such a system be "proper" or
not, so long as it is entirely clear? The answer is that a system of symbolic
logic may have this kind of abstractness, as will be demonstrated in the next
chapter. But it cannot be a criterion of valid inference unless the meaning,
or meanings, of "implies" which it involves are "proper". There are
two methods by which a system of symbolic logic may be developed: the
non-logistic method exemplified by the Boole-Schroder Algebra in Chapter
II, or the logistic method exhibited in Principia Mathematica and in the
development of Strict Implication in this chapter. The non-logistic method
takes ordinary logic for granted in order to state its proofs. This logic which
is taken for granted is either "proper" or the proofs are invalid. And if
the logic it takes for granted is not the logic it develops, then we have a
most curious situation. A symbolic logic, logistically developed — i. e.,
without assuming ordinary logic to validate its proofs — is peculiar among
mathematical systems in that its postulates and theorems have a double
use. They are used not only as premises from which further theorems are
deduced, but also as rules of inference by which the deductions are made.
A system of geometry, for example, uses its postulates as premises only;
it gets its rules of inference from logic. Suppose a postulate of geometry
The System of Strict Implication 325
to be perfectly acceptable as an abstract mathematical assumption, but
false of "our space". Then the theorems which spring from this assump
tion may be likewise false of uour space". But still the postulate will
truly imply these theorems. However, if a postulate of symbolic logic,
used as a rule of inference, be false, then not only will some of the theorems
be false, but some of the theorems will be invalidly inferred. The use of
the false postulate as a premise will introduce false 'theorems; its use as a
rule of inference will produce invalid proofs. " Abstractness " in mathe
matics has always meant neglecting any question of truth or falsity in
postulates or theorems; the peculiar case of symbolic logic has thus far
been overlooked. But we are hardly ready to speak of a "good" abstract
mathematical system whose proofs are arbitrarily invalid. Until we are,
it is requisite that the meaning of "implies" in any system of symbolic
logic shall be a "proper" one, and that the theorems — used as rules of
inference — shall be true of this meaning.
Unless "implies" has some "proper" meaning, there is no criterion of
validity, no possibility even of arguing the question whether there is one or
not. And yet the question What is the "proper" meaning of "implies"?
remains peculiarly difficult. It is difficult, first, because there is no common
agreement which is sufficiently self-conscious to decide, for example, about
"material implication" or "strict implication". Even those who feel quite
decided in the matter are easily confused by the subtleties of the problem.
And, second, it is difficult because argument on the topic is necessarily
petitio principii. One must make the Socratic presumption that one's
interlocutor already knows the meaning of "implies", and agrees with
one's self, and needs only to be made aware of that fact. One must sup
pose that the meaning in denotation is clear to all, as the meaning of "cat"
or "life" is clear, though the definition remains to be determined. If two
persons should really disagree about "implies"— should have different
"logical sense"— there would be nothing to hope for from their argument.
In consideration of this peculiar involution of logical questions, the
best procedure is to exhibit the alternatives in some detail. When the
nature of each meaning of "implies", and the consequences of taking it to
be the "proper" one have been exhibited, the case rests.
We have already drawn attention, both in this chapter and in Chapter II,
to the peculiar theorems which belong to all systems based on material
implication. Wre may repeat here a few of them:
(1) A false proposition implies any proposition; -p c (p c q)
326 A Survey of Symbolic Logic
(2) A true proposition is implied by any proposition; q c (p c q)
(3) If p does not imply q, then p is true; -(p c q) cp
(4) If p does not imply q, then (7 is false; -(p cq) c-g
(5) If p does not imply q, then p implies that q is false;
-(per/) c(pc-q)
(6) If p does not imply g, then 'p is false ' implies g; -(p c q) c (-29 c g)
(7) If p and q are both true, then p implies q and g implies p;
pqc(pcq)(qcp)
(8) If p and g are both false, then p implies q and q implies p\
-p-qc(pcq)(qcp)
These sufficiently characterize the relation of material implication. It is
obviously a relation between the truth-values of propositions, not between
any supposed content or logical import of propositions, "p materially
implies q" means "It is false that p is true and q false". All these the
orems, and an infinite number of others just as "peculiar" follow necessarily
from this definition. The one thing which this relation has in common
with other meanings of " implies "—a most important thing of course — is
that if p is true and q is false, then p does not materially imply q.
As has been said, there are any number of such "peculiar" theorems
in any calculus of propositions based on material implication.8 These the
orems do not admit of any application to valid inference. In a system of
material implication, logistically developed, there is nothing to prohibit
their being used as rules of inference, but when so used they give theorems
which are even more peculiar and quite as useless. If we apply these
theorems to non-symbolic propositions, we get startling results. "The
moon is made of green cheese" implies "2 + 2 == 4", — because q c (p c q).
Let q be "2 + 2 == 4" and p be "The moon is made of green cheese".
Then, since "2 + 2 == 4" is true, its consequence above is demonstrated.
"If the puppy's teeth are filled with zinc, tomorrow will be Sunday".
Because the puppy's teeth are not filled with zinc, and, anyway, it happens
to be Saturday as I write. A false proposition implies any, and a true
proposition is implied by any.9
There are, then, in the system of Material Implication, a class of propo
sitions, which do not admit of any application to valid inference. And
8 Every theorem gives others by substitution, as well as by being used as a rule of
inference. And there are ways whereby, for any such theorem, one other which is sure to
be "peculiar" also can be derived from it. And also, it can be devised so that no result of
one shall be the chosen result of any other. Hence the number is infinite.
9 Lewis Carroll wrote a Symbolic Logic. I shall never cease to regret that he had not
heard of material implication.
The System of Strict Implication 327
all the other, non-" peculiar ", theorems of Material Implication find their
analogues in other systems. Hence the presence of these peculiar and use
less theorems is a distinguishing mark of systems based upon material
implication.
There can be no doubt that the reason why the relation of material
implication is the basis of every calculus of propositions except MacColl's
and Strict Implication is a historical one. Boole developed his algebra for
classes; he then discovered that it could also be interpreted so as to cover
certain relations of propositions. Peirce modified Boole's algebra by intro
ducing the relation of inclusion, which we have symbolized by c . a cb
has all the properties of the relation between a and b when every member of
a is also a member of b. It has one notable peculiarity: if a is a class which
has no members — a "zero" class — then for any class x, a ex. Now the
idea of "zero" in any branch of mathematics seems a little more of an
arbitrary convention than the other numbers. The arithmetical fact that
0 <-8 seems "queer" to children, and it would, most likely, have seemed
"queer" to an ancient Roman. Once 0 is defined, its "queer" properties,
as well as the obvious one, 8 + 0 = 8, are inevitable. It is similar with
the "null class", a 0 = 0, "That which is both a and nothing is nothing",
is necessary. And (a b = a) = (a c6), "'That which is both a and 6,
is a' is equivalent to 'All a is &'", leads to the necessary consequence
0 c a. If there are no sea serpents, then " All sea serpents are arthropods "
necessarily follows. This consequence seems more "queer" and arbitrary
because it is a relation of extension with no analogue in intension. The
concept "sea serpent" does not imply the concept "arthropod" — as has
been pointed out, 0 -i a does not hold. And in our ordinary logical thinking
we pass from intension to extension and vice versa without noting the
difference, because the relations of the two are so generally analogous.
But once we make the necessary distinction of relations in extension from
relations in intension, it is clear that Oca in extension is a necessary conse
quence of the concept of the null-class. Entirely similar remarks apply to
the proposition a c 1, except that "a is contained in everything" does not
seem so "queer".
Boole suggested that the algebra of classes be reinterpreted as a calculus
of propositions by letting a, 6, c, etc., represent the times when the proposi
tions A, B, C, etc., are true. Then Peirce added the postulate which
holds for propositions but not for classes (or for propositional functions),
a = (a = i). \ proposition is either true in all cases, or true in none.
328 A Survey of Symbolic Logic
The class of cases in which any proposition is true is either 0 or 1. This
gives the characteristic property of the Two-Valued Algebra. If we add
to this the interpretation of a cb, "All cases in which A is true are cases in
which B is true", or loosely "If A, then B", we have the source of the
peculiar propositions of Material Implication. For Ocfr. acl, OcO,
0 c 1, and 1 c 1 follow from the laws which are thus extended from classes
to propositions. A false proposition [= 0] implies 6. And a implies any
true proposition [= 1]. Of any two true propositions [= 1] each implies
the other. Of any two false propositions [= 0] each implies the other.
And any false proposition implies any true one. "A false proposition
materially implies any proposition" means precisely "If there are no
cases in which A is true (if a = 0) then all cases in which A is true are
also cases in which B is true". It does not mean " B can be inferred from
any false proposition". "A true proposition is materially implied by any
proposition" means only, "If B is true [= 1], then the cases in which A is
true are contained among the cases (i. e., all cases) in which B is true".
It does not mean "Any true proposition can be inferred from A". Inference
depends upon meaning, logical import, intension, a c b is a relation purely
of extension. Is this material implication, a c b, a relation which can
validly represent the logical nexus of proof and demonstration?
Formal implication ILx(<pxc \f/x) is defined in terms of material implication.
It means "For every value of x, <px materially implies \f/x". Choose any
value of x, say for convenience z, and unless (pz is false, fa is true. Cer
tainly this relation approximates more closely to the usual meaning of
"implies". But the precisely accurate interpretation of Hx(<pxc\f/x)
depends upon what is meant by the "values of x". We have spoken of
them as "cases" or "individuals". It makes a distinct difference whether
the "cases" comprehended by H.x(<px c\f/x) are all the possible cases, all
conceivable individuals, or only all actual cases, all individuals which exist
(in the universe of discourse). Either interpretation may consistently be
chosen, but the consequences of the choice are important. Let us survey
briefly the more significant considerations on this point.
In the first place, supposing that the second choice is made and nx be
taken to signify "for all x's which exist", what shall we mean by "exist"?
This is entirely a matter of convenience, and logicians are by no means at
one in their use of the term. But any meaning of "exist" which confines
it to temporal and physical reality or to what is sometimes called "the
factual" is inconvenient because, for example, we may wish to distinguish
The System of Strict Implication 329
the status of curves without tangents in mathematics from the status of
the square of the circle. This distinction is usually made by saying that
the former "exist", since the general mathematical idea of a "curve"
admits such cases, and their equations may be given; while the square of
the circle is demonstrably impossible. Again, it is inconvenient to say that
Apollo exists in Greek mythology, whereas the god Agni does not. Now
the god Agni is not inconceivable in Greek mythology; we find no record
of him, that is all. Similarly, while the usual illustrations of mathematical
"non-existence" are impossibilities, there is still a difference between what
"does not exist" in a mathematical system and what is impossible. Sup
pose we have an "existence postulate" in a set which are consistent and
independent each of the others — the 0-postulate in the Boole-Schroder
Algebra, for example. Without this postulate, the remainder of the set
generate a system in which 0 does not exist. But it is possible, as the con
sistency of this postulate with the others demonstrates. The frequent
statement that "mathematical existence" is the same as "possibility" is a
very thoughtless one.
The most convenient use of "exist" in logic is, then, one which makes
the meaning depend upon the universe of discourse, but one which does
not, as is sometimes supposed, thereby identify the "existent" and the
possible. ("Possible" similarly varies its meaning with the universe of
discourse.) On the other hand, it is inconvenient to use "exist" so widely
that "existence" is a synonym for "conceivability ". This is so obvious
in the most frequent universe of discourse, "phenomena", that it hardly
needs to be pointed out.
Using "exists" in this sense, in which "existence" is narrower than
"possibility" but may, in some universes of discourse, be wider than "the
factual", it makes a difference whether 11* in Ut(^xcfa) denotes only
existent x's or all possible z's. All American silver coins dated 1915 have
milled edges. Let <px be "x is an American silver coin dated 1915", and
let fa be " x has a milled edge". There is no necessity about milled edges
for silver coins, unless one speak in the "legal" universe of discourse,
this illustration, Ux(<pxcfa) will be true if IIX denote only actual .r's;
false if it denote all possible x's. One illustration is as good as a hundred;
if Ilx(<px c fa) refer to all possible x's, Ux(<px c fa) means "It is impossible
that <px be true and fa false". If U^xcfa) be confined to actual x's,
then it signifies a relation of extension, "The class of things of which <px
is true is contained in the class of things of which fa is true".
330 A Survey of Symbolic Logic
It might be thought that the meaning of Hx((px c\f/x) is sufficiently
determined by saying that the "values of a;" in a function, px, are all the
entities for which <px is either true or false. But this is not the case, for
there is question whether, of an x which does not exist, <px is always true,
or always false, or sometimes true and sometimes false, or never either
true or false. Here again, the question is, in part, one of convention.
From the point of view of extension, it is obvious that if <px can be predi
cated at all of an x which does not exist, it will always be false. (Predicating
something, <p, of an "individual", x, which does not exist, should be dis
tinguished from asserting that an empty class, a, which exists though it
has no members, is included in some other, a c 6. "The King of France is
bald" is an example of the former; "All sea serpents have green wings",
of the latter.) And the point of view of extension is frequently that of
common sense. In this sense, "x is a man" is false of my non-existent twin
brother, and even identical propositions such as "My twin brother is my
twin brother" are false of the non-existent. But from the point of view of
intension, an identical proposition is always true, and <px may be true or
it may be false of a non-existent x. If the point of view of extension be
taken with reference to prepositional functions, then <px is either not-
significant or false of the non-existent, and \j/x is similarly not-significant
or false. If <px and \f/x are not significant of the non-existent, then Ux(<px
c\f/x) means "For every existent x, <px materially implies $x". If <px
and \l/x are significant and false of the non-existent, then <px c \f/x is true
of every non-existent x, since of two false propositions, each materially
implies the other. Hence on this interpretation, Ux(<px c\l/x) is significant
for all possible x's and true in case every existent x is such that <px c \j/x.
Hence its meaning will still be accurately rendered by "For every existent
x, <px materially implies $x". If the point of view of intension be taken
with reference to propositional functions, or if it be left open, then Hx(<px
cfa) may mean "For every possible x, <px materially implies $x", or we
may, by convention, still confine it to the meaning " For every existent x,
(px materially implies ^e".10
10 We would gladly have spared the reader these details, but we dared not. If logicians
do not consider one another's views, who will? In this connection we are reminded of a
passage in Lewis Carroll's Symbolic Logic (pp. 163-64) anent the controversy concerning
the existential import of propositions:
"The writers, and editors, of the Logical text-books which run in the ordinary
grooves— to whom I shall hereafter refer by the (I hope inoffensive) title ' The Logi
cians ' — take, on this subject, what seems to me to be a more humble position than is
at all necessary. They speak of the Copula of a Proposition 'with bated breath"
The System of Strict Implication 331
The ground being now somewhat cleared, we return to the simpler
considerations which are really more important. What are the conse
quences of taking Hx((px c\f/x) in one or the other meaning? The first
and most important is this. It is a desideratum that we should be able to
derive the calculus of classes from the calculus of prepositional functions.
