Introduction to Mathematical Logic
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(2) INTRODUCTION TO. MATHEMATIC AL LOGIC FIFTH EDITION.
(3) DISCRETE MATHEMATICS ITS APPLICATIONS Series Editor. Kenneth H. Rosen, Ph.D. Juergen Bierbrauer, Introduction to Coding Theory Francine Blanchet-Sadri, Algorithmic Combinatorics on Partial Words Richard A. Brualdi and Drago˘s Cvetkovi´c, A Combinatorial Approach to Matrix Theory and Its Applications Kun-Mao Chao and Bang Ye Wu, Spanning Trees and Optimization Problems Charalambos A. Charalambides, Enumerative Combinatorics Gary Chartrand and Ping Zhang, Chromatic Graph Theory Henri Cohen, Gerhard Frey, et al., Handbook of Elliptic and Hyperelliptic Curve Cryptography Charles J. Colbourn and Jeffrey H. Dinitz, Handbook of Combinatorial Designs, Second Edition Martin Erickson and Anthony Vazzana, Introduction to Number Theory Steven Furino, Ying Miao, and Jianxing Yin, Frames and Resolvable Designs: Uses, Constructions, and Existence Randy Goldberg and Lance Riek, A Practical Handbook of Speech Coders Jacob E. Goodman and Joseph O’Rourke, Handbook of Discrete and Computational Geometry, Second Edition Jonathan L. Gross, Combinatorial Methods with Computer Applications Jonathan L. Gross and Jay Yellen, Graph Theory and Its Applications, Second Edition Jonathan L. Gross and Jay Yellen, Handbook of Graph Theory Darrel R. Hankerson, Greg A. Harris, and Peter D. Johnson, Introduction to Information Theory and Data Compression, Second Edition Darel W. Hardy, Fred Richman, and Carol L. Walker, Applied Algebra: Codes, Ciphers, and Discrete Algorithms, Second Edition Daryl D. Harms, Miroslav Kraetzl, Charles J. Colbourn, and John S. Devitt, Network Reliability: Experiments with a Symbolic Algebra Environment Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words Leslie Hogben, Handbook of Linear Algebra Derek F. Holt with Bettina Eick and Eamonn A. O’Brien, Handbook of Computational Group Theory David M. Jackson and Terry I. Visentin, An Atlas of Smaller Maps in Orientable and Nonorientable Surfaces.
(4) Continued Titles Richard E. Klima, Neil P. Sigmon, and Ernest L. Stitzinger, Applications of Abstract Algebra with Maple™ and MATLAB®, Second Edition Patrick Knupp and Kambiz Salari, Verification of Computer Codes in Computational Science and Engineering William Kocay and Donald L. Kreher, Graphs, Algorithms, and Optimization Donald L. Kreher and Douglas R. Stinson, Combinatorial Algorithms: Generation Enumeration and Search C. C. Lindner and C. A. Rodger, Design Theory, Second Edition Hang T. Lau, A Java Library of Graph Algorithms and Optimization Elliott Mendelson, Introduction to Mathematical Logic, Fifth Edition Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone, Handbook of Applied Cryptography Richard A. Mollin, Algebraic Number Theory Richard A. Mollin, Codes: The Guide to Secrecy from Ancient to Modern Times Richard A. Mollin, Fundamental Number Theory with Applications, Second Edition Richard A. Mollin, An Introduction to Cryptography, Second Edition Richard A. Mollin, Quadratics Richard A. Mollin, RSA and Public-Key Cryptography Carlos J. Moreno and Samuel S. Wagstaff, Jr., Sums of Squares of Integers Dingyi Pei, Authentication Codes and Combinatorial Designs Kenneth H. Rosen, Handbook of Discrete and Combinatorial Mathematics Douglas R. Shier and K.T. Wallenius, Applied Mathematical Modeling: A Multidisciplinary Approach Jörn Steuding, Diophantine Analysis Douglas R. Stinson, Cryptography: Theory and Practice, Third Edition Roberto Togneri and Christopher J. deSilva, Fundamentals of Information Theory and Coding Design W. D. Wallis, Introduction to Combinatorial Designs, Second Edition Lawrence C. Washington, Elliptic Curves: Number Theory and Cryptography, Second Edition.
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(6) DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H. ROSEN. INTRODUCTION TO. MATHEMATIC AL LOGIC FIFTH EDITION. Elliott Mendelson Queens College Department of Mathematics F l u s h i n g , N ew Yo r k , U. S . A ..
(7) Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2010 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-58488-876-5 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Mendelson, Elliott. Introduction to mathematical logic / Elliott Mendelson. -- 5th ed. p. cm. -- (Discrete mathematics and its applications ; 48) Includes bibliographical references and index. ISBN-13: 978-1-58488-876-5 ISBN-10: 1-58488-876-8 1. Logic, Symbolic and mathematical. I. Title. QA9.M4 2009 511.3--dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com. 2009002833.
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(10) Contents Preface.................................................................................................................. xiii Introduction .......................................................................................................... xv 1. The Propositional Calculus ............................................................................ 1 1.1 Propositional Connectives. Truth Tables.......................................... 1 1.2 Tautologies ............................................................................................ 5 1.3 Adequate Sets of Connectives .......................................................... 18 1.4 An Axiom System for the Propositional Calculus ........................ 24 1.5 Independence. Many-Valued Logics ............................................... 34 1.6 Other Axiomatizations ...................................................................... 37 2. First-Order Logic and Model Theory......................................................... 41 2.1 Quantifiers........................................................................................... 41 2.1.1 Parentheses ............................................................................ 44 2.2 First-Order Languages and Their Interpretations. Satisfiability and Truth. Models.................................................... 48 2.3 First-Order Theories........................................................................... 61 2.3.1 Logical Axioms...................................................................... 62 2.3.2 Proper Axioms ...................................................................... 62 2.3.3 Rules of Inference ................................................................. 62 2.4 Properties of First-Order Theories ................................................... 64 2.5 Additional Metatheorems and Derived Rules ............................... 69 2.5.1 Particularization Rule A4 .................................................... 69 2.5.2 Existential Rule E4 ................................................................ 69 2.6 Rule C .................................................................................................. 73 2.7 Completeness Theorems ................................................................... 77 2.8 First-Order Theories with Equality.................................................. 88 2.9 Definitions of New Function Letters and Individual Constants ................................................................. 97 2.10 Prenex Normal Forms ..................................................................... 100 2.11 Isomorphism of Interpretations. Categoricity of Theories ......... 105 2.12 Generalized First-Order Theories. Completeness and Decidability ............................................................................... 107 2.12.1 Mathematical Applications.............................................. 111 2.13 Elementary Equivalence. Elementary Extensions........................ 117 2.14 Ultrapowers. Nonstandard Analysis.......................................... 123 2.14.1 Reduced Direct Products ................................................. 125 2.14.2 Nonstandard Analysis ..................................................... 131 2.15 Semantic Trees .................................................................................. 135 2.16 Quantification Theory Allowing Empty Domains...................... 141 ix.
