Table Of Content"
f6·
. Introduction,
· .E LEMENTARY' .
·.·MATHEMATICAL ·
:._·_. LO.GIC.
· .... Abram
.· · AroFlovich .
_;~
·. ...: ' s·tolyar . .· '
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Introduction to Elementary
MATHEMATICAL LOGIC
Introduction to Elementary
MATHEMATICAL LOGIC
Abram Aronovich Stolvar
TRANSLATED BY SCRIPTA TECHNICA, INC.
TRANSLATION EDITED BY ELLIOT MENDELSON
DOVER PUBLICATIONS, INC.
NEW YORK
Copyright I!) 1970 by The Massachuseus Institute of Technology.
All righlS reserved under Pan American and International Copyright
Conventions.
Published in Canada by General Publishing Company, Ltd., 30 Lesmill
Road, Don Mills, Toronto, Ontario.
Published in the United Kingdom by Constable and Company, Ltd.,
10 Orange Street, London WC2H 7EG.
This Dover edition, first published in 1983, is an unabridged and unaltered
republication of the work published by The MIT Press, Cambridge, Massa
chusetts, and London, England, in 1970. II was originally published in
Russian as Elementarnoye vvede11iJ•e v matematicl1esk1tJ'll logiku by Pros
veshcheniye Press, Moscow, in 196S. The present edition is published by
special arrangement with The MIT Press, 28 Carleton Street, Cambridge,
MA02142.
Manu£actured in the United States of America
Dover Publications, Inc., ISO Varick Street, New York, N.Y. 10014
Ubrary of Con1ress Calalo1tn1 In Publication Data
Stollar, A. A. (Abram Aronovich)
Introduction to elementary mathematical logic.
Translation of: Elementarnoe vvedenie v matematicheskuiu logiku.
Bibliography: p.
Includes index.
I. Logic, Symbolic and mathematical. J. Tille.
BCl3S.S7613 1983 SI 1.3 83-5223
ISBN 0-486-64561-4
By virtue of its numerous applications in the most varied fields of
science and technology (complicated problems in the foundations
of mathematics, linguistic problems, certain problems of synthesis
of automata, etc.), present-day mathematical logic is attracting
the attention of a large number of persons in various specialties,
including high school mathematics teachers.
The elements of mathematical logic and certain of its· applica
tions are included in the programs of high schools that place
emphasis on mathematics. They can serve as interesting material
for out-of-class work for students in the higher grades of any
high school.
Recently, a number of monographs and collections of articles
on mathematical logic have been published in the Soviet Union.
However, there is almost no literature on mathematical logic
available for high school mathematics pupils. The author of this
book has set himself the task of providing an elementary exposi
tion of the fundamentals of mathematical logic and certain of its
applications for this broad class of readers.
The term "mathematical logic" encompasses at the present
time an extremely broad and profound scientific study. In addition
to classical logic of propositions and predicates, the fundamentals
of which are expounded in this book, it includes a number of
other logical systems and theories (constructive logic, systems of
modal logic, many-valued logics, the theory of algorithms, etc.).
For this reason, this book cannot be regarded as an introduction
to mathematical logic as a whole. However, a familiarity with the
material expounded in it will make it easier for the reader who is
seriously interested in studying the subject to read the literature
recommended for this purpose at the end of the book.•
The book includes exercises for independent work on the part
of the reader.
The author expresses his deep appreciation to Yu. A. Gastev,
whose suggestions have proved of great help in the preparation of
the manuscript.
• The references lo this bibliography arc indicated by brackets enclosing the
number of the reference in question.
AUTHOR'S PREFACE V
Author's preface v
INTRODUCTION
1
PROPOSITIONAL LOGIC 29
1. Objects and operations. 30
2. Formulas. Equivalent formulas. Tautologies. 46
3. Examples of the application of the laws of the
logic of propositions in derivations. 62
4. Normal forms of functions. Minimal forms. 72
5. Application of the algebra of propositions to the
synthesis and analysis of discrete-action networks. 82
2
THE PROPOSITIONAL CALCULUS 95
I. The axiomatic method. The construction of
formalized languages. 96
2. Construction of a propositional calculus (alphabet,
formulas, derived formulas). 101
3. Consistency, independence, and completeness of a
system of axioms in the propositional calculus. 115
Vi CONTENTS