And in this calculus of classes, the inclusion relation of classes a c 0 or
£(<pz) c z(\j/z) can be defined by
z(<pz) c z(tz) = Ux(<px c I/M-)
or by some equivalent definition. If llz((px c\J/x) mean "For all x's
which exist, <pxc\}/x", then a c 0, or &(<pz) cz(\f/z), so defined, is the use
ful relation of extension, "All the existing things which are members of a
are also members of /3". Such a relation can represent such propositions as
"All American silver coins dated 1915 have milled edges". If, on the other
hand, we interpret Ux(<px c $x) to mean "For all possible z's, <pxc\l/x,
then two courses are open: (1) we can maintain that whatever is true of
all existent things is true of all possible— thus abrogating a useful and
probably indispensable logical distinction; or (2) we can allow that what
is true or false of the possible depends upon its nature as conceived or
defined. If we make the second choice here, the consequence is that
a c |S, or z(<pz) c z(\j/z), defined by
z(<pz) c z(tz) = nx(<t>x c t/a-)
or in any equivalent fashion, such as
o:C/3 = Hi(xe acxe )8)
is the relation of intension "The class-concept of a implies the class-
concept of |S". This relation does not symbolize such propositions as
almost as if it were a living, conscious Entity, capable of declaring for itself what it
chose to mean and that we, poor human creatures, had nothing to do but to ascertain
what was its sovereign will and pleasure, and submit to it.
" In opposition to this view, I maintain that any writer of a book is fully authorised
in attaching any meaning he likes to any word or phrase he intends to use. If I find
an author saying, at the beginning of his book, 'Let it be understood that by the
word "black" I shall always mean "white", and that by the word "white"
always mean "black",1 I meekly accept his ruling, however injudicious I may think ]
"And so, with regard to the question whether a Proposition is or is not to be
understood as asserting the existence of its Subject, I maintain that every writer may
adopt his own rule, provided of course that it is consistent with itself and with tl
accepted facts of Logic.
"Let us consider, one by one, the various views that may logically b
thus settle which of them may conveniently be held; after which I shall hold myse
free to declare which of them / intend to hold."
332 A Survey of Symbolic Logic
"All American silver coins dated 1915 have milled edges" or "It rained
every week in March", or in general, the frequent universal propositions
which predicate this relation of extension.
And whichever interpretation of Hi(<f>x c\f/x) be chosen, we can now
point out one interesting peculiarity of it. We quote from Principia
Mathematica: n "In the usual instances of implication, such as "'Socrates
is a man' implies 'Socrates is a mortal'", we have a proposition of the
form " <px c \j/x" in a case in which "ILx(<px c \f/x) " is true. In such a case,
we feel the implication as a particular case of a formal implication ". It
might be added that " ' Socrates is a man ' implies ' Socrates is a mortal ' '
is not a formal implication: it is a material implication and a strict impli
cation, but not formal. One may object: "But as a fact, in such cases
there is a tacit premise of the type ' All men are mortal ', and this is precisely
the formal implication,^^ c \j/x) ". Granted, of course. But add this
premise, and still the implication is strict and material, but not formal.
"All men are mortal and Socrates is a man" does not formally imply "Soc
rates is mortal".12 If the "proper" meaning of "implies" is one in which
"Socrates is a man" really and truly implies "Socrates is mortal", or one
in which "All men are mortal and Socrates is a man" really and truly
implies "Socrates is mortal", then formal implication is not that proper
meaning. However much any formal implication may lie behind and
support such an inference, it cannot state it.
One further consideration is worthy of note: If ILx(<px c \f/x) be restricted
to oj's which exist, then it will denote such propositions as "' x is an Ameri
can silver coin of 1915' implies 'x has a milled edge'"; "'x is a Monday
of last March' implies '# is a rainy day ' "; "x has horns and divided hoofs'
implies 'x chews a cud'". In other words it will denote relations which are
"contingent", and due to "coincidence". It may be doubted whether
such relations are "properly" implications. But upon this question the
reader will very likely find himself in doubt. What we regard as the
reason for this doubt will be pointed out later.
The strict implication, p -i q, means " It is impossible that p be true
11 1, p. 21. We render the symbolism of this passage in our own notation.
12 It may be objected that the calculus gives the formal implication
which is the formal implication of \J/z by Hx(<px c \f/x) x <pz, which is required. But this is
not what is required. The variables for all values of which this proposition is asserted are
<? and <A> not x and z. The reader will grasp the point if he specify <p and ^ here, and then
allow them to vary in his illustration.
The System of Strict Implication 333
and q false", or "p is inconsistent with the denial of q". Similarly $x -t fa
means "It is impossible that <?x be true and fa false", or "the assertion
of <px is inconsistent with the denial of fa". Some explanation of "im
possible" or "inconsistent" may seem called for here. These terms can
either of them be explained by the other, but one or the other must be taken
for granted. Yet the following observations may be of assistance: An
assemblage or set of propositions may be such that all of them can be true
at once. They are mutually compatible, compossible, consistent. There
may be more than one such set. Whoever denies this on metaphysical
grounds must assume the burden of proof. And whether, in fact, the
possible and the actual, the consistent and the concurrently true-in-fact
are identical, at least one must admit that our concept of the possible
differs from our concept of the actual: that we mean by "consistent" some
thing different from "concurrently true-in-fact". Any set of mutually
consistent propositions may be said to define a "possible situation" or
"case" or "state of affairs". And a proposition may be "true" of more
than one such possible situation — may belong to more than one such set.
Whoever understands "possible situation" thereby understands "con
sistent propositions", and vice versa. And whoever understands "im
possible situation" understands also "inconsistent propositions". In
these terms, we can translate p -J q by "Any situation in which p should be
true and q false is impossible".
But "situation" as here used should not be confused with Boole's
"times when A is true". A proposition, once true, is always true. A
proposition may be true of some possible "situations" and false of others,
but it must be in point of fact either simply true or simply false. This is
what constitutes the distinction between a proposition and a prepositional
function such as "x is a man". This last is, in point of fact, neither true
nor false.13
Of special interest are the cases of strict implication in which more
than two propositions are involved. We have already seen that strict
triadic relations take the form of strict dyads, one member of which is
itself a non-strict or material dyad. Where we might expect, p -i (q-*r),
we have instead p 4 (q c r) or p q 4 r. Instead of p o (q o r) we have
p o (q r) or (p q) o r. We may now discover the reason for this— the reason
13 On the other hand, it is impossible to deny that a proposition may be true of some
"situations'', and false of others unless one is prepared to maintain that whatever assertion
can be referred to different possible "circumstances", is not a proposition. And whoever
asserts this must, to be consistent, recognize that there is only one true proposition, the
whole of the truth, the assertion of all-fact, the Hegelian Idee.
334 A Survey of Symbolic Logic
not in mathematical-wise but in terms of common sense about inference.
Suppose that p, q, and the negation of r, form an inconsistent set. They
cannot all be true of any possible situation. We have symbolized this by
-(poqo-r).
- (p oqo -r) = -lp o (q -r)] = p -J -(q -r) = p-i(qcr) = p q -* r
If p, q, and -r form an inconsistent set and, in point of fact p and q are both
true, then r must be true also. So much is quite clear. The inference
from (p q) to r is strict. But suppose p, q, and -r cannot all be true in
any possible situation and suppose (in the actual situation) p only is known
to be true. We can then conclude that "If q is true, r is true".
-(poqo-r) = p-l(qcr)
This inference is also strict, but our symbolic equivalents tell us that this
"If . . . , then . . ."is not itself a strict implication; it is qcr, a
material implication. That is the puzzle; why is it not strict like the
other? The answer is simple. If p, q, and -r cannot all be true in any
possible situation and if p is true of the actual situation, it follows that
q and -r are not both true of the actual situation, that is, -(q-r), but it
does not follow that q and -r cannot both be true in some other possible
situation (in which p should be false) — it does not follow that q and -r are
inconsistent, that -(qo-r). Consequently it does not follow that qlr,
that q strictly implies r. If, then, we begin with an a priori truth (holding
for all possible situations), that p, q, and -r form an inconsistent set, and
to this add the (empirical) premise "p is true", we get, as a strict con
sequence, the proposition "If q is true, r is true". But the truth of this
consequence is confined to the actual situation, like the premise p. If, in
this case, we go on and infer r from q, our inference may be said to be valid
because the additional premise, p, required to make it strict, is taken for
granted. The inference depends on pq-ir. Or we may, if we prefer,
describe it as an inference based on material implication, which is valid
because it is confined to the actual situation. Much of our reasoning is
of this type. We state, or have explicitly in mind, only some of the premises
which are required to give the conclusion strictly. We have omitted or
forgotten the others, because they are true and are taken for granted.
In this sense, much of our reasoning may be said to make use of material
or formal implications. This is probably the source of our doubts whether
such propositions as f"x is an American silver coin dated 1915' implies
lx has a milled edge'", and '"x has horns and divided hoofs' implies 'x
The System of Strict Implication 335
chews a cud'" represent what are "properly" called implications. In
such cases, the reasoning is valid only if the missing premises, which would
render the implication strict, are capable of being supplied.
The case wrhere two premises are strictly required for inference is typi
cal of all those which require more than one. Where three, p, q, and r,
are required for a conclusion, s, we have
p(qr)*s = p*(qrc s) = p * [q c (r c *)]
And similar equations hold where four, five, etc., premises are required
for a conclusion. Only the main implication is strict. In other words, a
strict implication may be complex but is always dyadic.
Another significant property of strict implication, as opposed to material
implication or formal implication, is that if we have '"x is a man' strictly
implies 'x is a mortal'", we have likewise "' Socrates is a man' strictly
implies ' Socrates is a mortal ' ", and vice versa. Propositions strictly imply
each other when and only when any corresponding prepositional functions
similarly imply one another. According to this view of implication,
"'Socrates is a man' implies 'Socrates is a mortal'" is not simply felt to
be the kind of relation upon which most inference depends: it is the rela
tion upon which all inference does depend. Strict implication is the
symbolic representative of an inference which holds equally well whether
its terms are propositions or prepositional functions.
One further item concerning the properties of strict implication has to
do with the analogues of the "peculiar" propositions of Material Impli
cation. These analogues are themselves somewhat peculiar:
3-52 ~p -* (p 4 q) If p is impossible, then p implies any proposition, q.
3 . 55 ~-pi (q -j p) If p is necessarily true, then p is implied by any
proposition, q'.
These two are the critical members in this class of propositions: the re
mainder follow from them and are of similar import. In the "proper"
sense of "implies", does an absurd, not-self-consistent proposition imply
anything and everything? A part of the answer is contained in the
observation that "necessary" and "impossible" in every-day use are
commonly hyperbolical and no index. Xo proposition is "impossible"
in the sense of ~p except such as imply their own contradiction; and no
proposition is "necessary" in the sense of - -p unless its negation is self-
contradictory. Again, the implications of an absurd proposition are no
indication of what would be true if that absurd proposition were true.
336 A Survey of Symbolic Logic
It is the nature of an absurd proposition that it is not logically conceivable
that it should be true under any possible circumstances. And, finally,
we can demonstrate that, in the ordinary sense of "implies", an impossible
proposition implies anything and everything. It will be granted that in
the "proper" sense of ."implies", (1) "p and q are both true" implies "q is
true". And it will be granted that (2) if two premises p and q imply a
conclusion, r, and that conclusion, r, is false, while one of the premises,
say p, is true, then the other premise, g, must be false. That is, if "All
men are liars" and "John Blank is a man" together imply "John Blank
is a liar", but "John Blank is a liar" is false, while "John Blank is a man"
is true, then the other premise, "All men are liars", must be false. And it
will be granted that (3) If the two propositions, p and q, together imply r,
and r implies s, then p and q together imply s. These three principles being
granted, it follows that if q implies r, the impossible proposition " q is true
but r false" implies anything and everything. For by (1) and (3), if q
implies r, then "p and q are both true" implies r. But by (2), if "p and q
are both true" implies r, "q is true but r is false" implies "p is false".
Hence if q implies r, then " q is true but r is false" implies the negation of
any proposition, p. And since p itself may be negative, this impossible
proposition implies anything. "Today is Monday" implies "Tomorrow
is Tuesday". Hence "Today is Monday and the moon is not made of
green cheese " implies " Tomorrow is Tuesday ". Hence " Today is Monday
but tomorrow is not Tuesday" implies "It is false that the moon is not
made of green cheese", or "The moon is made of green cheese".
This may be taken as an example of the fact that an absurd proposition
implies any proposition. It should be noted that the principles of the
demonstration are quite independent of anything we have assumed about
strict implication, though they accord with our assumptions.
We shall now demonstrate: first, that there are a considerable class of
propositions which imply their own contradiction and are thus impossible,
and a class of propositions which are implied by their own denial and are
thus necessary; and second, that an impossible proposition implies any
proposition, and a necessary proposition is implied by any. These proofs
will be similarly free from any necessary appeal to symbolism, making use
only of indubitable principles of ordinary logic.
Any proposition which should witness to the falsity of a law of logic,
or of any branch of mathematics, implies its own contradiction and is
absurd. " p implies p" is a law of logic; and may be used as an example.
The System of Strict Implication 337
In general, any implication, "p implies </," is shown false by the fact that
"p is true and q is false". Thus the law "p implies p" would be disproved
by the discovery of any proposition p such that "p is true and p is false".
This is, then, an impossible proposition, "p is true and p is false" implies
its own negation, which is "At least one of the two, not-p and p, is true".
For "p is true and p is false" implies "p is true". And "p is true" implies
"At least one of the two, p and q, is true". And not-?;, "p is false," may
be this q. Hence " p is true " implies " At least one of the two, p and not-p,
is true ". Hence " p is true and p is false " implies " At least one of the two,
p and not-p, is true". The negation of "r/ is true and r is false" is "At
least one of the two, r and not-ry, is true". If p here replace both q and r,
we have as the negation of "p is true and p is false", "At least one of the
two, p and not-p, is true". And it is this which "p is true and p is false"
has been shown to imply.
Merely for purposes of comparison, we resume this proof in the symbols
of Strict Implication:
p H p, and p •* p = ~(p -p). Hence (p -p) is an impossible proposition.
