Many-Valued Logics

139 Pages·1993·2.373 MB·English

by  Grzegorz Malinowski

OXFORD LOGIC GUIDES: 25
General Editors
DOV GABBAY
ANGUS MACINTYRE
DANA SCOTT
OXFORD LOGIC GUIDES
1. Jane Bridge: Beginning model theory: the completeness theorem and some
consequences
2. Michael Dummett: Elements of intuitionism
3. A. S. Troelstra: Choice sequences: a chapter ofintuitionistic mathematics
4. J. L. Bell: Boolean-valued models and independence proofs in set theory
(1st edition)
5. Krister Segerberg: Classical propositional operators: an exercise in the
foundations of logic
6. G. C. Smith: The Boole-De Morgan correspondence 1842-1864
7. Alec Fisher: Formal number theory and computability: a work book
8. Anand Pillay: An introduction to stability theory
9. Η. E. Rose: Subrecursion: functions and hierarchies
10. Michael Hallett: Cantorian set theory and limitation of size
11. R. Mansfield and G. Weitkamp: Recursive aspects of descriptive set theory
12. J. L. Bell: Boolean-valued models and independence proofs in set theory
(2nd edition)
13. Melvin Fitting: Computability theory: semantics and logic programming
14. J. L. Bell: Toposes and local set theories: an introduction
15. Richard Kaye: Models ofPeano arithmetic
16. J. Chapman and F. Rowbottom: Relative category theory and geometric
morphisms
17. S. Shapiro: Foundations withoutfoundationalism
18. J. P. Cleave: A study of logics
19. R. M. Smullyan: GodeTs incompleteness theorems
20. T. E. Forster: Set theory with a universal set
21. C. McLarty: Elementary categories, elementary toposes
22. R. M. Smullyan: Recursion theory for metamathematics
23. Peter Clote and Jan Krajicek: Arithmetic, proof theoryt and computational
complexity
24. A. Tarski and J. Tarski: Introduction to logic and to the methodology of
deductive sciences (4th edition)
25. Grzegorz Malinowski: Many-valued logics
Many-Valued Logics
Grzegorz Malinowski
University of-Lodz
CLARENDON PRESS · OXFORD
1993
Oxford University Press, Walton Street, Oxford 0X2 6DP
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© Grzegorz Malinowski, 1993
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A catalogue record for this book is available from the British Library
Library of Congress Cataloging in Publication Data
Malinowski, Grzegorz.
Many-valued logics / Grzegorz Malinowski,
p. cm. — (Oxford logic guides; 25)
Includes bibliographical references and indexes.
1. Many-valued logic. I. Title. II. Series.
QA9.45.M35 1993
511.3—dc20 93-36413 CIP
ISBN0-19-853787-5
Typeset by the author
Printed in Great Britain on acid-free paper by
Bookcraft (Bath) Ltd, Midsomer Norton
Contents
Introduction 1
1. The classical logic 5
1.1 Truth-tables 5
1.2 Propositional tautologies 7
1.3 Functional completeness 8
1.4 Axiomatization of CPC 9
1.5 Predicate calculus 11
1.6 Algebraization 12
2. The third logical value of Lukasiewicz 16
2.1 Lukasiewicz and the Lvov-Warsaw school 16
2.2 Three-valued logic 17
2.3 Modality and three-valuedness 19
2.4 Interpretation difficulties 21
3. Logic algebras and matrices 24
3.1 Language and logic algebras 24
3.2 Functional completeness of finite algebras 26
3.3 Logical matrices 28
4. Many-valuedness 30
4.1 Two criteria 30
4.2 Structurality and many-valuedness 32
4.3 Finiteness and deduction 33
5. Lukasiewicz logics 36
5.1 Fundamental properties 36
5.2 Definability in Lukasiewicz matrices 38
5.3 Axiomatization 39
5.4 Algebraic interpretations 40
6. Post logics 44
6.1 Post matrices 44
VI Contents
6.2 Interpretation 46
6.3 Algebraic form of Post logics 47
6.4 Axiomatization of functionally complete systems
of n- valued logic 48
7. Three-valuedness of Kleene and Bochvar 51
7.1 Logic of indeterminacy 51
7.2 Bochvar logic 54
7.3 Partial logics 56
8. Standard properties of many-valued constructions 60
8.1 Standard conditions 61
8.2 Axiomatization 62
8.3 Standard matrix consequence 63
9. Probability and many-valuedness 66
9.1 Logical probability 66
9.2 Operationalistic conception of subjective probability 68
10. Classical characterization of many-valued logics 72
10.1 Suszko’s thesis 72
10.2 Scott’s method 74
10.3 Urquhart’s interpretation 77
11. Quantifiers in many-valued logic 79
11.1 Ordinary predicate calculi 79
11.2 Set theory and many-valued logic 81
11.3 Generalized quantifiers 83
12. Intuitionism and the modal logics of Lewis 87
12.1 Intuitionistic logic 87
12.2 Modal logics S4 and S5 91
12.3 Remarks on quantification 95
13. Fuzzy sets and Zadeh logic 98
13.1 Fuzzy sets and logics of imprecise concepts 98
13.2 Fuzzy logic 101
14. Applications and significance of the topic 105
14.1 Independence of axioms 105
14.2 Formalization of intensional functions 106
Contents vii
14.3 Many-valued algebras and switching theory 109
14.4 Many-valuedness in computer science 111
Bibliography 113
Author index 125
Subject index 127