(11) x. Contents. 3. Formal Number Theory .............................................................................. 149 3.1 An Axiom System .............................................................................. 149 3.2 Number-Theoretic Functions and Relations .................................. 166 3.3 Primitive Recursive and Recursive Functions ............................... 171 3.4 Arithmetization. Gödel Numbers .................................................... 188 3.5 The Fixed-Point Theorem. Gödel’s Incompleteness Theorem................................................................... 202 3.6 Recursive Undecidability. Church’s Theorem ............................... 214 3.7 Nonstandard Models ......................................................................... 224 4. Axiomatic Set Theory.................................................................................. 227 4.1 An Axiom System .............................................................................. 227 4.2 Ordinal Numbers ............................................................................... 242 4.3 Equinumerosity. Finite and Denumerable Sets ............................. 256 4.3.1 Finite Sets............................................................................... 261 4.4 Hartogs’ Theorem. Initial Ordinals. Ordinal Arithmetic ............ 266 4.5 The Axiom of Choice. The Axiom of Regularity........................... 279 4.6 Other Axiomatizations of Set Theory.............................................. 290 4.6.1 Morse–Kelley (MK) .............................................................. 291 4.6.2 Zermelo–Fraenkel (ZF) ........................................................ 291 4.6.3 The Theory of Types (ST).................................................... 293 4.6.3.1 ST1 (Extensionality Axiom).................................. 294 4.6.3.2 ST2 (Comprehension Axiom Scheme) ...................................................... 294 4.6.3.3 ST3 (Axiom of Infinity) ......................................... 295 4.6.4 Quine’s Theories NF and ML ............................................. 297 4.6.4.1 NF1 (Extensionality).............................................. 297 4.6.4.2 NF2 (Comprehension)........................................... 298 4.6.5 Set Theory with Urelements ............................................... 300 5. Computability ............................................................................................... 309 5.1 Algorithms. Turing Machines .......................................................... 309 5.2 Diagrams ............................................................................................. 315 5.3 Partial Recursive Functions. Unsolvable Problems....................... 322 5.4 The Kleene–Mostowski Hierarchy. Recursively Enumerable Sets ................................................................................. 338 5.5 Other Notions of Computability...................................................... 350 5.6 Decision Problems.............................................................................. 368 Appendix A: Second-Order Logic.................................................................. 375 Appendix B: First Steps in Modal Propositional Logic............................. 391.
(12) Contents. xi. Answers to Selected Exercises ........................................................................ 403 Bibliography....................................................................................................... 433 Notation............................................................................................................... 447 Index .................................................................................................................... 453.
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(14) Preface This is a compact introduction to some of the principal topics of mathematical logic. In the belief that beginners should be exposed to the easiest and most natural proofs, I have used free-swinging set-theoretic methods. The significance of a demand for constructive proofs can be evaluated only after a certain amount of experience with mathematical logic has been obtained. If we are to be expelled from ‘‘Cantor’s paradise’’ (as nonconstructive set theory was called by Hilbert), at least we should know what we would be missing. Readers who wish to learn about constructive mathematics and intuitionism can consult the books by van Atten (2003), van Atten, Boldini, Bourdeau, and Heinzmann (2008), Brouwer (1976), George and Velleman (2001), Heyting (1956), Mancosu (1997), and Troelstra and van Dalen (1988). The more important changes in this new edition are the following: 1. There is an entirely new exposition, Section 3.7, of some basic ideas and results about nonstandard models of number theory. 2. A second appendix, Appendix B, has been added as an introduction to modal propositional logic. The most important application in this area has to do with provability logic (see Boolos [1993]), but it is also of great interest elsewhere in logic. 3. The Bibliography has been greatly enlarged, although it is still not able to do full justice to the enormous expansion of research and scholarship in logic and related areas. 4. The exercises and the Answers to Selected Exercises have been expanded and necessary corrections have been made. The material in this book can be covered in two semesters, but Chapters 1–3 are quite adequate for a one-semester course (Sections 1.5, 1.6, 3.7, and 2.10–2.16 can be omitted in an abbreviated course). I have adopted the convention of prefixing a ‘‘D’’ to any section or exercise that will probably be difficult for a beginner, and an ‘‘A’’ to any section or exercise that presupposes familiarity with a topic that has not been carefully explained in the text. Bibliographic references are given to the best source of information, which is not always the earliest; hence these references give no indication as to priority. I believe that the essential parts of the book can be read with ease by anyone with some experience in abstract mathematical thinking. There is, however, no specific prerequisite.. xiii.
(15) xiv. Preface. This book owes an obvious debt to the standard works of Hilbert and Bernays (1934, 1939), Kleene (1952), Rosser (1953), and Church (1956). I am grateful to many people for their help, including those who helped with earlier editions, as well as Katalin Bimbo, Frank Cannonito, John Corcoran, George Hacken, Maidim Malkov, and Gordon McLean, Jr. I also want to thank my editor, Bob Stern, for his advice and patience. Elliott Mendelson.
(16) Introduction One of the popular definitions of logic is that it is the analysis of methods of reasoning. In studying these methods, logic is interested in the form rather than the content of the argument. For example, consider the two arguments: 1. All men are mortal. Socrates is a man. Hence, Socrates is mortal. 2. All cats like fish. Silvy is a cat. Hence, Silvy likes fish. Both have the same form: All A are B. S is an A. Hence, S is a B. The truth or falsity of the particular premisses and conclusions is of no concern to logicians. They want to know only whether the premisses imply the conclusion. The systematic formalization and cataloguing of valid methods of reasoning are a main task of logicians. If the work uses mathematical techniques or if it is primarily devoted to the study of mathematical reasoning, then it may be called mathematical logic. We can narrow the domain of mathematical logic if we define its principal aim to be a precise and adequate understanding of the notion of mathematical proof. Impeccable definitions have little value at the beginning of the study of a subject. The best way to find out what mathematical logic is about is to start doing it, and students are advised to begin reading the book even though (or especially if) they have qualms about the meaning and purpose of the subject. Although logic is basic to all other studies, its fundamental and apparently self-evident character discouraged any deep logical investigations until the late nineteenth century. Then, under the impetus of the discovery of nonEuclidean geometry and the desire to provide a rigorous foundation for calculus and higher analysis, interest in logic revived. This new interest, however, was still rather unenthusiastic until, around the turn of the century, the mathematical world was shocked by the discovery of the paradoxes— that is, arguments that lead to contradictions. The most important paradoxes are described here. 1. Russell’s paradox (1902). By a set, we mean any collection of objects— for example, the set of all even integers or the set of all saxophone players in Brooklyn. The objects that make up a set are called its members or elements. Sets may themselves be members of sets; for example, the set of all sets of integers has sets as its members. Most sets are not members of themselves; the set of cats, for example, is not a member of itself because the set of cats is not a cat. However, there may be sets that do belong to themselves—perhaps, for example, a set xv.