By the principle p q -J p, we have p -p -J p.
And by the principle q4p + q, we have p -1 (-p + p).
By the principle (p -J </)(</ -J r) H (p -J r), this gives (1) x (2)
p -P -* ("P + P)
But (-p + p) = -(p -p). Hence p -p -I -(p -p).
This is only one illustration of a process which might be carried out in
any number of cases. Take any one of the laws of Strict Implication and
transform it into a form which has the prefix, ~. For example,
The impossible proposition thus discovered, in the example [(p q) -(</ p)],
can always be shown to imply its own negation. The reader will easily see
how this may be done. Such illustrations are quite generally too complex
to be followed through without the aid of symbolic abbreviation, but only
the principles of ordinary logic are necessary for the proofs.
Wherever we find an impossible proposition, we find a necessary propo
sition, its negation. For example, "At least one of the two, p and not-p,
is true" is a necessary proposition. We have just demonstrated that
implied by its own denial.
(Some logicians have been inclined of late to deny the ex
23
338 A Survey of Symbolic Logic
necessary propositions and of impossible, or self-contradictory, propositions.
We beg their attention to the above, and request their criticisms.)
We shall now prove that every impossible proposition — i. e., every
proposition which implies its own negation — implies anything and every
thing. If p implies not-p, then p implies any proposition, q. WTe have
already shown that if q implies r, then "q is true and r is false" implies any
proposition. Hence if p implies not-p, "p is true but not-p is false",
that is, "p is true and p is true", implies any proposition, q. But p is
equivalent to "p is true and p is true". Hence if p implies not-p, p implies
any proposition, q.
Any necessary proposition, i. e., any proposition, q, whose denial, not-g,
implies its own negation, is implied by any proposition, r. This follows
from the above by the principle that if p implies q, then "q is false " implies
"p is false". In the theorem just proved, "If p implies not-p, then p
implies any proposition, g", let p be "not-g", and q be "not-r". We
then have "If not-g implies g, then not-g implies any proposition, not-r".
And if not-g implies not-r, then "not-r is false" implies "not-g is false",
i. e., r implies g. Hence if not-g implies g, then any proposition, r, implies g.
But "a man convinced against his will is of the same opinion still".
In what honest-to-goodness sense are the "necessary" principles of logic
and mathematics implied by any proposition? The answer is: In the
sense of presuppositions. And what, precisely, is that? Any principle, A,
may be said to be presupposed by a proposition, B, if in case A were false,
B must be false. If a necessary principle were false, anything to which
it is at all relevant would be false, because the denial of such a principle,
being an impossible proposition, implies the principle itself. And where a
principle and its negative are both operative in a system, anything which
is proved is liable to disproof. Imagine a system in which there are con
tradictory principles of proof. That the chaotic results which would ensue
are not, in fact, valid, requires as presuppositions the truth of the necessary
laws of the system. These laws— those strictly "necessary"— are always
logical in their significance. The logic of "presupposition" is, in fact, a
very pretty affair — we have no more than suggested its character here.
The time-honored principles of rationalism are thoroughly sound and
capable of the most rigid demonstration, however much the historic rational
ists have stretched them to cover what they did not cover, and otherwise
misused them.
In this respect, then, in which the laws of Strict Implication seemed
The System of Strict Implication 339
possibly not in accord with the " proper " sense of " implies ", we have demon
strated that they are, in fact, required by obviously sound logical principles,
though in ways which it is easy to overlook.
It may be urged that every demonstration we have given shows not
only that impossible propositions imply anything and necessary proposi
tions are implied by anything, but also that a false proposition implies
anything, and a true proposition is implied by anything. The answer is
that an impossible proposition is false, of course, and a necessary proposition
is true. But if anyone think that this validates the doubtful theorems of
Material Implication, it is incumbent upon him to show that some proposi
tion that is false but not impossible implies anything and everything, and
that some proposition which is true but not necessary is implied by all propo
sitions. And this cannot be done.
We shall not further prolong a tedious discussion by any special plea
for the " propriety" of strict implication as against material implication
and formal implication. Anyone who has read through so much technical
and uninteresting matter has demonstrated his right and his ability to
draw his own conclusions.
CHAPTER VI
SYMBOLIC LOGIC, LOGISTIC, AND MATHEMATICAL
METHOD
I. GENERAL CHARACTER OF THE LOGISTIC METHOD. THE " ORTHODOX"
VIEW
The method of any science depends primarily upon two factors, the
medium in which it is expressed and the type of operations by which it
is developed. "Logistic" may be taken to denote any development of
scientific matter which is expressed exclusively in ideographic language and
uses predominantly (in the ideal case, exclusively) the operations of sym
bolic logic. Though this definition would not explicitly include certain
cases of what would undoubtedly be called " logistic", and we shall wish
later to present an alternative view, it seems best to take this as our point
of departure.
"Modern geometry" differs from Euclid most fundamentally by the
fact that in modern geometry no step of proof requires any principle except
the principles of logic.1 It was the extra-logical principles of proof in
Euclidean geometry and other branches of mathematics which Kant
noted and attributed to the "pure intuition" of space (and time) as the
source of "synthetic judgments a priori" in science. The character of
space (or of time), as apprehended a priori, carries the proof over places
where the more general principles of logic — "analysis" — cannot take it.
Certain operations of thought are, thus, accepted as valid in geometry
because geometry is thought about space, and these transformations are
valid for spatial entities, though they might not be valid for other things.
The principal impetus to the modern method in geometry came from the
discovery of non-Euclidean systems which must necessarily proceed, to
some extent, without the aid of such space intuitions, a priori or otherwise.
And the perfection of the modern method is attained when geometry is
entirely freed from dependence upon figures or constructions or any appeal
1 In the opinion of most students, Euclid himself sought to give his proofs the rigorous
character which those of modern geometry have, and the difference of the two systems is
in degree of attainment of this ideal. But Euclid's successors introduced methods which
still further depended upon intuition.
340
Symbolic Logic, Logistic, and Mathematical Method 341
to the perceptual character of space. When geometry is thus freed from
this appeal to intuition or perception, the methods of proof are simply
those which are independent of the nature of the subject matter of the science —
that is, the methods of logic, which are valid for any subject matter.2
Coincidently with this alteration of method comes another change-
geometry is now abstract. If nothing in the proofs depends upon the fact
that the terms denote certain spatial entities, then, whatever may be meant
by "point", "plane", "triangle", "parallel", etc., if the assumptions be
true, then the theorems will be. Or in any sense in which the assumptions
can be asserted, in that same sense all the consequences of them can be
asserted. The student may carry in his mind any image of "triangle"
or "parallels" which is consistent with the propositions about them. More
than this, even the geometrical relations asserted to hold between "points",
"lines", etc., may be given any denotation which is consistent with the
properties assigned to them. In general, this means for relations, that any
meaning may be assigned which is consistent with the type of the relation—
e. g., transitive or intransitive, symmetrical or unsymmetrical, one-one or
one-many, etc. — and with the distributions of such relations in the system.
Essentially the same evolution has taken place in arithmetic, or "alge
bra". Any reference to the empirical character of tally marks or collections
of pebbles has become unnecessary and naive. The "indefinables" -of
arithmetic are specified, very likely, as "A class, K, of elements, a, 6, c,
etc., and a relation (or 'operation') +". Definitions have come to have
the character of what Kant called "transcendental definitions "—that is
to say, they comprehend those properties which differentiate the entity to
be defined by its logical relations, not those which distinguish it for sense
perception. The real numbers, for instance, no longer denote the possible
lengths of a line, but are the class of all the "cuts" that can be made (logi-
2 1 cannot pass over this topic without a word of protest against the widespread notion
that the development of modern geometry demonstrates the falsity of Kant's Transcendental
Aesthdik. It does indeed demonstrate the falsity of Kant's notion that such "synthetic"
principles are indispensable to mathematics. But, in general, it is accurate to say that
Kant's account is concerned with the source of our certainty about the world of nature,
not with the methods of abstract science which did not exist in his day. Nothing is mo
obvious than that the abstractness of modern geometry comes about through definitely
renouncing the thing which Kant valued in geometry— the certainty of its applicability
to our space. When geometry becomes abstract, the content of the science of space spin
into two distinct subjects: (1) geometry, and (2) the metaphysics of space, which is con
cerned with the application of geometry. This second subject has been much discui
since the development of modern geometry, usually in the skeptical or "pragmatic
(vide Poincare). But it is possible— and to me it seems a fact— that Kant's basic argu
ments are, with qualifications, capable of being rehabilitated as arguments concerning t)
certainty of our knowledge of the phenomenal world, i. e. as a metaphysics of space.
342 A Survey of Symbolic Logic
cally specified) in a dense, denumerable series, of the type of the series of
rationals.
Thus abstractness and the rigorously deductive method of development
have more and more prevailed in the most careful presentations of mathe
matics. When these are completely achieved, a mathematical system becomes
nothing more nor less than a complex logical structure.3 Consider any two
mathematical systems which have been given this ideal mathematical form.
They will not be distinguished by the entities which form their "subject
matter", for the terms of neither system have any fixed denotation. And
they will not be distinguished by the operations by means of which they
are developed, for the operations will, in both cases, be simply those of
logical demonstration.
A word of caution upon the meaning of "operation" is here necessary.
It is exactly by the elimination of all peculiarly mathematical operations
that a system comes to have the rigorously deductive form. For the
grocer who represents his putting of one sack of sugar with another sack
by 25 + 25 = 50, [+] is a symbol of operation. For the child who learns
the multiplication table as a means to the manipulation of figures, [X]
represents an operation, but in any rigorously deductive development of
arithmetic, in Dedekind's Was sind und was sollen die Zahlen, or Hunting-
ton's "Fundamental Laws of Addition and Multiplication in Elementary
Algebra", [+] and [X] are simply relations. An operation is something
done, performed. The only things performed in an abstract deductive system
are the logical operations — variables are not added or multiplied. But,
unfortunately, such relations as [+] and [X] are likely to be still spoken
of as "operations". Hence the caution.
Since abstract mathematical systems do not differ by any fixed meaning
of their terms, and since they are not distinguished through their operations,
they will be different from one another only with respect to the relations
of their terms, and probably also in certain relations of a higher order-
relations of relations. And the relations, being likewise abstract, will
differ, from system to system, only in type and in distribution in the systems;
that is, any two systems will differ only as types of logical order.
3M. Fieri, writing of "La Geometrie envisagee comme systeme purement logique",
says: "Je tiens pour assure que cette science, dans ces parties les plus elevees comme
dans les plus modestes, va en s'affirment et en consolidant de plus en plus comme I'etude
d'un certain ordre de relations logiques; en s'affranchissant peu a peu des liens qui 1'attachent
a 1'intuition, et en revetant par suite la forme et les qualites d'un science ideale purement
deductive et abstraite, comme V Arithmetique" . (Bibliotheque du congres Internationale de
Philosophic, in, 368.)
Symbolic Logic, Logistic, and Mathematical Method 343
The connection between abstract or "pure" mathematics and logistic
is, thus, a close one. But the two cannot be simply identified. For the
logical operations by which the mathematical system is generated from its
assumptions may not themselves be expressed in ideographic symbols.
Ordinarily they are not: there are symbols for "four" and "congruent",
"triangle" and "plus", but the operations of proof are expressed by "If
. . . then . . .", "Either . . . or . . .," etc. Only when the logical
operations also are expressed in ideographic symbols do we have logistic.
In other words, all rigorously deductive mathematics gets its principles of
operation from logic; logistic gets its principles of operation from symbolic
logic. Thus logistic, or the logistic development of mathematics, is a name
for abstract mathematics the logical operations of whose development
are represented in the ideographic symbols of symbolic logic.
Certain extensions of symbolic logic, as we have reviewed it, are needed
for the satisfactory expression of these mathematical operations — particu
larly certain further developments of the logic of relations, and the theory
of what are called "descriptions" in Principia Mathematica. But these
necessary additions in no wise affect what has been said of the relation
between symbolic logic and the logistic development of mathematics.
II. Two VARIETIES OF LOGISTIC METHOD: PEANO'S Fornmlaire AND
Principia Mathematica. THE NATURE OF LOGISTIC PROOF
The logistic method is, then, a universal method, applicable to any
sufficiently coordinated body of exact knowledge. And it gives, in mathe
matics, a most precise and compact development, displaying clearly the
type of logical order which characterizes the system. However, there are
certain variations of the logistic method, and systems so developed may
differ widely from one another in ways which have nothing directly to do
with the type and distribution of relations. One most important difference
has to do with the degree to which the analysis of terms is carried out.
"Number," for example, may be taken simply as a primitive idea, or it
may be defined in terms of more fundamental notions. And these notions
may, in turn, be defined. The length to which such analysis is carried, is
an important item in determining the character of the system. Correl-
atively, relations such as [+] and [X] may be taken as primitive, or they
may be defined. And, finally, the fundamental propositions which generate
the system may be simply assumed as postulates, or they may, by the analy
sis just mentioned, be derived from those of a more elementary discipline.
344 A Survey of Symbolic Logic
In general, the analysis of "terms" and of relations and the derivation of
fundamental propositions go together. And the use of this analytic
method requires, to some extent at least, a hierarchy of subjects, with
symbolic logic as the foundation of the whole.
To illustrate these possible differences between logistic systems, it will
be well to compare two notable developments of mathematics: Formulaire
de Mathematiques4 of Peano and his collaborators, and Principia Mathe-
matica of Whitehead and Russell. These two are by no means opposites
in the respects just mentioned. Principia Mathematica represents the
farthest reach of the analytic method, having no postulates and no primitive
ideas save those of the logic, while the Formulaire exhibits a partially hier
archic, partially independent, relation of various mathematical branches.5
For example, in the Formulaire, the following primitive ideas are assumed
for arithmetic, which immediately succeeds "mathematical logic".
X0 signifies 'number', and is the common name of 0, 1, 2, etc.
0 signifies 'zero'.
+ signifies 'plus'. If a is a number, a + indicates 'the number suc
ceeding a'.6
The primitive propositions, or postulates, are as follows: 7
1-0 No e Cls
1-1 OeN0
1-2 acNo.D.a + eNo
1 • 3 s e Cls • Oes:aes.D,,.a + €$SD.No€S
1-4 a, 6 e NO • a -f- =6 + .D.a = 6
l-5aeN0.3.a + -=0
The symbol D here represents ambiguously "implies" or "is contained
in"— the relation c of the Boole-Schroder Algebra. This and the idea of
a class, "Cls", and the e-relation, are defined and their properties demon
strated in the "mathematical logic". In terms of these, the above propo
sitions may be read:
4 All our references will be to the fifth edition, which is written in the proposed inter
national language, Interlingua, and entitled Formulario Mathemalico, Editio v (Tomo v de
Formulario complete).