Introduction
The vital part of the studies of logic seeks to determine structural cri­
teria for propositional validity and deals with formal inference relations.
A suitable starting point for any analysis of these problems consists in
the selection of a set of propositions from among all grammatically well-
formed sentences, the members of which satisfy some specified syntactical
and semantical conditions.
The assumption stating that to every proposition it may be ascribed
exactly one of the two logical values, truth or falsity, called the principle
of bivalence, constitutes the basis of classical logic. It determines both
the subject matter and the scope of applicability of the logic, the main
systems of which are the classical propositional calculus (CPC) and the
(first-order) predicate calculus (quantifier calculus). CPC is a theory of all
truth-functional propositional connectives, i.e. sentence-argument proposi­
tional functors having the property that the logical value of any complex
sentence formed with their use is determined uniquely by the logical values
of its components. The predicate calculus is formed by introducing to the
language system, with its semantics adequately extended, the symbols of
name-argument propositional functors representing the names of proper­
ties and relations and name quantifiers. It renders possible the profound
analysis of propositions within the principle of bivalence paradigm.
The most natural and straightforward step beyond the two-valued logic
is the introduction of more logical values, rejecting simultaneously the prin­
ciple of bivalence. The indirect way, on the other hand, consists either in the
revision of the ‘bunch’ of sentence connectives (then non-truth-functional
connectives are introduced into the language of propositional calculus) or,
after having questioned some classical laws, in modifying the connectives
characterized by them. The multiple-valued truth-tables constitute the ba­
sis of the first method, whereas in the other case they are procured as tools
for the procedures of decidability of logical theorems.
The roots of many-valued logics can be traced back to Aristotle (4th cen­
tury BC) who considered, within the modal framework, future contingents
sentences. In Ch. IX of his treatise De Interpretatione Aristotle provides
the time-honoured sentence-example representing this category: ‘There
will be a sea-battle tomorrow’. The philosopher from Stagira emphasizes
the fact that future contingents are neither actually true nor actually false,
which suggests the existence of a ‘third’ logical status of propositions. The
2 Introduction
prehistory of many-valued logics falls in the Middle Ages and was made
by Duns Scott, William of Ockham and Peter de Rivo (Louvain). At the
turn of the 19th century some attempts to create non-classical logical con­
structions, three-valued mainly, appeared: in 1897 Hugh MacColl’s inves­
tigations concentrated on the so-called ‘three-dimensional logic’, Charles
S. Peirce (1839-1914) worked on ‘trychotomic mathematics’ founded on
the ‘triadic logic’, whereas Nicolai A. Vasil’ev presented a system of ‘imag­
inary non-Aristotelian logic’ in which propositions may be ‘affirmative’,
‘negative’ or ‘indifferent’.
The final, thoroughly successful, formulation of many-valued logical
constructions was proposed as a result of the truth-table method applied to
the classical logic by Frege (1879), Peirce (1885) and others; and then the
logical matrices method (Lukasiewicz, Post). The ‘era of many-valuedness’
was finally inaugurated in 1920 by Lukasiewicz and Post. After many
years of investigation Lukasiewicz (1920) enriched the set of the classical
logical values with an intermediate value and laid down the principles of
a three-valued propositional calculus. Post (1920), on the other hand, de­
fined (finite) many-valued ‘logical algebras’. It is worth mentioning that in
the 1930s some non-classical logical constructions appeared formalizing in-
tensional (non-truth-functional) sentence connectives by means of axioms,
i.e. Lewis modal logics and intuitionistic logic which codifies the principles
of some significant trend in the philosophy of mathematics initiated by
Brouwer in 1907.
This book will present an elementary exposition of the topics connected
with many-valued logics. Our discussion will focus on the constructions
being ‘many-valued’ at their origin, i.e. on those obtained by the use of
matrices. Therefore, the matrix method has been chosen as the most suit­
able way of presenting the subject. The opening Chapter 1, ‘The classical
logic’, contains background material and is devoted to the fundamentals of
the classical truth-functional logic. First, it will serve as an introduction
for those who are not acquainted with elementary logic at all. Its second
role is to collect and organize the material in a way which makes it eas­
ier to compare and distinguish between classical and many-valued logic.
Chapter j2, ‘The third logical value of Lukasiewicz’, is an overview of the
origin and basic properties of the first three-valued system of propositional
logic. There, one may also find a discussion on the problem of intuitive
interpretation of Lukasiewicz’s third value and logic.
The next two chapters have a more general character. In Chapter 5,
‘Logic algebras and matrices’, interpretation structures of propositional
languages in the Fregean framework are derived. A discussion of functional
completeness, the algebraic property which warrants the biggest expressive
power of the corresponding bunch of connectives, then follows. In the
end, matrices, i.e. algebras with sets of designated elements and their basic
properties, are constructed. Chapter 4,;Many-valuedness’, aims at showing