(17) xvi. Introduction. containing all sets. Now, consider the set A of all those sets X such that X is not a member of X. Clearly, by definition, A is a member of A if and only if A is not a member of A. So, if A is a member of A, then A is also not a member of A; and if A is not a member of A, then A is a member of A. In any case, A is a member of A and A is not a member of A [See Link, (2004)]. 2. Cantor’s paradox (1899). This paradox involves the theory of cardinal numbers and may be skipped by those readers having no previous acquaintance with that theory. The cardinal number Y of a set Y is a measure of the size of the set; Y ¼ Z if and only if Y is equinumerous with Z (that is, there is a one–one correspondence between Y and Z). We define Y 4 Z to mean that Y is equinumerous with a subset of Z; by Y < Z we mean Y 4 Z and Y 6¼ Z. Cantor proved that, if p(Y) is the set of all subsets of Y, then Y < p(Y). Let V be the universal set— that is, the set of all sets. Now, p(V) is a subset of V; so it follows easily that p(V) 4 V. On the other hand, by Cantor’s theorem, V < p(V). Bernstein’s theorem asserts that, if Y 4 Z and Z 4 Y, then Y ¼ Z. Hence, V ¼ p(V), contradicting V < p(V). 3. Burali-Forti’s paradox (1897). This paradox is the analogue in the theory of ordinal numbers of Cantor’s paradox and requires familiarity with ordinal number theory. Given any ordinal number, there is a still larger ordinal number. But the ordinal number determined by the set of all ordinal numbers is the largest ordinal number. 4. The liar paradox. A man says, ‘‘I am lying’’. If he is lying, then what he says is true and so he is not lying. If he is not lying, then what he says is true, and so he is lying. In any case, he is lying and he is not lying.* 5. Richard’s paradox (1905). Some phrases of the English language denote real numbers; for example, ‘‘the ratio between the circumference and diameter of a circle’’ denotes the number p. All the phrases of the English language can be enumerated in a standard way: order all phrases that have k letters lexicographically (as in a dictionary) and then place all phrases with k letters before all phrases with a larger number of letters. Hence, all phrases of the English language that denote real numbers can be enumerated merely by omitting all other phrases in the given standard enumeration. Call the nth real * The Cretan ‘‘paradox,’’ known in antiquity, is similar to the liar paradox. The Cretan philosopher Epimenides said, ‘‘All Cretans are liars.’’ If what he said is true, then, since Epimenides is a Cretan, it must be false. Hence, what he said is false. Thus, there must be some Cretan who is not a liar. This is not logically impossible; so we do not have a genuine paradox. However, the fact that the utterance by Epimenides of that false sentence could imply the existence of some Cretan who is not a liar is rather unsettling..
(18) Introduction. xvii. number in this enumeration the nth Richard number. Consider the phrase: ‘‘the real number whose nth decimal place is 1 if the nth decimal place of the nth Richard number is not 1, and whose nth decimal place is 2 if the nth decimal place of the nth Richard number is 1.’’ This phrase defines a Richard number—say, the kth Richard number; but, by its definition, it differs from the kth Richard number in the kth decimal place. 6. Berry’s paradox (1906). There are only a finite number of symbols (letters, punctuation signs, etc.) in the English language. Hence, there are only a finite number of English expressions that contain fewer than 200 occurrences of symbols (allowing repetitions). There are, therefore, only a finite number of positive integers that are denoted by an English expression containing fewer than 200 occurrences of symbols. Let k be the least positive integer that is not denoted by an English expression containing fewer than 200 occurrences of symbols. The italicized English phrase contains fewer than 200 occurrences of symbols and denotes the integer k. 7. Grelling’s paradox (1908). An adjective is called autological if the property denoted by the adjective holds for the adjective itself. An adjective is called heterological if the property denoted by the adjective does not apply to the adjective itself. For example, ‘‘polysyllabic’’ and ‘‘English’’ are autological, whereas ‘‘monosyllabic’’ and ‘‘French’’ are heterological. Consider the adjective ‘‘heterological.’’ If ‘‘heterological’’ is heterological, then it is not heterological. If ‘‘heterological’’ is not heterological, then it is heterological. In either case, ‘‘heterological’’ is both heterological and not heterological. 8. Löb’s paradox (1955). Let A be any sentence. Let B be the sentence: ‘‘If this sentence is true, then A.’’ So, B asserts: ‘‘If B is true, then A.’’ Now consider the following argument: Assume B is true; then, by B, since B is true, A holds. This argument shows that, if B is true, then A. But this is exactly what B asserts. Hence, B is true. Therefore, by B, since B is true, A is true. Thus, every sentence is true. (This paradox may be more accurately attributed to Curry [1942].) All of these paradoxes are genuine in the sense that they contain no obvious logical flaws. The logical paradoxes (1–3) involve only notions from the theory of sets, whereas the semantic paradoxes (4–8) also make use of concepts like ‘‘denote,’’ ‘‘true,’’ and ‘‘adjective,’’ which need not occur within our standard mathematical language. For this reason, the logical paradoxes are a much greater threat to a mathematician’s peace of mind than the semantic paradoxes. Analysis of the paradoxes has led to various proposals for avoiding them. All of these proposals are restrictive in one way or another of the ‘‘naive’’ concepts that enter into the derivation of the paradoxes. Russell noted the self-reference present in all the paradoxes and suggested that every object.
(19) xviii. Introduction. must have a definite nonnegative integer as its ‘‘type.’’ Then an expression ‘‘x is a member of the set y’’ is to be considered meaningful if and only if the type of y is one greater than the type of x. This approach, known as the theory of types and systematized and developed in Principia Mathematica by Whitehead and Russell (1910–1913), is successful in eliminating the known paradoxes,y but it is clumsy in practice and has certain other drawbacks as well. A different criticism of the logical paradoxes is aimed at their assumption that, for every property P(x), there exists a corresponding set of all objects x that satisfy P(x). If we reject this assumption, then the logical paradoxes are no longer derivable.z It is necessary, however, to provide new postulates that will enable us to prove the existence of those sets that are needed by the practicing mathematician. The first such axiomatic set theory was invented by Zermelo (1908). In Chapter 4 we shall present an axiomatic theory of sets that is a descendant of Zermelo’s system (with some new twists given to it by von Neumann, R. Robinson, Bernays, and Gödel). There are also various hybrid theories combining some aspects of type theory and axiomatic set theory—for example, Quine’s system NF. A more radical interpretation of the paradoxes has been advocated by Brouwer and his intuitionist school (see Heyting, 1956). They refuse to accept the universality of certain basic logical laws, such as the law of excluded middle: P or not-P. Such a law, they claim, is true for finite sets, but it is invalid to extend it on a wholesale basis to all sets. Likewise, they say it is invalid to conclude that ‘‘There exists an object x such that not-P(x)’’ follows from the negation of ‘‘For all x, P(x)’’; we are justified in asserting the existence of an object having a certain property only if we know an effective method for constructing (or finding) such an object. The paradoxes are not derivable (or even meaningful) if we obey the intuitionist strictures, but so are many important theorems of everyday mathematics, and, for this reason, intuitionism has found few converts among mathematicians. Exercises 0.1 Use the sentence (*) This entire sentence is false or 2 þ 2 ¼ 5 to prove that 2 þ 2 ¼ 5. Comment on the significance of this proof. 0.2 Show how the following has a paradoxical result. The smallest positive integer that is not denoted by a phrase in this book.. y. z. Russells’s paradox, for example, depends on the existence of the set A of all sets that are not members of themselves. Because, according to the theory of types, it is meaningless to say that a set belongs to itself, there is no such set A. Russell’s paradox then proves that there is no set A of all sets that do not belong to themselves. The paradoxes of Cantor and Burali-Forti show that there is no universal set and no set that contains all ordinal numbers. The semantic paradoxes cannot even be formulated, since they involve notions not expressible within the system..