5 The independence of various branches in the Formulaire is somewhat greater than a
superficial examination reveals. Not only are there primitive propositions for arithmetic
and geometry, but many propositions are assumed as "definitions" which define in that
discursive fashion in which postulates define, and which might as well be called postulates.
Observe, for example, the definitions of + and X, to be quoted shortly.
6 Section n, § 1, p. 27.
7 Ibid.
Symbolic Logic, Logistic, and Mathematical Method 345
1-0 No is a class, or 'number' is a common name.
1-1 0 is a number.
1-2 If a is a number, then the successor of a is a number.
1-3 If s is a class, and if 0 is contained in s, and if, for every a, .'a is
contained in s' implies 'the successor of a is contained in s', then No is
contained in s (every number is a member of the class s).
(1-3 is the principle of ''mathematical induction".)
1-4 If a and b are numbers, and if the successor of a =* the successor
of b, then a = b.
1-5 If a is a number, then the successor of a 4= 0.
The numbers are then defined in the obvious way: 1 = 0 -f, 2 = 1 +,
3 = 2+, etc.8 The relation +, which differs from the primitive idea, a + ,
is then defined by the assumptions: 9
3 • 1 a e No . 3 . a + 0 = a
(If a is a number, then a -f 0 = a.)
3-2 a, fceNo.D.o + (6 +) = (a + b) +
(If a and b are numbers, then a + 'the successor of 6' 'the successor of
a + 6'.)
The relation X is defined by: 10
1-0 a, 6, ceN0.3.a X 0 = 0
1 -01 a, b, c e No - s - a X (b + 1) = (a X 6) + a
It will be clear that, except for the expression of logical relations, such
as e and D , in ideographic symbols, these postulates and definitions are of
the same general type as any set of postulates for abstract arithmetic.
A class, X0, of members a, b, c, etc., is assumed, and the idea of a +, "suc
cessor of a". The substantive notions, "number" and "zero", the de
scriptive function, "successor of," the relations + and X, are not analysed
but are taken as simple notions.11 However, the properties which numbers
have by virtue of being members of a class, X0, are not taken for granted, as
would necessarily be done in a non-logistic treatise— they are specifically
set forth in propositions of the "mathematical logic" which precedes.
And the other principles by which proof is accomplished are similarly
demonstrated. Of the specific differences of method to which this explicit-
ness of the logic leads, we shall speak shortly.
8 See ibid., p. 29.
9 Ibid.
10 See ibid., § 2, p. 32.
" Peano does not suppose them to be unanalyzdble. He says (p. 27):
si nos pote defini N0, significa si nos pote scribe aequalitate de forma, N0 = expressione
composite per signos noto ~ ~ 1 . . . -, quod non est facile". (This was written after the
publication of Russell's Principles of Mathematics, but before Principia Mathem
346 A Survey of Symbolic Logic
In Principia Mathematica, there are no separate assumptions of arith
metic, except definitions which express equivalences of notation and make
possible the substitution of a single symbol for a complex of symbols.
There are no postulates, except those of the logic, in the whole work. In
other words, all the properties of numbers, of sums, products, powers, etc.,
are here proved to be what they are, solely on account of what number is,
what the relations + and X are, etc. Postulates of arithmetic can be
dispensed with because the ideas of arithmetic are thoroughly analysed.
The lengths to which such analysis must go in order to derive all the proper
ties of number solely from definitions is naturally considerable. We should
be quite unable, within reasonable space, to give a satisfactory account of
the entities of arithmetic in this manner. In fact, the latter half of Volume
I and the first half of Volume II of Principia Mathematica may be said to
do nothing but just this. However, we may, as an illustration, follow out
the analysis of the idea of "cardinal number". This will be tedious but,
with patience, it is highly instructive.
We shall first collect the definitions which are involved, beginning with
the definition of cardinal number and proceeding backward to the definition
of the entities in terms of which cardinal number is defined, and then to
the entities in terms of which these are defined, and so on.12
*100-02 NC - D'Xc. Df
"Cardinal number" is the defined equivalent of "the domain of (the rela
tion) Xc".
*33-01 D = aR[a = .r{(32/l . x R y]. Df
"D" is the relation of (a class) a to (a relation) R, when a and R are such
that a is (the class) x which has the relation R to (something or other) y.
That is, "D" is the relation of a class of .T'S, each of which has the relation
R to something or other, to that relation R itself.
*30-01 R'y = (ix)(xRy). Df
" R'y" means "the x which has the relation R to ?/".
Putting together this definition of the use of the symbol ' and the
definition of "D", we see that "D'Xc" is "the x which has the relation
D to Nc", and this x is a class a such that every member of a has the rela-
12 The place of any definition quoted, in Principia, is indicated by the reference number.
The " translations" of these definitions are necessarily ambiguous and sometimes inaccurate,
and, of course, any "translation" must anticipate what here follows — but in Principia
precedes.
Symbolic Logic, Logistic, and Mathematical Method 347
tion Nc to something or other. "D'/J" is "the domain of the relation R".
If "R" be "precedes", then "D7T will be "the class of all those things
which precede anything". "Cardinal number", "NC," is defined as
"D'Nc", "the domain of the relation Xc".
We now turn to the meaning of "Nc".
*100-01 Nc = sm. Df
" Nc " is the relation of the class of referents of " sm " to " sm " itself. First,
let us see the meaning of the arrow over "sm".
*32-01 R = ay[a = $(xRy)}. Df
" R" is "the relation of a to y, where a and y are such that a is the class
of x's, each of which has the relation R to y" . If " R" be "precedes",
" R" will be the relation of the class "predecessors of y7' to y itself.
Now for "sm". We shall best not study its definition but a somewhat
simpler proposition.
*73-l asm/3. = . (3/0 .flcl ->1 .« = I)'/? . 0 = G7?
"a sm 0" is equivalent to "For some relation R, R is a one-to-one relation,
while a is the domain of R and 0 is the converse-domain of /?".
We have here anticipated the meaning of "G'/f" and of " 1 — > 1".
*33-02 Q = M[0 = y{(Xx).xRy}]. Df
"G" is "the relation of (a class) 0 to (a relation) R, when 0 and R are
such that 0 is the class of #'s, for each of which (something or other) x has
the relation R to y". Comparing this with the definition of "D" and of
"D'R" above, we see that "Q'/?", the converse-domain of /?, is the class
of those things to which something or other has the relation R. If " R"
be "precedes", "G*/?" will be the class of those things which are preceded
by something or other.
*71 -03 !->!== R(R"d'R c 1 . R"D'R c 1). Df
This involves the meaning of "#", of ", and of " 1 "
*32-02 R = 0.r|0 = $(xRy)}. Df
"5" signifies "the relation of 0 to x, when 0 and .r are such that 0 is the
class of i/'s to which x has the relation R.
^37-01 fl"0 = if-((3y) .yep.xRy}. Df
"#"0" is "the class of .r's such that, for some ?/, ?/ is a member of 0, and z
has the relation R to ?/. In other words, "#"0" (the R's of the 0's) is the
348 A Survey of Symbolic Logic
class of things which have the relation R to some member or other of the
class 0. If "R" be "precedes", "fl"0" will be the class of predecessors of
all (any) members of /?.
With the help of this last and of preceding definitions, we can now read
•#71-03. "1 -» 1" is "the class (of relations) R, such that whatever has
the relation R to any member of the class of things-to-which-anything-has-
the-relation-72, is contained in 1; and whatever is such that any member
of the class of those-things-which-have-the-relation-#-to-anything has the
relation R to it, is contained in 1 . " Or more freely and intelligibly : " 1 -» 1 "
is the class of relations, R, such that if a R 13 is true, then a is a class of
one member and (3 is a class of one member: " 1 — > 1 " is the class of all
one-to-one correspondences. Hence "asmjS" means "There is a one-to-
one correspondence of the members of a. with the members of (3. "sm"
is the relation of classes which are (cardinally) similar.
The analysis of the idea of cardinal number has now been carried out
until the undefined symbols, except "1", are all of them logical symbols; —
of relations, R; of classes, a, /3, etc.; of individuals, x, y, etc.; of propo-
sitional functions such as x R y [which is a special case of <p(x, y)]\ of
" (px for some x", (3.x) . <px; the relations e, c, and =; and the idea
(1 x)(<px), "the x for which px is true". This last notion occurs in various
special cases, such as D^R, R"0, etc.
" 1 " is also defined in terms which reduce to these, but the definitions
involved are incapable of precise translation — more accurately, ordinary
language is incapable of translating them.
*52-01 I == a{&x) .a = i'x}. Df
*ol-01 1 = 7. Df
*50-01 7 = xy (x = y). Df13
"7" is the relation of identity; "i" is the class of those things which have
the relation of identity to something or other; and " 1 " is the class of such
classes, i. e., the class of all classes having only a single member. Thus the
definition of " 1 " is given in terms of the idea of individuals, x and yt of
the relation = , of classes, and the idea involved in the use of the arrow
over 7, which has already been analyzed. This definition of 1 is in no
wise circular, however much its translation may suggest that it is; nor is
there any circularity involved in the fact that the definition of cardinal
number requires the previous definition of " 1 ".
13 Strictly, analysis of =, which differs from the defining relation, [. . . = ... Df], is
required. But the lack of this does not obscure the analysis, so we omit it here.
Symbolic Logic, Logistic, and Mathematical Method 349
We have now completely accomplished the analysis of the idea of cardinal
number into constituents all of which belong to mathematical logic. The
important significance of this analysis for the method involved we must
postpone for a moment to discuss the definition itself.
If we go back over these definitions, we find that the notion of cardinal
number can now be defined as follows: "Cardinal number" is the class of
all those classes the members of which have a one-to-one correspondence
(with members of some other class). "Cardinal number" is the class of
all the cardinal numbers; and a cardinal number is the class of all those
classes whose members have a one-to-one correlation with the members of a
given class. This is definition "in extension". We most frequently think
of the cardinal number of a class, a, as a property of the class. Definition
in extension determines any such property by logically exhibiting the class
of all those things which have that property. Thus if a be the class com
posed of Henry, Mary and John, the cardinal number of a will be deter
mined by logically exhibiting all those classes which have a one-to-one
correlation with the members of a — i. e., all the classes with three members.
"3" will, then, be the class of all classes having three members; "4", the
class of all classes of four, etc. And "cardinal number" in general will be
the class of all such classes of classes.
It may be well to observe here also that, by means of ideographic sym
bols, we can represent exactly, and in brief space, ideas which could not
possibly be grasped or expressed or carried in mind in any other terms.
Perhaps the reader has not grasped those presented: we can assure him it
is not difficult once the symbolism is clear. And if the symbolism appals
by its unfamiliarity, we would call attention to the fact that the number of
different symbols is not greater, nor is their meaning more obscure than
those of the ordinary algebraic signs. It is the persistent accuracy of the
analysis that has troubled him; far be it from us to suggest that we do not
like to think accurately.
So much analysis may appeal to us as unnecessary and burdensome.
But observe the consequences of it for the method. When "cardinal
number" is defined as "D'Xc," all the properties of cardinal number follow
from the properties of "D" and "Xc" and the relation between these
represented by '. And when these in turn are defined in terms of "sm"
and the idea expressed by the arrow, and so on, their properties follow from
the properties of the entities which define them. And finally, when all the
constituents of "cardinal number", and the other ideas of arithmetic
350 A Survey of Symbolic Logic
have been analyzed into ideas which belong to symbolic logic, all the propo
sitions about cardinal number follow from these definitions. When analysis
of the ideas of arithmetic is complete, all the propositions of arithmetic
follow from the definitions of arithmetic together with the propositions of
logic. Now in Principia Mathematica it is found possible to so analyze all
the ideas of mathematics. Hence the whole of mathematics is proved
from its definitions together with the propositions of logic. And, except
the logic, no branch of mathematics needs any primitive ideas or postulates of
its own. It is thus demonstrated by this analysis that the only postulates
and primitive ideas necessary for the whole of mathematics are the postu
lates and primitive ideas of logic.
In the light of this, we can understand Mr. Russell's definition of
mathematics: 14
"Pure Mathematics is the class of all propositions of the form 'p im
plies q where p and q are propositions containing one or more variables,
the same in the two propositions, and neither p nor q contains any constants
except logical constants. And logical constants are all notions definable
in terms of the following: Implication, the relation of a term to the class
of which it is a member, the notion of such that, the notion of relation, and
such further notions as may be involved in the general notion of propositions
of the above form."
The content of mathematics, on this view, is the assertion that certain
propositions imply certain others, and these propositions are all expressible
in terms of "logical constants", that is, the primitive ideas of symbolic
logic. These undefined notions, as the reader is already aware, need not
be numerous: ten or a dozen are sufficient. And from definitions in terms,
finally, of these and from the postulates of symbolic logic, the whole of
mathematics is deducible.
The logistic development of a mathematical system may, like the arith
metic of the Formulaire, assume certain undefined mathematical ideas and
mathematical postulates in terms of these ideas, and thus differ from an
ordinary deductive system of abstract mathematics only by expressing the
logical ideas which occur in its postulates by ideographic symbols and by
using principles of proof supplied by symbolic logic. Or it may, like
arithmetic in Principia Mathematica, assume no undefined ideas beyond
those of logic, define all its mathematical ideas in terms of these, and thus
require no postulates except, again, those of logic. Or it may pursue an
14 Principles of Mathematics, p. 3.
Symbolic Logic, Logistic, and Mathematical Method 351
intermediate course, assuming some of its ideas as primitive but defining
others in terms of a previously developed logic, and thus require some
postulates of its own but still dispense with others which would have been
necessary in a non-logistic treatment.
But whichever of these modes of procedure is adopted, the general
method of proof in logistic will be the same, and will differ from any non-
logistic treatment. A non-logistic development will proceed from postulates
to theorems by immediate inference or the use of syllogism, or enthymeme,
or the reductio ad absurdum, and such general logical methods. Or it may,
upon occasion, make use of methods of reasoning the validity of which
depends upon the subject matter. It may make use of "mathematical
induction", which requires the order of a discrete series with a first term.
Or if proofs of consistency and independence of the postulates are offered,
these will make use of logical principles which are most complex and difficult
of comprehension — principles of which no thoroughly satisfactory account
has ever been given. The principles of all this reasoning will not be men
tioned; it will be supposed that they are understood, though sometimes
they are clear neither to the reader nor to the mathematician who uses
them, and they may even be such that nobody really understands them.
