An Introduction to Symbolic Logic

387 Pages·1967·18.27 MB·English

by  Susanne K. Langer

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SUSANNE K
LANGER
AN INTRODUCTION TO
SYMBOLIC LOGIC
by
SUSANNE K. LANGER
Third Revised Edition
DOVER PUBLICATIONS, INC.
NEW YORK
Copyright © 1967 by Dover Publications, Inc.
Copyright © 1953 by Susanne K. Langer.
All rights reserved under Pan American and
International Copyright Conventions.
Published in Canada by General Publishing Com­
pany, Ltd., 30 Lesmill Road, Don Mills, Toronto,
Ontario.
Published in the United Kingdom by Constable
and Company, Ltd., 10 Orange Street, London
WC Z
This is the third revised (1967) edition of An
Introduction to Symbolic Logic. The work was orig­
inally published by George Allen & Unwin, Ltd., in
1937. The second revised and enlarged edition was
first published by Dover Publications, Inc., in 1953.
International Standard Book Number: 0-486-60164-1
Library of Congress Catalog Card Number: 66-29834
Manufactured in the United States of America
Dover Publications, Inc.
180 Varick Street
New York, N.Y. 10014
TO MY MOTHER
ELSE U. KNAUTH

PREFACE TO THE THIRD EDITION
This new edition incorporates a number of corrections which
were recently brought to the author’s attention and which
were published together with the second edition as a list of
errata. The author is grateful to Mr. James Forrester of
Baltimore, Maryland, for his assiduous efforts in reviewing
the book and compiling these many errata. Also, in this edi­
tion the bibliographic references have been brought up to
date in an attempt to provide the reader with more current­
ly available sources.
October 1966. S. K. L.
PREFACE TO THE SECOND EDITION
When this book was written, there was no systematic
textbook of symbolic logic in English except the Symbolic
Logic of Lewis and Langford. Perhaps Couturat's little
summary, The Algebra of Logic, could also be regarded as
a text, in that it set forth a system developed by various
persons, notably De Morgan, Boole, and Schroeder, making
no claim for itself to originality; but that was, after all, an
outline rather than an introduction. For the rest, every
exposition was still in connection with a contribution:
Lewis's Survey of Symbolic Logic had been written mainly
to propose the system of “strict implication," Russell raised
many questions he did not claim to settle, Quine was almost
completely original; and even the most “digested" work
(Lewis and Langford) entered into no elementary discussion
of the basic logical notions—generalization, abstraction,
relation, form, system. Such fundamental concepts were
taken for granted.
They have always been taken for granted in that paragon
of pure sciences, mathematics. It is the exceptional student
of mathematics who knows why “+ ” is classed as an
operation and “=" as a relation, or, indeed, why his exercises
are called “examples"; or who understands, even at a fairly
advanced stage of technical proficiency, just how algebra is
related to arithmetic. His training is entirely in techniques
and their application to problems; arithmetic, geometry and
algebra are related for him only by the fact that their several
techniques may converge on one and the same problem;
that is to say, they are related practically but not intellec­
tually. The mathematics books used in schools dwell almost
exclusively on rules of operation.
The prestige of mathematics is so great that logic, in
emulating its method, tended to follow its pedagogy as well.
The excellent textbooks of logic that have appeared during
PREFACE TO THE SECOND EDITION 7
the last fifteen years (see the supplementary new list appended
to “Suggestions for further reading") tend similarly to
emphasize techniques—transformation, inference, consis­
tency-tests, decision-procedures—without detailed explana­
tions of the concepts involved. Now, mathematics has such
obvious practical uses that to learn its tricks without
understanding their significance is not entirely silly; but
may the same be said of symbolic logic? Is the manipulation
of its symbols of such practical importance that students
should learn to perform logical operations even without
knowing or questioning their conceptual foundations? It
seems to me that despite its practical uses, which are still
coming to light in unexpected quarters, its chief value is con­
ceptual. Through the influence of the various positivistic
schools of philosophy, which are certainly the most promi­
nent and perhaps the most promising schools to-day, our
scholars and educators are so imbued with methodology that
they value the new logic primarily for its codification of the
rules of inference. But that is only one of its contributions to
human thought, and even to science; method can be over­
emphasized, and tends to be so in our intellectual life.
Symbolic logic is an instrument of exact thought, both
analytic and constructive; its mission, accordingly, is not
only to validate scientific methods, but also to clarify the
semantic confusions that beset the popular mind as well as
the professional philosopher at the present time. “Semantics"
(blessed word!) is in dire need of responsible analysis and
skilful handling, and symbolic logic is the most effective
preparation I can think of for a frontal attack on the pathetic
muddles of modern philosophical thought. It blasts natural
misconceptions with every move, not by a process of “de­
bunking," but by purposeful and lucid construction of ideas.
Because this book seeks to present in clear, stepwise
fashion the elementary concepts of logic, it cannot encompass
as much technical material as other textbooks do. But to-day
that is not a serious embarrassment, because it may be
8 AN INTRODUCTION TO SYMBOLIC LOGIC
supplemented by a standard text. One of the most useful
devices now generally taught—the construction of “truth-
tables” to test the legitimacy of constructs in a truth-value
system—is included in this edition as Appendix C. The
reading list for further study has been brought up to date.
For the rest, no revisions have been made except to correct
errors, especially a major error in Chapter IV, which
mysteriously escaped the several readers of the script, and
has necessitated a little actual rewriting. Apart from these
details, the book is unchanged—a book for the student who
has no teacher, or for the teacher who has to meet too many
naïve questions for his comfort. As such it is still alone in its
class, at least for English readers; and in the belief that there
is always a real need for at least one such book, I send my
Introduction out anew.
July 1952. S. K. L.
PREFACE TO THE FIRST EDITION
Symbolic Logic is a relatively new subject, and the easiest
methods of approach have not yet been determined. In
point of arrangement, therefore, this Introduction has no
predecessor. That is just why it was written: the need of
some systematic guide, from the state of perfect innocence
to a possible understanding of the classical literature, has
become acute and commanding.
But, although my text has no predecessor, it has had at
least one inspirer: my debt to Professor Sheffer of Harvard
is too great to be expressed in any detailed acknowledgments.
The underlying ideas of the book—its emphasis upon system,
its progress from the specific to the general, from the general
to the abstract, its whole treatment of logic as a science of
forms—all this is due to his influence. How many lesser ideas
also derive from him I am unable to say; it is the mark of a
great teacher that one cannot render to him the things
that are his. They become part of one’s own mentality, and
ultimately of the intellectual commonwealth.
To my friend Professor Paul Henle I wish to express my
heartiest thanks for his kindness and patience in reading the
entire manuscript, uncovering various errors, and checking
all the algebraic formulae. He has suggested many improve­
ments, and sometimes found simpler or briefer explanations
or formal demonstrations. I would also thank Dr. Henry S.
Leonard for some excellent suggestions, and for reading a
part of the proof.
S. K. L.