(20) Introduction. xix. Whatever approach one takes to the paradoxes, it is necessary first to examine the language of logic and mathematics to see what symbols may be used, to determine the ways in which these symbols are put together to form terms, formulas, sentences, and proofs, and to find out what can and cannot be proved if certain axioms and rules of inference are assumed. This is one of the tasks of mathematical logic, and, until it is done, there is no basis for comparing rival foundations of logic and mathematics. The deep and devastating results of Gödel, Tarski, Church, Rosser, Kleene, and many others have been ample reward for the labor invested and have earned for mathematical logic its status as an independent branch of mathematics. For the absolute novice a summary will be given here of some of the basic notations, ideas, and results used in the text. The reader is urged to skip these explanations now and, if necessary, to refer to them later on. A set is a collection of objects.* The objects in the collection are called elements or members of the set. We shall write ‘‘x 2 y’’ for the statement that x is a member of y. (Synonymous expressions are ‘‘x belongs to y’’ and ‘‘y contains x’’.) The negation of ‘‘x 2 y’’ will be written ‘‘x 2 = y.’’ By ‘‘x y’’ we mean that every member of x is also a member of y (synonymously, that x is a subset of y, or that x is included in y). We shall write ‘‘t ¼ s’’ to mean that t and s denote the same object. As usual, ‘‘t 6¼ s’’ is the negation of ‘‘t ¼ s.’’ For sets x and y, we assume that x ¼ y if and only if x y and y x—that is, if and only if x and y have the same members. A set x is called a proper subset of a set y, written ‘‘x y’’ if x y but x 6¼ y. (The notation x 6 y is often used instead of x y.) The union x [ y of sets x and y is defined to be the set of all objects that are members of x or y or both. Hence, x [ x ¼ x, x [ y ¼ y [ x, and (x [ y) [ z ¼ x [ (y [ z). The intersection x \ y is the set of objects that x and y have in common. Therefore, x \ x ¼ x, x \ y ¼ y \ x, and (x \ y) \ z ¼ x \ (y \ z). Moreover, x \ (y [ z) ¼ (x \ y) [ (x \ z) and x [ (y \ z) ¼ (x [ y) \ (x [ z). The relative complement x y is the set of members of x that are not members of y. We also postulate the existence of the empty set (or null set) ;—that is, a set that has no members at all. Then x \ ; ¼ ;, x [ ; ¼ x, x ; ¼ x, ; x ¼ ;, and x x ¼ ;. Sets x and y are called disjoint if x \ y ¼ ;. Given any objects b1 , . . . , bk , the set that contains b1 , . . . , bk as its only members is denoted fb1 , . . . , bk g. In particular, {x, y} is a set having x and y as its only members and, if x 6¼ y, is called the unordered pair of x and y. The set {x, x} is identical with {x} and is called the unit set of x. Notice that {x, y} ¼ {y, x}. By hb1 , . . . , bk i we mean the ordered k-tuple of b1 , . . . , bk . The basic property of ordered k-tuples is that hb1 , . . . , bk i ¼ hc1 , . . . , ck i if and only if * Which collections of objects form sets will not be specified here. Care will be exercised to avoid using any ideas or procedures that may lead to the paradoxes; all the results can be formalized in the axiomatic set theory of Chapter 4. The term ‘‘class’’ is sometimes used as a synonym for ‘‘set,’’ but it will be avoided here because it has a different meaning in Chapter 4. If a property P(x) does determine a set, that set is often denoted {xjP(x)}..
(21) xx. Introduction. b1 ¼ c1 , b2 ¼ c2 , . . . , bk ¼ ck . Thus, hb1 , b2 i ¼ hb2 , b1 i if and only if b1 ¼ b2 . Ordered 2-tuples are called ordered pairs. The ordered 1-tuple hbi is taken to be b itself. If X is a set and k is a positive integer, we denote by Xk the set of all ordered k-tuples hb1 , . . . , bk i of elements b1 , . . . , bk of X. In particular, X1 is X itself. If Y and Z are sets, then by Y Z we denote the set of all ordered pairs hy, zi such that y 2 Y and z 2 Z. Y Z is called the Cartesian product of Y and Z. An n-place relation (or a relation with n arguments) on a set X is a subset of Xn—that is, a set of ordered n-tuples of elements of X. For example, the 3-place relation of betweenness for points on a line is the set of all 3-tuples hx, y, zi such that the point x lies between the points y and z. A 2-place relation is called a binary relation; for example, the binary relation of fatherhood on the set of human beings is the set of all ordered pairs hx, yi such that x and y are human beings and x is the father of y. A 1-place relation on X is a subset of X and is called a property on X. Given a binary relation R on a set X, the domain of R is defined to be the set of all y such that hy, zi 2 R for some z; the range of R is the set of all z such that hy, zi 2 R for some y; and the field of R is the union of the domain and range of R. The inverse relation R1 of R is the set of all ordered pairs hy, zi such that hz, yi 2 R. For example, the domain of the relation < on the set v of nonnegative integers* is v, its range is v {0}, and the inverse of < is >. Notation: Very often xRy is written instead of hx, yi 2 R. Thus, in the example just given, we usually write x < y instead of hx, yi 2 <. A binary relation R is said to be reflexive if xRx for all x in the field of R; R is symmetric if xRy implies yRx; and R is transitive if xRy and yRz imply xRz. Examples: The relation on the set of integers is reflexive and transitive but not symmetric. The relation ‘‘having at least one parent in common’’ on the set of human beings is reflexive and symmetric, but not transitive. A binary relation that is reflexive, symmetric, and transitive is called an equivalence relation. Examples of equivalence relations are: (1) the identity relation IX on a set X, consisting of all pairs hx, xi, where x 2 X; (2) given a fixed positive integer n, the relation x y (mod n), which holds when x and y are integers and x y is divisible by n; (3) the congruence relation on the set of triangles in a plane; (4) the similarity relation on the set of triangles in a plane. Given an equivalence relation R whose field is X, and given any y 2 X, define [y] as the set of all z in X such that yRz. Then [y] is called the R-equivalence class of y. Clearly, [u] ¼ [v] if and only if uRv. Moreover, if [u] 6¼ [v], then [u] \ [v] ¼ ;; that is, different R-equivalence classes have no elements in common. Hence, the set X is completely partitioned into the R-equivalence classes. In example (1) above, the equivalence classes are just the unit sets {x}, where x 2 X. In example (2), there are n equivalence classes, the kth equivalence class (k ¼ 0, 1, . . . , n 1) being the set of all integers that leave the remainder k upon division by n.. * v will also be referred to as the set of natural numbers..
(22) Introduction. xxi. A function f is a binary relation such that hx, yi 2 f and hx, zi 2 f imply y ¼ z. Thus, for any element x of the domain of a function f, there is a unique y such that hx, yi 2 f ; this unique y is denoted f(x). If x is in the domain of f, then f(x) is said to be defined. A function f with domain X and range Y is said to be a function from X onto Y. If f is a function from X onto a subset of Z, then f is said to be a function from X into Z. For example, if the domain of f is the set of integers and f (x) ¼ 2x for every integer x, then f is a function from the set of integers onto the set of even integers, and f is a function from the set of integers into the set of integers. A function whose domain consists of n-tuples is said to be a function of n arguments. A total function of n arguments on a set X is a function f whose domain is Xn. It is customary to write f ðx1 , . . . , xn Þ instead of f ðhx1 , . . . , xn iÞ, and we refer to f ðx1 , . . . , xn Þ as the value of f for the arguments x1 , . . . , xn . A partial function of n arguments on a set X is a function whose domain is a subset of Xn. For example, ordinary division is a partial, but not total, function of two arguments on the set of integers, since division by 0 is not defined. If f is a function with domain X and range Y, then the restriction fz of f to a set Z is the function f \ (Z Y). Then fZ (u) ¼ v if and only if u 2 Z and f (u) ¼ v. The image of the set Z under the function f is the range of fz. The inverse image of a set W under the function f is the set of all u in the domain of f such that f (u) 2 W. We say that f maps X onto (into) Y if X is a subset of the domain of f and the image of X under f is (a subset of) Y. By an n-place operation (or operation with n arguments) on a set X we mean a function from Xn into X. For example, ordinary addition is a binary (i.e., 2-place) operation on the set of natural numbers {0, 1, 2, . . . }. But ordinary subtraction is not a binary operation on the set of natural numbers. The composition f g (sometimes denoted fg) of functions f and g is the function such that (f g)(x) ¼ f (g(x)); (f g)(x) is defined if and only if g(x) is defined and f(g(x)) is defined. For example, if g(x) ¼ x2 and f (x) ¼ x þ 1 for every integer x, then (f g)(x) ¼ x2 þ 1 and p(gffiffiffi f )(x) ¼ (x þ 1)2 . Also, if h(x) ¼ x for every real number x and f (x) ¼ x for every nonnegative real number px,ffiffiffiffiffiffithen (f h)(x) is defined only for x 4 0, and, for such ffi x, (f h)(x) ¼ x. A function f such that f (x) ¼ f (y) implies x ¼ y is called a 1–1 (one–one) function. For example, the identity relation IX on a set X is a 1–1 function, since IX (y) ¼ y for every y 2 X; the function g with domain v, such that g(x) ¼ 2x for every x 2 v, is 1–1 (one–one); but the function h whose domain is the set of integers and such that h(x) ¼ x2 for every integer x is not 1–1, since h(1) ¼ h(1). Notice that a function f is 1–1 if and only if its inverse relation f 1 is a function. If the domain and range of a 1–1 function f are X and Y, then f is said to be a 1–1 correspondence between andY; then f 1 X 1 is a 1–1 correspondence between Y and X, and f f ¼ IX and f f 1 ¼ IY . If f is a 1–1 correspondence between X and Y and g is a 1–1 correspondence between Y and Z, then g f is a 1–1 correspondence between X and Z. Sets X and Y are said to be equinumerous (written X ffi Y) if and only if there is a 1–1 correspondence between X and Y. Clearly, X ffi X, X ffi Y implies Y ffi X, and X ffi Y and Y ffi Z implies X ffi Z. It is somewhat harder to.