(This is not to say that such proofs are unsound. Proofs by " mathematical
induction" were valid before Frege and Peano showed that they are strictly
deductive in all respects. But in mathematics as in other matters, the
assurance or recognition of validity rests upon familiarity and upon prag
matic sanctions more often than upon consciously formulated principles.)
As contrasted with this, the logistic method requires that every principle
of proof be explicitly given, because these principles are required to state
each step of proof.
The method of proof in logistic is sufficiently illustrated by any extended
proof of Chapter V. Proofs in arithmetic or geometry do not differ in
method from proofs in the logic, and the procedures there illustrated are
universal in logistic. An examination of these proofs will show that postu
lates and previously established theorems are used as principles of proof
by substituting for the variables p, q, r, etc., in these propositions, other
expressions which can be regarded as values of their variables. The general
principle
(p -i </) -* (-? -* -P)
can thus be made to state
352 A Survey of Symbolic Logic
by substituting p q for p and p for q. Or if /* e XC be substituted for p
and n e D'Xc for q, it states
[(/z e NC) -i (M e D'Nc)] •< [-(M e D'Xc) -* -(M e XC)]
Thus any special case which comes under a general logical principle is
stated by that principle, when the proper substitutions are made. This is
exactly the manner in which the principles of proof which belong to sym
bolic logic state the various steps of any particular proof in the logistic
development of arithmetic or geometry.
Returning to our first example, we discover that in
the first half, p q •* p, is itself a true proposition. Suppose this already
proved as, in fact, it is in the last chapter. We can then assert what
p q -i p is stated by the above to imply, that is,
-P -* "(p q)
We thus prove this new theorem by using p q H p as a premise. To use a
previous proposition as a premise means, in the logistic method, exactly
this: to make such substitutions in a general principle of inference, like
(p -J q) -I (-q -* -p)
that the theorem to be used as a premise appears in the first half of the
expression — the part which precedes the main implication sign. That
part of the expression which follows the main implication sign may then
be asserted as a consequence of this premise.
There are two other operations which may be used in the proofs of
logistic — the operation of substituting one of a pair of equivalent expressions
for the other, and the operation of combining two previously asserted
propositions into a single assertion.15 The first of these is exemplified
whenever we make use of a definition. For example, we have, in the system
from which our illustration is borrowed, the definition
p+q = -(-p-q)
and the theorem
15 The operation of combining two propositions, p and q, into the single assertion,
p q, is not required in systems based on material implication, because we have
pqcr = pc(qcr)
Symbolic Logic, Logistic, and Mathematical Method 353
If in the definition, we substitute -p for p and -q for q, it states
-p + -q = -[-(-p)._(-r/)]
And then, making the substitutions which the theorem p = -(-p) allows,
we have
-p + -<i = -(P </)
which may be asserted as a theorem. Again, if we return to the theorem
proved above,
"P -* -(p </)
we are allowed, by this last equivalence, to make the substitution in it of
-p + -q for -(p q). Thus we prove
-p -* -p + -q
This sufficiently illustrates the part played in proof by the substitution of
equivalent expressions.
We may now see exactly what the mechanics of the logistic method is.
The only operations required, or allowable, in proof are the following:
(1) In some postulate or theorem of symbolic logic, other, and usually
more complex, propositions are substituted for the variables p, q, r, etc.,
which represent propositions. The postulate or theorem in which these
substitutions are made is thereby used as a principle of proof which states,
in this particular case, the proposition which results when these substi
tutions are made.
(2) The postulate or theorem of logic to be used as a principle of proof
may, and in most cases does, state that something implies something else.
In that event, we may make such substitutions as will produce an expression
in which that part which precedes the main implication sign becomes
identical with some postulate or previously proved theorem— of logic, of
arithmetic, of geometry, or whatever. That part of the expression which
follows the main implication sign may then be separately asserted as a new
theorem, or lemma, which is thus established. The postulate or previously
proved theorem which is identical with what precedes the main implication
sign, in such a case, is thus used as a premise.
It should here be noted that propositions of logic, of geometry, of any
logistic system, may be used as premises; but only propositions of symbolic
logic, which state implications, are used as general principles of inference.
(3) At any stage of a demonstration, one of a pair of equivalent expres
sions may be substituted for the other.
24
354 A Survey of Symbolic Logic
(4) If, for example, two premises are required for a certain desired
consequence, and each of these premises has been separately proved, then
the two may be combined in a single assertion.
These are all the operations which are strictly allowable in demonstra
tions by the logistic method. To their simplicity and definiteness is
attributable a large part of the precision and rigor of the method. Proof
is here not a process in which certain premises retire into somebody's reason
ing faculty, there to be transformed by the alchemy of thought and emerge
in the form of the conclusion. The whole operation takes place visibly in
the successive lines of work, according to definite rules of the simplest
possible description. The process is as infallible and as mechanical as the
adding machine — except in the choice of substitutions to be made, for which,
as the reader may discover by experiment, a certain amount of intelligence
is required, if the results are to be of interest.
III. A "HETERODOX" VIEW OF THE NATURE OF MATHEMATICS AND OF
LOGISTIC
We have now surveyed the general character of logistic and have set
forth what may be called the " orthodox" view of it. As was stated earlier
in the chapter, the account which has now been given is such as would
exclude certain systems which would almost certainly be classified as
logistic in their character. And these excluded systems are most naturally
allied with another view of logistic, which we must now attempt to set
forth. The differences between the "orthodox" and this "heterodox"
view have to do principally with two questions: (1) What is the nature of
the fundamental operations in mathematics; are they essentially of the
nature of logical inference and the like, or are they fundamentally arbitrary
and extra-logical? (2) Is logistic ideally to be stated so that all its assertions
are metaphysically true, or is its principal business the exhibition of logical
types of order without reference to any interpretation or application?
The two questions are related. It will appear that the systems which the
previous account of logistic did not cover are such as have been devised
from a somewhat different point of departure. One might characterize
the logistic of Principia Mathematica roughly by saying that the order of
logic is assumed, and the order of the other branches then follows from the
meaning of their terms. On the other hand, the systems which remain to be
discussed might, equally roughly, be characterized by saying that they
attempt to set up a type of logical order, which shall be general and as
Symbolic Logic, Logistic, and Mathematical Method 355
inclusive as possible, and to let the meaning of terms depend upon their
properties of order and relation. Thus this "heterodox" view of logistic
is one which takes it to cover all investigations and developments of types
of logical order which involve none but ideographic symbols and proceed
by operations which may be stated with precision and generality.
In any case, it must be granted that the operations of the logistic method
are themselves pre-logical, in the sense that they underlie the proofs of
logic as well as of other branches. The assumption of these operations —
substitution, etc. — is the most fundamental of all the assumptions of
logistic. It is possible to view the subject in a way which makes such
pre-logical principles the fundamentally important thing, and does not
regard as essential the use of symbolic logic as a foundation. The pro
priety of the term logistic for such studies may be questioned. But if
such a different view is consistent and useful, it is of little consequence
what the method ought to be called.
We see at once that, if such a view can be maintained, Mr. Russell's
definition of mathematics, quoted above, is arbitrary, for by that definition
any "logistic" development which is not based upon logic as # foundation
will not be mathematics at all. As a fact, it will be simplest to present this
"heterodox" view of logistic by first presenting and explaining the cor
relative view of mathematics. If to the reader we seem here to wander
from the subject, we promise to return later and draw the moral.
A mathematical system is any set of strings of recognizable marks in which
some of the strings are taken initially and the remainder derived from these
by operations performed according to rules which are independent of any mean
ing assigned to the marks. That a system should consist of marks instead
of sounds or odors is immaterial, but it is convenient to discuss mathe
matics as written. The string-like arrangement is due simply to our habits
of notation. And there is no theoretical reason why a single mark may not,
in some cases, be recognized as a "string".
The distinctive feature of this definition lies in the fact that it regards
mathematics as dealing, not with certain denoted things — numbers, tri
angles, etc. — nor with certain symbolized "concepts" or "meanings",
but solely with recognizable marks, and dealing with them in such wise
that it is wholly independent of any question as to what the marks repre
sent. This might be called the " external view of mathematics " or " mathe
matics without meaning". It distinguishes mathematics from other sets
of marks by precisely those criteria which the external observer can always
356 A Survey of Symbolic Logic
apply. Whatever the mathematician has in his mind when he develops a
system, what he does is to set down certain marks and proceed to manipulate
them in ways which are capable of the above description.
This view is, in many ways, suggested by growing tendencies in mathe
matics. Systems become "abstract", entities with which they deal "have
no properties save those predicated by postulates and definitions", and
propositions lose their phenomenal reference. It becomes recognized
that any procedure the only ground for which lies in the properties of the
things denoted — as "constructions" in geometry — is defective and un-
mathematical. Demonstrations must take no advantage of the names
by which the entities are called. But if Mr. Russell is right, the mathe
matician has given over the metaphysics of space and of the infinite only
to be plunged into the metaphysics of classes and of functions. Questions
of empirical possibility and factual existence are replaced by questions of
"logical" possibility — questions about the "existence" of classes, about
the empty or null-class, about the class of all classes, about "individuals",
about " descriptions ", about the relation of a class of one to its only member,
about the "values" of variables and the "range of significance" of func
tions, about material and formal implication, about "types" and "system
atic ambiguities" and "hierarchies of propositions". And we may be
pardoned for wondering if the last state of that mathematician is not worse
than the first. It is possible to think that these logico-metaphysical
questions are essentially as non-mathematical as the earlier ones about
empirical possibility and phenomenal existence. One may maintain that
nothing is essential in a mathematical system except the type of order.
And the type of order may be viewed as a question solely of the distri
bution of certain marks and certain complexes of marks in the system.
The question of logical meaning, like the question of empirical denotation,
may be regarded as one of possible applications and not of anything internal
to the system itself.
Before discussing the matter further, it may prove best to give an illus
tration. Let us choose a single mathematical system and see what we shall
make of it by regarding it simply as a set of strings of marks.
We take initially the following eight strings:
(P D tf) = (~JP v 0)
(pxq) = ~(~pv~q)
(P = q) = ((P*q) *(<7=>p))
Symbolic Logic, Logistic, and Mathematical Method 357
((p v(ry vr))D(r/v(p vr)))
((r/Dr)D((^ vry)D(p vr)))
We must now state rules according to which other strings can be derived
from the above. In stating these rules, we shall refer to quids and quods:
these words are to have no connotation; they serve merely for abbreviation
in referring to certain marks.
(1) The marks +, x, D, ==, and =, are quods.
(2) The marks p, q, r, are quids; and any recognizable mark not appear
ing in the above may be taken arbitrarily as a quid.
(3) Any expression consisting of two quids, one quod, and the marks
) and (, in the order (quid quod quid), may be treated as a quid.
(4) The combination of any quid preceded immediately by the mark ~
may be treated as a quid.
(5) Any string in the set may be repeated.
(0) Any quid which is separated only by the mark == from some other
quid, in any string in the set, may be substituted for that other quid any
where.
(7) In any string in the initial set, or in any string added to the list
according to rule, any quid whatever may be substituted for p or q or r,
or for any quid consisting of only one mark. When a quid is substituted
for any mark in a string, the same quid must also be substituted for that
same mark wherever it appears in the string.
(8) The string resulting from the substitution of a quid consisting of
more than one mark for a quid of one mark, according to (7), may be added
to the list of strings.
(9) In any string added to the list, according to (8), if that portion of
the string which precedes any mark D is identical with some other string
in the set, preceded by (, then the portion of that string which follows the
mark D referred to may be separately repeated, with the omission of the
final mark ), and added to the set.
These rules are unnecessarily awkward. In the illustration, it was
important not to refer to "propositions", "relations", "variables", "paren
theses," etc., lest it should not be clear that the rules are independent of
the meanings of the marks. But though cumbersome, they are still precise.
The original eight strings of marks are, with minor changes of notation,
358 A Survey of Symbolic Logic
definitions and postulates of divisions #1 to * 5 in Principia Mathematica.
By following the rules given, anyone may derive all the theorems of these
divisions and all other consequences of these assumptions, without knowing
anything about symbolic logic — either before or after. In fact, these
rules formulate exactly what the authors have done in proving the theorems
from the postulates.16 For this reason, it is unnecessary to carry our illus
tration further and actually derive other strings of marks from the initial
set. The process may be observed in detail in Principia Mathematica:
it is, in all important respects, the same with the process of proof exhibited
in our Chapter V.
The method of development in Principia Mathematica differs from the
one wre have suggested, not in the actual manipulation of the strings of
marks, but most fundamentally in that the reasons why — the principles —
of their operations are to be found, not in explicitly stated rules, but in
discussions and assumptions concerning the conceptual content of the
system. In fact, the rules of operation are contained in explanations of
the meaning of the notation — in discussions of the nature and properties
of "elementary propositions", "elementary propositional functions", and
so forth. For example, instead of stating that certain substitutions may
be made for p, q, r, etc., they assume as primitive ideas the notions of
"elementary propositions" — p, q, r, etc. — the notion of "elementary
propositional functions" — px, \J/z, etc. — and the idea of "negation", indi
cated by writing ~ immediately before the proposition. And in part, the
rules of operation are contained in certain postulates, distinguished by
their non-symbolic form: "If p is an elementary proposition, ~p is an
elementary proposition", "If p and q are elementary propositions, pvq
is an elementary proposition", and "If (pp and ij/p are elementary propo
sitional functions which take elementary propositions as arguments, <pp v $p
is an elementary propositional function. The warrant for the substitution
of various complexes for p, q} r, etc., is contained in these. The operation
which requires our complicated rule (9), whiich states precisely what may
be done, is covered by their assumptions: "Anything implied by a true
elementary proposition is true", and "When <px can be asserted, where x
is a real variable, and <px D \f/x can be asserted, where x is a real variable,
then ipx can be asserted, where x is a real variable". To make the con
nection between these and our rule, we must remember that 3 is the
16 With the single and unimportant except ion that they do not add every new string
which they arrive at, to the list of strings. Many such are simply asserted as lemmas,
used immediately for one further proof and not listed as theorems.
Symbolic Logic, Logistic, and Mathematical Method
symbol for " implies", that if what precedes this sign is identical with
some other string in the set, that means that what precedes is true or is
asserted, and that the number of ' open ' and ' close ' parenthetic marks will
indicate whether the implication in question is the main, or asserted, impli
cation.
We have chosen this particular system to illustrate the requirements
of " mathematics without meaning" for a special reason, which will appear
shortly. But the same sort of modifications would be sufficient to bring
any good mathematical system into this form; and in most cases such
modifications would be necessary.