(23) xxii. Introduction. show that, if X ffi Y1 Y and Y ffi X1 X, then X ffi Y (see Bernstein’s theorem in Chapter 4). If X ffi Y, one says that X and Y have the same cardinal number, and if X is equinumerous with a subset of Y but Y is not equinumerous with a subset of X, one says that the cardinal number of X is smaller than the cardinal number of Y.* A set X is denumerable if it is equinumerous with the set of positive integers. A denumerable set is said to have cardinal number Q0 , and any set equinumerous with the set of all subsets of a denumerable set is said to have the cardinal number 2Q0 (or to have the power of the continuum). A set X is finite if it is empty or if it is equinumerous with the set {1, 2, . . . , n} of all positive integers that are less than or equal to some positive integer n. A set that is not finite is said to be infinite. A set is countable if it is either finite or denumerable. Clearly, any subset of a denumerable set is countable. A denumerable sequence is a function s whose domain is the set of positive integers; one usually writes sn instead of s(n). A finite sequence is a function whose domain is the empty set or {1, 2, . . . , n} for some positive integer n. Let Pðx, y1 , . . . , yk Þ be some relation on the set of nonnegative integers. In particular, P may involve only the variable x and thus be a property. If Pð0, y1 , . . . , yk Þ holds, and, if, for every n, Pðn, y1 , . . . , yk Þ implies Pðn þ 1, y1 , . . . , yk Þ, then Pðx, y1 , . . . , yk Þ is true for all nonnegative integers x (principle of mathematical induction). In applying this principle, one usually proves that, for every n, Pðn, y1 , . . . , yk Þ implies Pðn þ 1, y1 , . . . , yk Þ by assuming Pðn, y1 , . . . , yk Þ and then deducing Pðn þ 1, y1 , . . . , yk Þ; in the course of this deduction, Pðn, y1 , . . . , yk Þ is called the inductive hypothesis. If the relation P actually involves variables y1 , . . . , yk other than x, then the proof is said to proceed by induction on x. A similar induction principle holds for the set of integers greater than some fixed integer j. An example is: to prove by mathematical induction that the sum of the first n odd integers 1 þ 3 þ 5 þ. þ (2n 1) is n2, first show that 1 ¼ 12 (that is, P(1)), and then, that if 1 þ 3 þ 5 þ. þ (2n 1) ¼ n2 , then 1 þ 3 þ 5 þ. þ (2n 1) þ (2n þ 1) ¼ (n þ 1)2 (that is, if P(n) then P(n þ 1)). From the principle of mathematical induction one can prove the principle of complete induction: If, for every nonnegative integer x the assumption that Pðu, y1 , . . . , yk Þ is true for all u < x implies that Pðx, y1 , . . . , yk Þ holds, then, for all nonnegative integers x, Pðx, y1 , . . . , yk Þ is true, (Exercise: Show by complete induction that every integer greater than 1 is divisible by a prime number.) A partial order is a binary relation R such that R is transitive and, for every x in the field of R, xRx is false. If R is a partial order, then the relation R0 that is the union of R and the set of all ordered pairs hx, xi, where x is in the field of R, we shall call a reflexive partial order; in the literature, ‘‘partial order’’ is used for either partial order or reflexive partial order. Notice that (xRy and yRx) is * One can attempt to define the cardinal number of a set X as the collection [X] of all sets equinumerous with X. However, in certain axiomatic set theories, [X] does not exist, whereas in others [X] exists but is not a set..
(24) Introduction. xxiii. impossible if R is a partial order, whereas (xRy and yRx) implies x ¼ y if R is a reflexive partial order. A (reflexive) total order is a (reflexive) partial order such that, for any x and y in the field of R, either x ¼ y or xRy or yRx. Examples: (1) the relation < on the set of integers is a total order, whereas is a reflexive total order; (2) the relation on the set of all subsets of the set of positive integers is a partial order but not a total order, whereas the relation is a reflexive partial order but not a reflexive total order. If B is a subset of the field of a binary relation R, then an element y of B is called an R-least element of B if yRz for every element z of B different from y. A well-order (or a well-ordering relation) is a total order R such that every nonempty subset of the field of R has an R-least element. Examples: (1) the relation < on the set of nonnegative integers is a well-order; (2) the relation < on the set of nonnegative rational numbers is a total order but not a well-order; (3) the relation < on the set of integers is a total order but not a wellorder. Associated with every well-order R having field X there is a complete induction principle: if P is a property such that, for any u in X, whenever all z in X such that zRu have the property P, then u has the property P, then it follows that all members of X have the property P. If the set X is infinite, a proof using this principle is called a proof by transfinite induction. One says that a set X can be well-ordered if there exists a well-order whose field is X. An assumption that is useful in modern mathematics but about the validity of which there has been considerable controversy is the well-ordering principle: every set can be wellordered. The well-ordering principle is equivalent (given the usual axioms of set theory) to the axiom of choice: for any set X of nonempty pairwise disjoint sets, there is a set Y (called a choice set) that contains exactly one element in common with each set in X. Let B be a nonempty set, f a function from B into B, and g a function from B2 into B. Write x0 for f(x) and x \ y for g(x, y). Then hB, f , gi is called a Boolean algebra if B contains at least two elements and the following conditions are satisfied: 1. x \ y ¼ y \ x for all x and y in B 2. (x \ y) \ z ¼ x \ (y \ z) for all x, y, z in B 3. x \ y0 ¼ z \ z0 if and only if x \ y ¼ x for all x, y, z in B. Let x [ y stand for ðx0 \ y0 Þ0 , and write x 4 y for x \ y ¼ x. It is easily proved that z \ z0 ¼ w \ w0 for any w and z in B; we denote the value of z \ z0 by 0. Let 1 stand for 00 . Then z [ z0 ¼ 1 for all z in B. Note also that is a reflexive partial order on B, and hB, f , [i is a Boolean algebra. (The symbols \, [ , 0, 1 should not be confused with the corresponding symbols used in set theory and arithmetic.) An ideal J in hB, f , gi is a nonempty subset of B such that (1) if x 2 J and y 2 J, then x [ y 2 J, and (2) if x 2 J and y 2 B, then x \ y 2 J. Clearly, {0} and B are ideals. An ideal different from B is called a proper ideal. A maximal ideal is a proper ideal that is included in no other proper ideal. It can be shown.
(25) xxiv. Introduction. that a proper ideal J is maximal if and only if, for any u in B, u 2 J or u0 2 J. From the axiom of choice it can be proved that every Boolean algebra contains a maximal ideal, or, equivalently, that every proper ideal is included in some maximal ideal. For example, let B be the set of all subsets of a set X; for Y 2 B, let Y0 ¼ X Y, and for Y and Z in B, let Y \ Z be the ordinary set-theoretic intersection of Y and Z. Then hB,0 , \i is a Boolean algebra. The 0 of B is the empty set ;, and 1 is X. For each element u in X, the set Ju of all subsets of X that do not contain u is a maximal ideal. For a detailed study of Boolean algebras, see Sikorski (1960), Halmos (1963), and Mendelson (1970)..
(26) 1 The Propositional Calculus. 1.1 Propositional Connectives. Truth Tables Sentences may be combined in various ways to form more complicated sentences. We shall consider only truth-functional combinations, in which the truth or falsity of the new sentence is determined by the truth or falsity of its component sentences. Negation is one of the simplest operations on sentences. Although a sentence in a natural language may be negated in many ways, we shall adopt a uniform procedure: placing a sign for negation, the symbol :, in front of the entire sentence. Thus, if A is a sentence, then :A denotes the negation of A. The truth-functional character of negation is made apparent in the following truth table: A T F. :A F T. When A is true, :A is false; when A is false, :A is true. We use T and F to denote the truth values true and false. Another common truth-functional operation is the conjunction: ‘‘and.’’ The conjunction of sentences A and B will be designated by A ^ B and has the following truth table: A T F T F. B A^B T T T F F F F F. A ^ B is true when and only when both A and B are true. A and B are called the conjuncts of A ^ B. Note that there are four rows in the table, corresponding to the number of possible assignments of truth values to A and B. In natural languages, there are two distinct uses of ‘‘or’’: the inclusive and the exclusive. According to the inclusive usage, ‘‘A or B’’ means ‘‘A or B or both,’’ whereas according to the exclusive usage, the meaning is ‘‘A or B, but 1.
(27) 2. Introduction to Mathematical Logic. not both,’’ We shall introduce a special sign, _, for the inclusive connective. Its truth table is as follows: A T F T F. B A_B T T T T F T F F. Thus, A _ B is false when and only when both A and B are false. ‘‘A _ B’’ is called a disjunction, with the disjuncts A and B. Another important truth-functional operation is the conditional: ‘‘if A, then B.’’ Ordinary usage is unclear here. Surely, ‘‘if A, then B’’ is false when the antecedent A is true and the consequent B is false. However, in other cases, there is no well-defined truth value. For example, the following sentences would be considered neither true nor false: 1. If 1 þ 1 ¼ 2, then Paris is the capital of France. 2. If 1 þ 1 6¼ 2, then Paris is the capital of France. 3. If 1 þ 1 6¼ 2, then Rome is the capital of France. Their meaning is unclear, since we are accustomed to the assertion of some sort of relationship (usually causal) between the antecedent and the consequent. We shall make the convention that ‘‘if A, then B’’ is false when and only when A is true and B is false. Thus, sentences 1–3 are assumed to be true. Let us denote ‘‘if A, then B’’ by ‘‘A ) B.’’ An expression ‘‘A ) B’’ is called a conditional. Then ) has the following truth table: A T F T F. B A)B T T T T F F F T. This sharpening of the meaning of ‘‘if A, then B’’ involves no conflict with ordinary usage, but rather only an extension of that usage.* * There is a common non-truth-functional interpretation of ‘‘if A, then B’’ connected with causal laws. The sentence ‘‘if this piece of iron is placed in water at time t, then the iron will dissolve’’ is regarded as false even in the case that the piece of iron is not placed in water at time t—that is, even when the antecedent is false. Another non-truth-functional usage occurs in so-called counterfactual conditionals, such as ‘‘if Sir Walter Scott had not written any novels, then there would have been no War Between the States.’’ (This was Mark Twain’s contention in Life on the Mississippi: ‘‘Sir Walter had so large a hand in making Southern character, as it existed before the war, that he is in great measure responsible for the war.’’) This sentence might be asserted to be false even though the antecedent is admittedly false. However, causal laws and counterfactual conditions seem not to be needed in mathematics and logic. For a clear treatment of conditionals and other connectives, see Quine (1951). (The quotation from Life on the Mississippi was brought to my attention by Professor J.C. Owings, Jr.).
(28) 3. The Propositional Calculus. A justification of the truth table for ) is the fact that we wish ‘‘if A and B, then B’’ to be true in all cases. Thus, the case in which A and B are true justifies the first line of our truth table for ), since (A and B) and B are both true. If A is false and B true, then (A and B) is false while B is true. This corresponds to the second line of the truth table. Finally, if A is false and B is false, (A and B) is false and B is false. This gives the fourth line of the table. Still more support for our definition comes from the meaning of statements such as ‘‘for every x, if x is an odd positive integer, then x2 is an odd positive integer.’’ This asserts that, for every x, the statement ‘‘if x is an odd positive integer, then x2 is an odd positive integer’’ is true. Now we certainly do not want to consider cases in which x is not an odd positive integer as counterexamples to our general assertion. This supports the second and fourth lines of our truth table. In addition, any case in which x is an odd positive integer and x2 is an odd positive integer confirms our general assertion. This corresponds to the first line of the table. Let us denote ‘‘A if and only if B’’ by ‘‘A , B.’’ Such an expression is called a biconditional. Clearly, A , B is true when and only when A and B have the same truth value. Its truth table, therefore is: A T F T F. B A,B T T T F F F F T. The symbols :, ^ , _, ), and , will be called propositional connectives.* Any sentence built up by application of these connectives has a truth value that depends on the truth values of the constituent sentences. In order to make this dependence apparent, let us apply the name statement form to an expression built up from the statement letters A, B, C, and so on by appropriate applications of the propositional connectives. 1. All statement letters (capital italic letters) and such letters with numerical subscriptsy are statement forms. 2. If b and c are statement forms, then so are (:b), (b ^ c), (b _ c), (b ) c), and (b , c). 3. Only those expressions are statement forms that are determined to be so by means of conditions 1 and 2.z * We have been avoiding and shall in the future avoid the use of quotation marks to form names whenever this is not likely to cause confusion. The given sentence should have quotation marks around each of the connectives. See Quine (1951, pp. 23–27). y For example, A1 , A2 , A17 , B31 , C2 , . . .. z This can be rephrased as follows: c is a statement form if and only if there is a finite sequence b1, . . . , bn (n 1) such that bn ¼ c and, if 1 i n, bi is either a statement letter or a negation, conjunction, disjunction, conditional, or biconditional constructed from previous expressions in the sequence. Notice that we use script letters a, b, c, . . . to stand for arbitrary expressions, whereas italic letters are used as statement letters..