If, for example, the system in question were one of the better-known
algebras, wre should probably have "a class, K, of elements, a, b, c, etc.",
and such assumptions as "If a and b are elements in K, a + b is an element
in K". These would do duty as the principles according to which, for
example, x + y would be substituted for a in any symbolic postulate or
theorem. The changes in such a system would be less radical, hardly
more than alterations in phraseology, but still necessary.
Ileliance upon meanings for the validity of the method has obvious
advantages. It is simple and natural and clear. (So is measuring two
line-segments with a foot rule to prove equality.) It also has disadvantages.
Besides the logico-metaphysical questions into which this reliance upon
meanings plunges us, there is the disadvantage that it works a certain
confusion of the form of the system with its content. The clear separation
of these is the ideal set by "mathematics without meaning". Not only
must mathematical procedure be free from all appeal to intuition or to
empirical data; it should also be independent of the meaning of any special
concepts which constitute the subject matter of the system. No alteration
or abridgment of mathematical procedure anywhere should be covered by
the names which are given to the terms. Only those relations or other
properties which determine a system as a particular type of order should
be allowed to make a difference in its manner of development.
To secure complete separation of form from special content, and to
present the system as purely formal and abstract, means precisely to use
principles of operation which are capable of statement as rules for the
manipulation of marks— though, in general, the meticulous avoidance of
any reference to " meanings " would be a piece of pedantry. The important
consideration is the fact that the operations of any abstract and, really rigorous
mathematical system are capable of formulation without any reference to truth
360 A Survey of Symbolic Logic
or meanings.11 We are less interested in any superiority of this "external
view of mathematics", or in the conjectured advantages of such procedure
as has been suggested, than in its bare possibility. If the considerations here
presented are not wholly mistaken, then the ideal of form which requires
17 It is possible to regard such manipulation of marks, the discovery of sufficiently
precise rules and of initial strings which will, together, determine certain results, and the
exhibition of the results which such systems give, as the sole business of the mathematician.
Mathematics, so developed, achieves the utmost economy of assertion. Nothing is
asserted. There are no primitive ideas. Since no meanings are given to the characters,
the strings are neither true nor false. Nothing is assumed to be true, and nothing is asserted
as "proved". It is not even necessary to assert that certain operations upon certain marks
give certain other marks. The initial strings are set down: the requirements of pure
mathematics are satisfied if the others are got and recorded. Yet these initial strings and
the rules of operation determine a definite set of strings of marks — determine unambiguously
and absolutely a certain mathematical system.
To many, such a view will seem to exclude from mathematics everything worthy of
the name. These will urge that the modern developments of mathematics have aimed
at exact analysis into fundamental concepts; that this analysis does, as a fact, bring about
such simplification of the essential operations as to make possible mechanical manipulation
of the system without reference to meanings; but that it is absurd to take this shell of
refined symbolism for the meat of mathematics. To any such, it might be replied that
the development of kinematics as an abstract mathematical system does not remove the
physics of matter in motion from the field of experimental investigation; that abstract
geometry still leaves room for all sorts of interesting inquiry about the nature of our space :
that for every system which is freed from empirical denotations there is created the separate
investigation of the possible applications of this system. Correspondingly, for every
system which is made independent of classes, individuals, relations, and so on, there is
created the separate investigation of the metaphysical status of the classes, individuals,
and relations in question — of the application of the system of marks to systems of more
special "concepts", i. e. to systems of logical and metaphysical entities. That we are
more interested in the applications of a system than in its rigorous development, more
interested .in its "meaning" than in its structure, should not lead to a confusion of meaning
with structure, of applications with method of development.
It may be further objected that this view seems to remove mathematics from the
field of science altogether and make it simply an art; that the computer would, by this
definition, be the ideal mathematician. But there is one feature of mathematics, even as a
system of marks, which is not, and cannot be made, mechanical. Valid results may be
obtained by mechanical operations, and each single step may be essentially mechanical,
yet the derivation of "required" or "interesting" or "valuable" results will need an in
telligent and ingenious manipulator. Gulliver found the people of Brobdingnag (?) feed
ing letters into a machine and waiting for it to turn out a masterpiece. Well, master
pieces are combinations achieved by placing letters in a certain order! However mechanical
the single operation, it will take a mathematician to produce masterpieces of mathematics.
A machine, or machine-like process, will start from something given, take steps of a deter
mined nature, and render the result, whatever it is; but it will not choose its point of
departure and select, out of various possibilities, the steps to be taken in order to achieve a
desired result. Is not just this ingenuity in controlling the destination of simple operations
the peculiar skill which mathematics requires? The mathematician, like any other sci
entific investigator, is largely engaged upon what are, from the point of view of the finished
science, inverse processes: he gets, by trial and error, or intuition, or analogy, what he
presents finally as rigidly necessary. To produce or reveal necessities previously un
noticed — this is the peculiar artistry of his work.
Symbolic Logic, Logistic, and Mathematical Method 361
that mathematics abstract not only from possible empirical meanings but
also from logical or metaphysical meanings is a wholly attainable ideal.
And if this is possible, then Mr. Russell's view that "Pure Mathematics
is the class of all propositions of the form ' p implies q', etc.", is an arbitrary
definition, and the ideal of form which it imposes is not a necessary one,
but must take its chances with other such ideals. The decision among
these will, then, be a matter of choice, dependent upon the advantages to
be gained by one or the other form. There is no a priori reason why
systems which are generated by " mathematical" operations, some of which
may be peculiar to the system and meaningless in logic, are not just as
"sound" and "good" and even "ideal" as systems developed by the com
pletely analytical method of Principia Mathematica which reduces all
operations to those of logic. And "extra-logical" modes of development
may be just as universal as the "logical ", since symbolic logic itself may be
developed by the "extra-logical" method. It was to make this clear that
we chose the particular system which we did for our illustration.
In fact, symbolic logic, or that branch of it which is developed first as a
basis for others, must be developed by operations the validity of which is
presumed apart from the logic so developed. It may, indeed, be the case
that logic is developed by methods which it validates by its own theorems,
when these are proved; it may thus be "self-critical", or "circular" in a
sense which means consistency rather than fallacy. But this is not really
to the point: if the validity of certain operations is presupposed, then that
validity is presupposed, whether it is afterward proved valid or not. There
is, then, a certain advantage in the explicit recognition that a system of
symbolic logic is merely a set of strings of marks, manipulated by certain
arbitrary and "extra-logical" principles. It is, in fact, only on this view
that symbolic logic can be abstract. For symbolic logic, as has already
been pointed out, is peculiar among mathematical systems in that its postu
lates and theorems are used to state proofs. If, then, the proofs are to be
logically valid, these postulates and theorems must be true, and the system
cannot be abstract. But if the "proofs" are required to be "valid" only
in the sense that certain arbitrary and extra-logical rules for manipulation
have been observed, then it matters no more in logic than in any other
branch whether the propositions be true, or even what they mean. There
is the same possibility of choice here that there is in the case of other mathe
matical systems— the choice which is phrased most sharply as the alterna
tive between the Russellian view and the "external view of mathematics
^ It may be noted that if mathematics consists of "propositions of the form (p im-
362 A Survey of Symbolic Logic
If we take this view of mathematics, or any view which regards arbitrary
mathematical operations as equally fundamental with the operations of
logic, we shall then give a different account of logistic and of its relation
to logic. We shall, in that case, regard symbolic logic as one mathematical
system, or type of order, among others. We shall recognize the possibility
of generating all other types of order from the order of logic, but we shall
find no necessity in this proceeding. We may, possibly, find some other
very general type of order from which the order of logic may be derived.
And the question of any hierarchic arrangement of systems will then depend
upon convenience or simplicity or some other pragmatic consideration.
Logistic will, then, be defined not by any relation to symbolic logic but as
the study of types of order as such, or as any development of mathematics
which seeks a high degree of generality and complete independence of
any particular subject matter.
IV. THE LOGISTIC METHOD OF KEMPE AND ROYCE
We should not care to insist upon the "external view of mathematics"
and the consequent view of logistic which has been outlined. Other con
siderations aside, it seems especially dubious to dogmatize about the ideal
of mathematical form when there is no common agreement on the topic
among mathematicians. But we can now answer the questions which
prefaced this discussion: Are the fundamental operations of mathematics
those of logic or are they extra-logical? And is logistic ideally to be so
stated that all its assertions are metaphysically true or is it concerned simply
to exhibit certain general types of order? The answer is that it is entirely a
matter of choice, since either view can consistently be maintained and
mathematics be developed in the light of it.19 This is especially important
for us, since, as has been mentioned, there are certain studies which would
most naturally be called logistic which would not be covered by the "ortho-
plies <?', where p and q are propositions containing one or more variables, etc.", and the
theorems about "implies" are required to be true if proofs are to be valid, all mathematics
must be true in order to be valid. On this view, " abstractness " can reside only in the
range of the variables contained in p and q.
19 One case in which the "external view of mathematics" is highly convenient, is of
especial interest to us. There are various symbolic "logics" which differ from one another
both in method and in content. Discussion of the correctness and relative values of these
is almost impossible unless we recognize that the order of logic can be viewed quite apart
from its content — that a symbolic logic may be abstract, just like any other branch of
mathematics — and thus separate the question of mathematical consistency (of mere ob
servance of arbitrary and precise principles of operation) from questions of applicability
of a system to valid reasoning. The difficulty of making this separation hampered our
discussion of "implies" in the last chapter.
Symbolic Logic, Logistic, and Mathematical Method 363
dox" view, since they are based, not upon logic, but upon an order prior
to or inclusive of logic. These studies exemplify a method which differs
in notable respects both from that of Peano and that of Principia Mathe-
matica. And it seems highly desirable that we should discuss this alterna
tive method without initial prejudice.
It is characteristic of this alternative method that it seeks to define
initially a field, or class of entities, and an order in this field, which shall be
mathematically as inclusive as possible, so that more special orders may be
specified by principles of selection amongst the entities. It is distinguished
from the method followed by Peano in the Formidaire by the fact that it
seeks to get special orders, such as that of geometry, without further
"existence postulates", and from the method of Principia Mathematica by
the attempt to substitute selection ivithin an initial order for analysis (defini
tion by previous ideas) of newly introduced terms. The result is that this
method is particularly adapted to exhibit the analogies of different special
fields — the partial identities of various types of order.
The application of this method has not been carried out extensively
enough so that we may feel certain either of its advantages or of its limita
tions. The method is, in a certain sense, exemplified wherever we have
various mathematical systems all of which satisfy a given set of postulates,
but each — or, say, all but one — satisfying some one or more of the postulates
"vacuously". For here we have an ordered field within which other and
more limited systems are specified by a sort of selection. ("Selection" is
not the proper word, but no better one has occurred to us.) It is particu
larly in two studies of the relation of geometry to logic that the method has
been consciously followed: 20 in a paper by A. B. Kempe, "On the Relation
between the Logical Theory of Classes and the Geometrical Theory of
Points,"21 and in Josiah lloyce's study, "The Relation of the Princi
ples of Logic to the Foundations of Geometry".22 We shall hardly wish to
go into these studies in detail, but something of the mode of procedure and
general character of the results achieved may be indicated briefly.
Kempe enunciates the principle that "... so far as processes of exact
thought are concerned, the properties of any subject matter depend solelj
on the fact that it possesses 'form'— i. e., that it consists of a number of
2°Peirce's system of "logical quaternions" (see above, pp. 102-04) also exhibits
something of this method.
21 Proc. London Math. Soc., xxi (1890), 147-82.
22 Trans. Amer. Math. Soc., vi (1905), 353-415.
Some portions of the discussion of this paper and Kempe's are here reprinted
article, "Types of Order and the System S," in Phil. Rev. for May, 1916.
364 A Survey of Symbolic Logic
entities, certain individuals, pairs, triads, &c., certain of which are exactly like
each other in all their relations, and certain not; these like and unlike indi
viduals, pairs, triads, &c., being distributed through the whole system of
entities in a definite way".23 In illustration of this theory, he seeks to
derive the order both of logical classes and of geometrical sets of points
from assumptions in terms of a triadic relation, a c • b, which may be read
"b is 'between' a and c". The type of this relation may be illustrated as
follows: Let a, b, c represent areas; then ac-b symbolizes the fact that b
includes whatever area is common to a and c, and is itself included in the
area which comprises what is either a or c or common to both. Or it
may be expressed in the Boole-Schroder Algebra as
(ac) cb c (a + c), or a-bc + -ab-c = 0
The essential properties of serial order may be formulated in terms of this
relation. If ac-b and ad-c, then also ad-b and bd-c. If b is between
a and c and c is between a and d, then also b is between a and d, and c is
between b and rf.24
abed
Thus the relation gives the most fundamental property of linear sets.
If a be regarded as the origin with reference to which precedence is deter-
23 "On the Relation between, etc.", loc. cit., p. 147.
24 Assuming ac • b to be expressed in the Boole-Schroder Algebra (as above) by
(ac) c b c (a + c)
this deduction is as follows:
ac-b is equivalent to (a c) c 6 c (a + c).
ad-c is equivalent to (ad) cc c (a + d).
By the laws of the algebra,
(a c) c b is equivalent to a -b c = 0.
b c (a + c) is equivalnt to -(a + c) b = -a b -c = 0.
(ad) c cis equivalent to a -c d = 0.
c c (a + d) is equivalent to -(a + d) c = -a c -d = 0.
Combining these premises, i. e. adding the equations, we have
a -b c + -a b -c + a -c d + -a c -d = 0.
Expanding each term of the left-hand member with reference to that one of the
elements, a, b, c, d, not already involved in it,
a -b c d + a -b c -d + -a b -c d + -a b -c -d + a b -c d + a -b -c d
+ -a b c -d + -a -b c -d = 0
By the law "If a + b = 0, a = 0", we get from this,
(1) a -b d (c + -c) + -a b -d (c + -c) = Q = a-bd + -ab -d,
and (2) b -c d (a + -a) + -b c -d (a + -a) = 0=b-cd + -bc -d.
(1) is equivalent to (ad) cb c(a+d), or ad-b,
and (2) is equivalent to (b d) c c c (b + d), or bd-c.
Symbolic Logic, Logistic, and Mathematical Method 365
mined, ac-b will represent "6 precedes c", and a d-c that "c precedes d".
Since ac-b and a d-c together give a d-b, we have: if "b precedes c" and
"c precedes d", then "6 precedes rf". Hence this relation has the essential
transitivity of serial order, with the added precision that it retains reference
to the origin from which " precedes" is determined.