(29) 4. Introduction to Mathematical Logic. Some examples of statement forms are B, ð:C2 Þ, ðD3 ^ (:B)Þ, ððð:B1 Þ _ B2 Þ ) ðA1 ^ C2 ÞÞ, and (((:A) , A) , (C ) (B _ C))). For every assignment of truth values T or F to the statement letters that occur in a statement form, there corresponds, by virtue of the truth tables for the propositional connectives, a truth value for the statement form. Thus, each statement form determines a truth function, which can be graphically represented by a truth table for the statement form. For example, the statement form (((:A) _ B) ) C) has the following truth table: A T F T F T F T F. B T T F F T T F F. C T T T T F F F F. (:A) F T F T F T F T. ((:A) _ B) T T F T T T F T. (((:A) _ B) ) C) T T T T F F T F. Each row represents an assignment of truth values to the statement letters A, B, and C and the corresponding truth values assumed by the statement forms that appear in the construction of (((:A) _ B) ) C). The truth table for ((A , B) ) ((:A) ^ B)) is as follows: A T F T F. B (A , B) T T T F F F F T. (:A) F T F T. ((:A) ^ B) F T F F. ((A , B) ) ((:A) ^ B)) F T T F. If there are n distinct letters in a statement form, then there are 2n possible assignments of truth values to the statement letters and, hence, 2n rows in the truth table. A truth table can be abbreviated by writing only the full statement form, putting the truth values of the statement letters underneath all occurrences of these letters, and writing, step by step, the truth values of each component statement form under the principal connective of the form.* As an example, for ((A , B) ) ((:A) ^ B)), we obtain ((A T F T F. , B) ) T T F F T T F F T T F F. ((:A) FT TF FT TF. ^ F T F F. B)) T T F F. * The principal connective of a statement form is the one that is applied last in constructing the form..
(30) 5. The Propositional Calculus. Exercises 1.1 Let designate the exclusive use of ‘‘or.’’ Thus, A B stands for ‘‘A or B but not both.’’ Write the truth table for . 1.2 Construct truth tables for the statement forms ((A ) B) _ (:A)) and ((A ) (B ) C)) ) ((A ) B) ) (A ) C))). 1.3 Write abbreviated truth tables for ((A ) B) ^ A) and ((A _ (:C)) , B). 1.4 Write the following sentences as statement forms, using statement letters to stand for the atomic sentences—that is, those sentences that are not built up out of other sentences. (a) If Mr Jones is happy, Mrs Jones is not happy, and if Mr Jones is not happy, Mrs Jones is not happy. (b) Either Sam will come to the party and Max will not, or Sam will not come to the party and Max will enjoy himself. (c) A sufficient condition for x to be odd is that x is prime. (d) A necessary condition for a sequence s to converge is that s be bounded. (e) A necessary and sufficient condition for the sheikh to be happy is that he has wine, women, and song. (f) Fiorello goes to the movies only if a comedy is playing. (g) The bribe will be paid if and only if the goods are delivered. (h) If x is positive, x2 is positive. (i) Karpov will win the chess tournament unless Kasparov wins today.. 1.2 Tautologies A truth function of n arguments is defined to be a function of n arguments, the arguments and values of which are the truth values T or F. As we have seen, any statement form containing n distinct statement letters determines a corresponding truth function of n arguments.* * To be precise, enumerate all statement letters as follows: A, B, . . . , Z; A1 , B1 , . . . , Z1 ; A2 , . . . ,. If a statement form contains the i1th , . . . , inth statement letters in this enumeration, where i1 < < in , then the corresponding truth function is to have xi1 , . . . , xin , in that order, as its arguments, where xij corresponds to the ijth statement letter. For example, (A ) B) generates the truth function: x1 T F T F. x2 T T F F. f ðx1 , x2 Þ T T F T. whereas (B ) A) generates the truth function: x1 T F T F. x2 T T F F. gðx1 , x2 Þ T F T T.
(31) 6. Introduction to Mathematical Logic. A statement form that is always true, no matter what the truth values of its statement letters may be, is called a tautology. A statement form is a tautology if and only if its corresponding truth function takes only the value T, or equivalently, if, in its truth table, the column under the statement form contains only Ts. An example of a tautology is (A _ (:A)), the so-called law of the excluded middle. Other simple examples are (:(A ^ (:A))), (A , (:(:A))), ((A ^ B) ) A), and (A ) (A _ B)). b is said to logically imply c (or, synonymously, c is a logical consequence of b) if and only if every truth assignment to the statement letters of b and c that makes b true also makes c true. For example, (A ^ B) logically implies A, A logically implies (A _ B), and (A ^ (A ) B)) logically implies B. b and c are said to be logically equivalent if and only if b and c receive the same truth value under every assignment of truth values to the statement letters of b and c. For example, A and (:(:A)) are logically equivalent, as are (A ^ B) and (B ^ A).. PROPOSITION 1.1 (a) b logically implies c if and only if (b ) c) is a tautology. (b) b and c are logically equivalent if and only if (b , c) is a tautology. Proof (a) (i) Assume b logically implies c. Hence, every truth assignment that makes b true also makes c true. Thus, no truth assignment makes b true and c false. Therefore, no truth assignment makes (b ) c) false, that is, every truth assignment makes (b ) c) true. In other words, (b ) c) is a tautology. (ii) Assume (b ) c) is a tautology. Then, for every truth assignment, (b ) c) is true, and, therefore, it is not the case that b is true and c false. Hence, every truth assignment that makes b true makes c true, that is, b logically implies c. (b) (b , c) is a tautology if and only if every truth assignment makes (b , c) true, which is equivalent to saying that every truth assignment gives b and c the same truth value, that is, b and c are logically equivalent. By means of a truth table, we have an effective procedure for determining whether a statement form is a tautology. Hence, by Proposition 1.1, we have effective procedures for determining whether a given statement form logically implies another given statement form and whether two given statement forms are logically equivalent. To see whether a statement form is a tautology, there is another method that is often shorter than the construction of a truth table..
(32) 7. The Propositional Calculus. Examples 1. Determine whether ((A , ((:B) _ C)) ) ((:A) ) B)) is a tautology. Assume that the statement form sometimes is F (line 1). Then (A , ((:B) _ C)) is T and ((:A) ) B) is F (line 2). Since ((:A) ) B) is F, (:A) is T and B is F (line 3). Since (:A) is T, A is F (line 4). Since A is F and (A , ((:B) _ C)) is T, ((:B) _ C) is F (line 5). Since ((:B) _ C) is F, (:B) and C are F (line 6). Since (:B) is F, B is T (line 7). But B is both T and F (lines 7 and 3). Hence, it is impossible for the form to be false.. ((A , ((:B) _ C)) ) ((:A) ) B)) F 1 2 T F T F 3 F F 4 F 5 F F 6 T 7. 2. Determine whether ((A ) (B _ C)) _ (A ) B)) is a tautology. Assume that the form is F (line 1). Then (A ) (B _ C)) and (A ) B) are F (line 2). Since (A ) B) is F, A is T and B is F (line 3). Since (A ) (B _ C)) is F, A is T and (B _ C) is F (line 4). Since (B _ C) is F, B and C are F (line 5). Thus, when A is T, B is F, and C is F, the form is F. Therefore, it is not a tautology.. ((A ) (B _ C)) _ (A ) B)) F 1 F F 2 T F 3 T F 4 F F 5. Exercises 1.5 Determine whether the following are tautologies. (a) (((A ) B) ) B) ) B) (b) (((A ) B) ) B) ) A) (c) (((A ) B) ) A) ) A) (d) (((B ) C) ) (A ) B)) ) (A ) B)) (e) ((A _ (:(B ^ C))) ) ((A , C) _ B)) (f) (A ) (B ) (B ) A))) (g) ((A ^ B) ) (A _ C)) (h) ((A , B) , (A , (B , A))) (i) ((A ) B) _ (B ) A)) (j) ((:(A ) B)) ) A).