The last-mentioned property of this relation makes possible an inter
pretation of it for logical classes in which it becomes more general than
the inclusion relation of ordinary syllogistic reasoning. If there should
be inhabitants of Mars whose logical sense coincided with our own — so
that any conclusion which we regarded as valid would seem valid to them,
and vice versa — but whose psychology was somewhat different from ours,
these Martians might prefer to remark that "b is 'between' a and c",
rather than to note that "All a is b and all b is c". These Martians might
then carry on successfully all their reasoning in terms of this triadic ' between'
relation. For ac-b, meaning
-a b -c + a -b c = 0
is a general relation which, in the special case where a is the "null" class
contained in every class, becomes the familiar "b is contained in c" or
" All b is c ". By virtue of the transitivity pointed out above, 0 c • b and 0 d-c
together give 0 d-b, which is the syllogism in Barbara, "If all b is c and all
c is d, then all b is d". Hence these Martians would possess a mode of
reasoning more comprehensive than our own and including our own as a
special case.
The triadic relation of Kempe is, then, a very powerful one, and capable
of representing the most fundamental relations not only in logic but in
all those departments of our systematic thinking where unsymmetrical
transitive (serial) relations are important.25 In terms of these triads,
Kempe states the properties of his "base system", from whose order the
relations of logic and geometry both are derived. The "base system"
consists of an infinite number of homogeneous elements, each having an
infinite number of equivalents. It is assumed that triads are disposed in
this system according to the following laws: 26
2* It should be pointed out that while capable of expressing such relations, this triadic
relation is itself not necessarily unsymmetrical: ac-b and ab-c may both be true
that case, b = c, as may be verified by adding the equations for these two triads
for any a and b, ab-a and ab-b always hold-& is always contained in itself,
triadic relation represents serial order with the qualification that any term
itself or is "between" itself and any other-an entirely intelligible and .
vention.
26 See Kempe's paper, loc. cit., pp. 148-49.
366 A Survey of Symbolic Logic
1. If we have a b • p and c b-q, r exists such that we have a q • r and c p • r.
2. If we have a b • p and c p • r, q exists such that we have a q - r and c b • r.27
3. If we have a b-c and a = b, then c = a = b.
4. If a = b, then we have a c-b and b c-a, whatever entity of the system
c may be.
To these, Kempe adds a fifth postulate which he calls the Law of Con
tinuity: "No entity is absent from the system which can consistently be
present". From these assumptions and various definitions in terms of
the triadic relation, he is able to derive the laws of the symbolic logic of
classes and fundamental properties of geometrical sets of points. But
further and most important properties of geometrical sets depend upon the
selection of such sets within the "base system" by the law: 28
If we have a-p-q,
b-p-q,
and a-b-p does not hold;
then p = q.
'a-p-q' here represents a relation of a, p, and q, such that some one at
least of ap-q, aq-p, pq-a will hold. If we call ab-c a "linear triad",
then the set or locus selected by the above law will be such that no two
linear triads of the ' points' comprised in it can have two non-equivalent
' points' in common. Of such a geometric set, Kempe says:29 "It is
precisely the set of entities which is under consideration by the geometrician
when he is considering the system of points which make up flat space of
unlimited dimensions".
But there are certain dubious features of Kempe's procedure. As Pro
fessor Royce notes, the Law of Continuity makes postulates 1 and 2 super
fluous. And there are other objections to it also. Moreover, in spite of
the fact that Kempe has assumed an infinity of elements in the "base set",
there are certain ambiguities and difficulties about the application of his
principles to infinite collections.
In Professor Royce's paper, we have no such 'blanket assumption' as
the Law of Continuity, and the relations defined may be extended without
difficulty to infinite sets. We have here, in place of the "base system"
and triadic relations, the "system 2", the " ^-relation " and the "0-rela-
27 If the reader will draw the triangle, a be, and put in the " betweens" as indicated,
the geometrical significance of these postulates will be evident. I have changed a little
the order of Kempe's terms so that both 1 and 2 will be illustrated by the same triangle.
28 See Kempe's paper, loc. dt., pp. 176-77.
29 Ibid., p. 177.
Symbolic Logic, Logistic, and Mathematical Method 367
tion". The ^-relation is a polyadic relation such that F(ab. . ./.rz/. . .) is
expressible in the Boole-Schroder Algebra as
a b ... -x -y . . . + -a -6 ... x y . . . =0
This is the generalization of Kempe's a c-b, which is F(ac/b). The 0-rela-
tion is a polyadic symmetrical relation which expresses simultaneously a
whole set of equivalent F-relations. 0(abc. . .) is expressible as
a b c . . . + -a -b -c . . . = 0
We have used the algebra of classes to express these relations, but in
Professor Royce's paper, this order is, of course, reversed. In terms of 0-
relations, the ideas of the logic of classes are defined, and from the postula-
tion of certain 0 -relations, the laws of the symbolic logic of classes are de
rived. And, in most interesting ways which we cannot here discuss, the
order of the system 2 is also shown to possess all the fundamental proper
ties of geometric sets of points. The system 2 has a structure such that it
might be called "the logical continuum", and there are good grounds for
presuming that types of order in the greatest variety may be specified within
the system simply by selection. In the words of Professor Royce: 30
" Wherever a linear series is in question, wherever an origin of coordi
nates is employed, wherever ' cause and effect', 'ground and consequence,'
orientation in space or direction of tendency in time are in question, the
diadic asymmetrical relations involved are essentially the same as the rela
tion here symbolized by p -< ,, q, [' q is "between" y and p'; or, with y
as origin, 'p precedes </'; or, where y is the null-class, 'p is contained
in q'; or, in terms of propositions, 'p implies q1]. This expression, then,
is due to certain of our best established practical instincts and to some of
our best fixed intellectual habits. Yet it is not the only expression for the
relations involved. It is in several respects inferior to the more direct
expression in terms of 0-relations. . . . When, in fact, we attempt to de
scribe the relations of the system S merely in terms of the antecedent-
consequent relation, we not only limit ourselves to an arbitrary choice of
origin [y in p —< „ q], but miss the power to survey at a glance relations
of more than a diadic, or triadic character."
V. SUMMARY AND CONCLUSION
There are, then, in general, three types of logistic procedure. There is,
first, the "simple logistic method", as we may call it— the most obvious
30 "The Relations of the Principles of Logic, etc.," loc. tit., pp. 381-82.
368 A Survey of Symbolic Logic
one, in which the various branches of pure mathematics, taken in the non-
logistic but abstract form, are simply translated into the logistic terms which
symbolize ideographically the relations involved in proof. When this
translation is made, the proofs in arithmetic, or geometry, etc., will be
simply special cases of the propositions of symbolic logic. But other
branches than logic will have their own primitive or undefined ideas and
their own postulates in terms of these. We have used Peano's Formulaire
as an illustration of this method, although the Formulaire has, to an extent,
the characters of the procedure to be mentioned next. Second, there is
the hierarchic method, or the method of complete analysis, exemplified by
Principia Mathematica. Here the calculus of propositions (or implications)
is first developed, because by its postulates and theorems all the proofs of
other branches are to be stated. And, further, all the terms and relations
of other branches are to be so analyzed, i. e., defined, that from their defini
tion and the propositions of the logic alone, without additional primitive
ideas or postulates, all the properties of these terms will follow. And,
third, there is the method of Kempe and Royce. This method aims to
generate initially an order which is not only general, as is the order of logic,
but inclusive, so that the type of order of various special fields (in as large
number and variety as possible) may be derived simply by selection — i. e.,
by postulates which determine the class which exhibits this special order as
a selection of members of the initially ordered field.31 For this third method,
other types of order will not necessarily be based upon the order of logic:
in the only good examples which we have of the method, logic is itself
derived from a more inclusive order. The sense in which such a procedure
may still be regarded as logistic has been made clear in what precedes.
Which of these methods will, in the end, prove most powerful, no one
can say at present. The wThole subject of logistic is too new and un
developed. But certain characters of each, indicating their adaptability,
or the lack of it, to certain ends, can be pointed out. The hierarchic or
completely analytic method has a certain imposing quality which right
fully commands attention. One feels that here, for once, we have got to
the bottom of things. Any work in which this method is extensively carried
out, as it is in Principia Mathematica, is certainly monumental. Further,
the method has the advantage of setting forth various branches of the
subject investigated in the order of their logical simplicity. And the step
31 Professor Royce used to say facetiously that the system S had some of the properties
of a junk heap or a New England attic. Almost everything might, be found in it: the ques
tion was, how to get these things out.
Symbolic Logic, Logistic, and Mathematical Method 3G9
from one such division to another based upon it is always such as to make
clear the connection between the two. The initial analyses — definitions—
which make such steps possible are, indeed, likely to tax our powers, but
once the initial analysis is correctly performed, the theorems concerning the
derived order will be demonstrable by processes which have already become
familiar and even stereotyped. The great disadvantage of this completely
analytic method is its great complexity and the consequent tediousness of
its application. It is fairly discouraging to realize that the properties of
cardinal number require some four hundred pages of prolegomena — in a
symbolism of great compactness — for their demonstration. To those whose
interests are simply "mathematical" or "scientific" in the ordinary sense,
it is forbidding.
The simple logistic method offers an obvious short-cut. It preserves
the notable advantages of logistic in general — the brevity and precision
of ideographic symbols, and the consequent assurance of correctness. And
since it differs from the non-logistic treatment in little save the introduction
of the logical symbols, it makes possible the presentation of the subject in
hand in the briefest possible form. When successful, it achieves the acme
of succinctness and clearness. Its shortcoming lies in the fact that, having
attempted little which cannot be accomplished without logistic, it achieves
little more than is attained by the ordinary abstract and deductive presenta
tion. For what it is, it cannot be improved upon; but those who are inter
ested in the comparison of types of order, or the precise analysis of mathe
matical concepts, will ask for something further.
No one knows how far the third method — that of Kempe and Royce—
can be carried, or whether the system 2, or some other very inclusive type
of order, will be found to contain any large number, or all, of the various
special orders in which we are interested. But we can see that, so far as
it works, this method gives a maximum of useful result with a minimum of
complication. It avoids the complexities of the completely analytic method,
yet it is certain to disclose whatever analogies exist between various systems,
by the fact that its terms are allowed to denote ambiguously anything which
has the relations in question, or relations of precisely that type. In another
important respect, also, advantage seems to lie with this method. One
would hardly care to invent a new geometry by the analytical procedure ;
it is difficult enough to present one whose properties are already familiar.
Nor would one be likely to discover the possibility of a new system by
the simple logistic procedure. With either of these two methods, we need
25
370 A Survey of Symbolic Logic
to know where we are going, or we shall go nowhere. By contrast, the
third method is that of the pathfinder. The prospect of the novel is here
much greater. The system S may, probably does, contain new continents
of order whose existence we do not even suspect. And some chance trans
formation may put us, suddenly and unexpectedly, in possession of such
previously unexplored fields. The outstanding difficulty of the method,
apart from our real ignorance of its possibilities, seems to be that it must
rely upon devices which are not at all obvious. It may not tax severely
the analytical powers, but it is certain to tax the ingenuity. Having set
up, for example, the general order of geometrical points, one may be at a
loss how to specify "lines" having the properties of Euclidean parallels.
In this respect, the analytic method is superior. But the prospect of
generality without complexity, which the third method seems to offer, is
most enticing.
We have spoken of symbolic logic, logistic and mathematics. It may
well be questioned whether the method of logistic does not admit of useful
application beyond the field of mathematics. Symbolic logic is an instru
ment as much more flexible and more powerful than Aristotelian logic as
modern science is more complex than its medieval counterpart. Some of
the advantages which might have accrued to alchemy, had the alchemists
reduced their speculations to syllogisms, might well accrue to modern sci
ence through the use of symbolic logic. The use of ideographic symbolism
is capable of making quite the same difference in the case of propositions and
reasoning that it has already made in the case of numbers and reckoning.
It is reported that the early Australian settlers could buy sheep from the
Bushmen only by holding up against one sheep the coins or trinkets repre
senting the price, then driving off that sheep and repeating the process.
It might be reported of the generality of our thinking that it is possible to
get desired conclusions only by holding up one or two propositions, driving
off the immediate consequences, and then repeating the process. Symbolic
logic is capable of working the same transformation in the latter case that
arithmetic does in the former. Those unfamiliar with logistic may not
credit this — but upon this point we hesitate to press the analogy. Certain
it is, that for the full benefit that symbolic logic is capable of giving, we
should need to be brought up in it, as we are in the simpler processes of
arithmetic. What the future may bring in the widespread use of this
new instrument, one hardly ventures to prophesy.
Some of the advantages which would be derived from the wider use of
Symbolic Logic, Logistic, and Mathematical Method 371
logistic in science, one can make out. The logistic method is applicable
wherever a body of fact or of theory approaches that completeness and
systematic character which belongs to mathematical systems. And by the
use of it, the same assurance of correctness which belongs to the mathe
matical portions of scientific subjects may be secured for those portions
which are not stateable in terms of ordinary mathematics.
Dare we make one further suggestion of the possible use of logistic in
science? Since it seems to us important, we shall venture it, with all due
apologies for our ignorance and our presumption. A considerable part
seems to be played in scientific investigation by imagery which is more or
less certainly extraneous to the real body of scientific law. The scientist is
satisfied to accept a certain body of facts — directly or indirectly observed
phenomena, "laws," and hypotheses which, for the time being at least,
need not be questioned. But beyond this, he finds a use for what is neither
directly nor indirectly observed, but serves somehow to represent the situ
ation. A physicist, for example, will indulge in mechanical models of the
ether, or mechanical models of the atom which, however much he may hope
to verify them, he knows to run beyond established fact. The value of
such imagery is, in part at least, its concreteness. The established relations,
simply in terms of mathematics and logic, do not come to possess their full
significance unless they are vested in something more palpable. A great
deal of what passes for "hypothesis" and "theory" seems to have, in part
at least, this character and this value; if it were not for the greater "sug-
gestiveness" of the concrete, much of this would have no reason for being.
Now whoever has worked with the precise and terse formulations of logistic
realizes that it is capable of performing some of the offices of concrete
imagery. Its brevity enables more facts to be "seen" at once, thought of
together, treated as a single thing. And a logistic formulation can be free
from the unwarranted suggestions to which other imagery is liable. Perhaps
a wider use of logistic would help to free science from a considerable body of
"hypotheses" whose value lies not in their logical implications but in their
psychological "suggestiveness". But the reader will take this conjecture
only for what it is worth. What seems certain is that for the presentation
of a systematic body of theory, for the comparison of alternative hypotheses
and theories, and for testing the applicability of theory to observed facts,
logistic is an instrument of such power as to make its eventual use almost
certain.
Merely from the point of view of method, the application of logistic to
372 A Survey of Symbolic Logic
subjects outside the field of mathematics needs no separate discussion.
For when mathematics is no longer viewed as the science of number and
quantity, but as it is viewed by Mr. Russell or by anyone who accepts the
alternative definition offered in this chapter, then the logistic treatment
of any subject becomes mathematics. Mathematics itself ceases to have
any peculiar subject matter, and becomes simply a method. Logistic is
the universal method for presenting exact science in ideographic symbols.
It is the "universal mathematics" of Leibniz.
Finis
BIBLIOGRAPHY
The following bibliography contains titles of all the positive contributions to symbolic
logic and logistic in the strict sense with which I am acquainted, as well as some, taken
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But it is not complete. A few of the numerous general discussions of symbolic logic and
logistic have also been listed, and certain mathematical books and papers which, though
not strictly logistic, are of special interest to the student of that subject.
What are considered the most important contributions to symbolic logic, are indicated
by an asterisk (*) preceding the title, while certain studies which should be especially helpful
to students of this book are indicated by a "dagger" (f). Volume numbers of periodicals
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Who discovered the principle of the quantification of the predicate? Contemp.
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*BOOLE, G. The mathematical analysis of logic. Cambridge, Macmillan, 1847.
*_ The calculus of logic. Camb. and Dublin Math. Jour., 3 (1848), pp. 183-98.
On the theory of probabilities, and in particular on Mitchell's problem of the
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Further observations on the theory of probabilities. Ibid., ser. 4, 2 (1851),
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389
390 A Survey of Symbolic Logic
* — An investigation of the laws of thought. London, Walton, 1854. Also re
printed as vol. 2 of Boole's Collected logical works; ed. by Jourdain, Chicago,
Open Court Publ. Co., 1916.
• Solution of a question in the theory of probabilities. Phil. Mag., ser. 4, 7
(1854), pp. 29-32.
Reply to some observations published by Mr. Wilbraham . . . Ibid., ser. 4,
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abilities are limited. Ibid., pp. 91-98.
Further observations relating to the theories of probabilities . . . Ibid., pp.
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• On a general method in the theory of probabilities. Ibid., pp. 431-44.
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On the application of the theory of probabilities to the question of the combina
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* — On the theory of probabilities. Phil. Trans. Roy. Soc. (London), 152 (1862),
pp. 225-52.
• On the theory of probabilities. Phil. Mag., ser. 4, 25 (1863), pp. 313-17.
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BOOLE, MARY. Symbolical methods of study. London, Kegan, 1884.
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BRUNSCHWIEG, L. Les etapes de la philosophic mathematique. Paris, Alcan, 1912.
BRYANT, MRS. S. On the nature and functions of a complete symbolic language. Mind,
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The relation of mathematics to general formal logic. Proc. Aristot. Soc.,
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Sur les rapports logique des concepts et des propositions. Ibid., 24 (1917),
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La logique algorithmique et le calcul de probability. Ibid., pp. 291-313.
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392 A Survey of Symbolic Logic
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Weg zur Wahrheit. 1776.
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INDEX
(References here given are in addition to, not inclusive of, those readily suggested by
the Table of Contents).
Absorption, law of, 74.
Absurd, see Impossible.
Addition, arithmetical, 343; definition of
arithmetical by Peirce, 81 ff., by Peano,
345; relative addition, 91, 95, 275 ff.
See also Sum.
Alphabet, logical, 74.
Aristotle, 231.
Bayne, A., 36.
Bentham, G., 36.
Bernoulli, J., 18.
Bernstein, B., 119
Boole, G., 3, 4, 31, 78-79, 118, 137, 207,
211, 212, 217; Jevons on system of, 72;
system of, compared with Peirce 's, 80
ff.; his general problem, 162.
Calculus, 4, 6, 8. See also Classes, Proposi
tions, Propositional functions, Rela
tions.
Cantor, G., 4, 5.
Carroll, Lewis, see Hodgson, S. G.
Castillon, G. F., 4, 18, 32-35.
Classes, conception of, 261 ; application of
Boole-Schroder Algebra to, 121-22;
calculus of, derived from Strict Impli
cation, 321 ff.
Coefficients, in Boole's system, 59-63; in
the Boole-Schroder Algebra, 137 ff.
Consistency, relation of, in Mrs. Ladd-
Franklin's svstem, 109; in the system of
Strict Implication, 293; meaning of,
333.
Contained in (the relation C), 43, 16, 47,
65, 118-19, 262, 270; Peirce on the
meaning of, 83-84, 96.
Contradictory propositions, 189; as treated
by De Morgan, 40, Jevons, 76-77.
Contrary propositions, 189; as treated by
Jevons, 76-77.
Converse, propositions, 191; Converse
relations, 46, 74, 91, 273-74.
Dalgarno, G., 5.
Dedekind, R., 4,5, 114,342.
del Re, A., 119.
DeMorgan, A., 4, 5, 79, 209; DoMorgan s
Theorem, 125, 237, 283.
Descriptions, 290, 343.
Distribution of terms, 43.
Division, in Lambert's system, 21 ff.; in
Holland's, 30 ff.; in Boole's, 61 ff.; in
Peirce's, 81 ff.
Duality, law of, in terminology of Boole,
58/in Boole-Schroder Algebra, 126.
Either or, meaning of, 213.
Elective symbols, 52.
Elimination, in Boole's system, 59-60, 64;
in Jevons's, 75-76; in the Boolo-
Schroder Algebra, 153 ff., 194.
e-relation, 16, 261-62, 270, 345.
Equivalence, in the Boole-Schroder Al
gebra, 120; of classes, 262; of relations,
270; of propositions, strict, and mater
ial, 292-93.
Euclid, 342.
Euler's diagrams, 176.
Existence, 12, 14, 186-88, 328-29.
Exponents, in the logic of relatives, 87 ff.
Extension, Chap. I, Sects. II and III,
passim; in Leibniz's system, 13-14;
Boole's logic based on, 52; classes in,
184-86; relations in, 219; propositions
in, 230-31, 328; definition in, 349.
See also Intension.
Formal implication, see Implication.
Formulaire de Mathtmatiques, v, 5, 7,
115-16, 278, 28078!, 368.
Fractions, see Division.
Frege, G., 4, 5, 50, 114-15, 273, 351.
F-relation, 366.
Functions, as developed by Boole, 58, 82;
in the Boole-Schroder Algebra, 125, 133
ff. See also Pro positional Functions.
Geometry, 10, 340-42; 363-67.
Grassmann, H., 4.
Grassmann, R., 107-08.
Hamilton, Sir W., 4, 36-37.
Hamilton, W. R., 4.
Hilbert, D., 5.
Hodgson, S. G., 326; his diagrams, 180 ff.
Holland, G. J. von, 18, 29-32, 323.
Huntington, E. V., 119, 342.
Ideographic language, 2, 6-8, 340.
Implication, no symbol for, in Boole's sys
tem, 65; material, 84-85, 214-15, 231,
Chap. V, passim, see esp. 303-04 and
Sect. V; formal, 243, 328 ff.; four types
of, 259. See also Contained in.
Impossible, 32, 292. 336-39. See also Con
sistency.
Inclusion, see Contained in.
Inconsistent triad, 19.5-97.
Indeterminate, see Undetermined.
Index law, 54.
Individual, 324. See also e-relation.
Induction, see Mathematical induction.
Index
Inference, 57-58, Chap. Ill, Sects. II and
III, passim; in the logistic method,
352-53; immediate, 14, 33, 39, 41, 77,
190 ff.
Intension, 8, 13 ff., Chap. I, Sects. II and
III, passim, 73; postulates for calculus
of propositions in, 318; calculus of
classes in, 323.
Jevons, W. S., 4, 118, 149, 205.
Kant, I., 341.
Kempe, A. B., 198.
Kircher, A., 5.
Ladd-Franklin, Mrs. C., 78, 108-10; 118,
195, 205, 211, 231.
Lambert, J. H., 4, 18-29, 32, 204, 323.
Leibniz, G. W., 3, 4, 79, 372.
Linear Sets, 365, 367.
Logical product, see Product.
Logical sum, see Sum.
Logistic, vi, 5-7, 11.
Lully, R., 5.
MacColl, H., 4, 108, 119, 327.
Mathematical induction, 29, 131, 236, 351.
Multiplication, arithmetical, denned by
Peirce, 101-02, by Peano, 345; relative
multiplication '86 ff., 275 ff. See also
Product.
Necessary, 17, Chap. V, passim.
Negative, in the Boole-Schroder Algebra,
119, 124; terms, 38, 53, 73; classes, 121,
185, 263: relations, 46, 220, 271; propo
sitions, 14-15, 25-26, 30-31, 32-33, 40,
57, 108, 188-89, 213. 292.
Null-class, 185-90, 327. See also 0.
Null-proposition, see 0.
Null-relation, see 0.
Number, 80, 101; in Peano's Formulaire,
344; denned in Principia Mathematica,
346 ff.
1, see Universe of discourse; see also under 0
Operation, meaning of, 342; nature of, in
logistic, 358 ff.
Order, logical, 3, 342, 364.
Peano, G., 50, 115. See also Formulaire de
Mathematiques .
Peirce, C. S., 4, 261, 279.
Plato, 4.
Ploucquet, C., 4, 18.
n and 2 operators, 79, 97 ff., 140, 234.
Fieri, M., 5, 342.
Poretsky, P., 114, 145-46, 163-66, 200.
Possible, 15, 329. See aho Impossible.
Premise, in logistic, 352-53.
Primitive concepts, in Leibniz, 7.
Primitive ideas and propositions, of Prin
cipia Mathematica, 282, 287-88; of
arithmetic in Peano's Formula-ire, 344-
45.
Principia Mathematica; v, 5, 7, 8, 102, 116,
222, 261, 277, 279, 281, 314-16, 324,
361.
Probability, Boole's treatment of, 67 ff.;
Peirces' treatment of, 105-06.
Product, in Leibniz's system, 12 ff. ; in
Lambert's, 19; in Boole's, 52; in
Peirce's, 81; in the Boole-Schroder
Algebra, 119; of classes, 120, 185, 262;
of regions in space, 175; of relations,
86 ff., 219, 271, 275; of two functions,
143; strict logical, 293. See also Multi
plication and IT and 2 operators.
Prepositional functions, 94, 113; meaning
of, 232-33; range of significance of, 233,
242, 254; in Principia Mathematica,
287; calculus of, derived from Strict
Implication, 320 ff.
Quantification of the predicate, 19, 24, 36,
38 ff., 56. See also Undetermined co
efficient.
Quaternions, 4; Peirce's logical, 103 ff.
Reductio ad absurdum, 166-67.
Regions in a plane, application of the
Boole-Schroder Algebra to, 120-21,
175.
Relations, as treated by Lambert, 28-29,
by DeMorgan, 37, 45 ff., by Peirce,
85 ff., 102-05, by Schroder, 111 ff.;
Peirce's paper on logic of, 100; mean
ing of, in extension, 219, 269; calculus
of, compared with calculus of classes,
271; converse, 91, 273-74, 276; powers
of, 29, 277; domain, converse domain,
and field of, 277-78; calculus of, de
rived from Strict Implication, 323.
Relative terms, 277-78; in DeMorgan's
system, 45 ff. ; Peirce's treatment of,
85 ff.; Schroder's treatment of, 111 ff.
See also Relations.
Royce, J., vi, 195.
Schroder, E., v, 4, 5, 78, 110 ff., 211, 246,
261, 279.
Science, exact, 7, 370-71.
Segner, J. A., 18.
Self-contradictory, see Impossible.
Sheffer, H. M., 119.
S operator, see II and 2 operators.
Solly, 7, 29, 36.
Solutions, in Boole's system, 60-63; in
Jevons's, 76-77; in Peirce's, 98-100; by
means of diagrams, 77, 181 ff. ; of some
logical problems, 201-12, 215-19. See
also Equations and Inequations in
Table of Contents.
Square of opposition, 190.
Subcontrary propositions, 190.
Substitution of similars, in Jevons, 75.
Subtraction, arithmetical, treated by
Peirce, 80-81; in Leibniz's system, 17-
18; in Lambert's, 19; in Castillon's, 32;
in Boole's, 53; in Peirce's, 81 ff.
Sum, in Leibniz's system, 16 ff. ; in Lam
bert's, 19; in Castillon's, 32 ff.; in
Boole's, 52-53; in Jevons's, 73; in
Peirce's, 81-82; in Schroder's, 111; in
the Boole-Schroder Algebra, 119; of
classes, 121, 185, 263; of propositions,
213; of relations, 271; of two functions,
143; of prepositional functions, 94;
strict logical, 291, 301-02. See also
Addition.
Index
Syllogisms, in Lambert's system, 26 ff.;
in Holland's, 31-32; in Castillon's, 34;
in DeMorgan's, 41, 49 ff.; in Boole's,
57-58; in Jevons's, 75; in Mrs. Ladd-
Franklin's, 109-10, 195-97; Peirce's
principle of, 85; application of the
Boole-Schroder Algebra to, 181-82,
193-95; in Barbara, 245, interpreted
by Kempe, 365;" conditional, 197;
limitation of, 1, 198-201.
System-S, 366.
Thomson, W., 36.
Tonnies, I. H., 18.
Truth value of propositions, 227, 230, 294.
Two-valued Algebra, defects of, as a calcu
lus of propositions, 281.
Undetermined class, in Holland's system,
30; in Castillon's, 32. See also Undeter
mined coefficient.
Undetermined coefficient, in Leibniz's
system, 15; in Lambert's, 24 ff.; in
Boole's, 50-51; in Jevons's, 75; in
Peirce's, 82; in the Boole-Schroder
Algebra, 186.
Universe of discourse, 37; diagrams of, 177
ff. See aho refs. under 0.
Variables, 3, 232 ff.; Peirce on, 93; in
Principia Mathematica, 289.
Venn, J., v, 18, 201, 203, 211; diagrams,
77, 176 ff.
Whitchead, A. N., v, 118. See also Prin
cipia Mathematica.
Wilkins, J., 5.
0: in Boole's system, 52 ff.; in Jevons's,
73-74; in Peirce's, 82; in Schroder's,
111; in the Boole-Schroder Algebra,
119; in the Calculus of classes, 121,
185 ff., 263; in the system of regions in
a plane, 181; in the calculus of proposi
tions, 213-14, 223 ff.; in the calculus of
relations, 218-19, 271; Boole's algebra
is an algebra of 0 and 1, 52.
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