(33) 8. 1.6. 1.7. 1.8 1.9 1.10 1.11. 1.12 1.13 1.14. Introduction to Mathematical Logic. Determine whether the following pairs are logically equivalent. (a) ((A ) B) ) A) and A (b) (A , B) and ((A ) B) ^ (B ) A)) (c) ((:A) _ B) and ((:B) _ A) (d) (:(A , B)) and (A , (:B)) (e) (A _ (B , C)) and ((A _ B) , (A _ C)) (f) (A ) (B , C)) and ((A ) B) , (A ) C)) (g) (A ^ (B , C)) and ((A ^ B) , (A ^ C)) Prove: (a) (A ) B) is logically equivalent to ((:A) _ B). (b) (A ) B) is logically equivalent to (:(A ^ (:B))). Prove that b is logically equivalent to c if and only if b logically implies c and c logically implies b. Show that b and c are logically equivalent if and only if, in their truth tables, the columns under b and c are the same. Prove that b and c are logically equivalent if and only if (:b) and (:c) are logically equivalent. Which of the following statement forms are logically implied by (A ^ B)? (a) A (b) B (c) (A _ B) (d) ((:A) _ B) (e) ((:B) ) A) (f) (A , B) (g) (A ) B) (h) ((:B) ) (:A)) (i) (A ^ (:B)) Repeat Exercise 1.11 with (A ^ B) replaced by (A ) B) and by (:(A ) B)), respectively. Repeat Exercise 1.11 with (A ^ B) replaced by (A _ B). Repeat Exercise 1.11 with (A ^ B) replaced by (A , B) and by (:(A , B)), respectively.. A statement form that is false for all possible truth values of its statement letters is said to be contradictory. Its truth table has only Fs in the column under the statement form. One example is (A , (:A)): A T F. (:A) F T. (A , (:A)) F F. Another is (A ^ (:A)). Notice that a statement form b is a tautology if and only if (:b) is contradictory, and vice versa..
(34) The Propositional Calculus. 9. A sentence (in some natural language like English or in a formal theory)* that arises from a tautology by the substitution of sentences for all the statement letters, with occurrences of the same statement letter being replaced by the same sentence, is said to be logically true (according to the propositional calculus). Such a sentence may be said to be true by virtue of its truth-functional structure alone. An example is the English sentence, ‘‘If it is raining or it is snowing, and it is not snowing, then it is raining,’’ which arises by substitution from the tautology (((A _ B) ^ (:B)) ) A). A sentence that comes from a contradictory statement form by means of substitution is said to be logically false (according to the propositional calculus). Now let us prove a few general facts about tautologies.. PROPOSITION 1.2 If b and (b ) c) are tautologies, then so is c. Proof Assume that b and (b ) c) are tautologies. If c took the value F for some assignment of truth values to the statement letters of b and c, then, since b is a tautology, b would take the value T and, therefore, (b ) c) would have the value F for that assignment. This contradicts the assumption that (b ) c) is a tautology. Hence, c never takes the value F.. PROPOSITION 1.3 If t is a tautology containing as statement letters A1 , A2 , . . . , An , and b arises from t by substituting statement forms s1, s2, . . . , sn for A1 , A2 , . . . , An , respectively, then b is a tautology; that is, substitution in a tautology yields a tautology. Example Let t be ((A1 ^ A2 ) ) A1 ), let s1 be (B _ C) and let s2 be (C ^ D). Then b is (((B _ C) ^ (C ^ D)) ) (B _ C)). Proof Assume that t is a tautology. For any assignment of truth values to the statement letters in b, the forms s1, . . . , sn have truth values x1 , . . . , xn (where each xn is T or F). If we assign the values x1 , . . . , xn to A1 , . . . , An , * By a formal theory we mean an artificial language in which the notions of meaningful expressions, axioms, and rules of inference are precisely described (see page 25)..
(35) 10. Introduction to Mathematical Logic. respectively, then the resulting truth value of t is the truth value of b for the given assignment of truth values. Since t is a tautology, this truth value must be T. Thus, b always takes the value T.. PROPOSITION 1.4 If c1 arises from b1 by substitution of c for one or more occurrences of b, then ((b , c) ) (b1 , c1)) is a tautology. Hence, if b and c are logically equivalent, then so are b1 and c1. Example Let b1 be (c _ D), let b be c, and let c be (:(:c)). Then c1 is ((:(:c)) _ D). Since c and (:(:c)) are logically equivalent, (c _ D) and ((:(:c)) _ D) are also logically equivalent. Proof Consider any assignment of truth values to the statement letters. If b and c have opposite truth values under this assignment, then (b , c) takes the value F, and, hence, ((b , c) ) (b1 , c1)) is T. If b and c take the same truth values, then so do b1 and c1, since c1 differs from b1 only in containing c in some places where b1 contains b. Therefore, in this case, (b , c) is T, (b1 , c1) is T, and, thus, ((b , c) ) (b1 , c1)) is T. Parentheses It is profitable at this point to agree on some conventions to avoid the use of so many parentheses in writing formulas. This will make the reading of complicated expressions easier. First, we may omit the outer pair of parentheses of a statement form. (In the case of statement letters, there is no outer pair of parentheses.) Second, we arbitrarily establish the following decreasing order of strength of the connectives: :, ^, _, ), ,. Now we shall explain a step-by-step process for restoring parentheses to an expression obtained by eliminating some or all parentheses from a statement form. (The basic idea is that, where possible, we first apply parentheses to negations, then to conjunctions, then to disjunctions, then to conditionals, and finally to biconditionals.) Find the leftmost occurrence of the strongest connective that has not yet been processed. (i) If the connective is : and it precedes a statement form b, restore left and right parentheses to obtain (:b). (ii) If the connective is a binary connective C and it is preceded by a statement form b and followed by a statement form d, restore left and right parentheses to obtain (b C d)..
(36) The Propositional Calculus. 11. (iii) If neither (i) nor (ii) holds, ignore the connective temporarily and find the leftmost occurrence of the strongest of the remaining unprocessed connectives and repeat (i)–(iii) for that connective. Examples Parentheses are restored to the expression in the first line of each of the following in the steps shown: 1. A , (:B) _ C ) A A , ((:B) _ C) ) A A , (((:B) _ C) ) A) (A , (((:B) _ C) ) A)) 2. A ) :B ) C A ) (:B) ) C (A ) (:B)) ) C ((A ) (:B)) ) C) 3. B ) ::A B ) :(:A) B ) (:(:A)) (B ) (:(:A))) 4. A _ :(B ) A _ B) A _ :(B ) (A _ B)) A _ (:(B ) (A _ B))) (A _ (:(B ) (A _ B)))) Not every form can be represented without the use of parentheses. For example, parentheses cannot be further eliminated from A ) (B ) C), since A ) B ) C stands for ((A ) B) ) C). Likewise, the remaining parentheses cannot be removed from :(A _ B) or from A ^ (B ) C). Exercises 1.15 Eliminate as many parentheses as possible from the following forms. (a) ((B ) (:A)) ^ C) (b) (A _ (B _ C)) (c) (((A ^ (:B)) ^ C) _ D) (d) ((B _ (:C)) _ (A ^ B)) (e) ((A , B) , (:(C _ D))) (f) ((:(:(:(B _ C)))) , (B , C)) (g) (:((:(:(B _ C))) , (B , C))) (h) ((((A ) B) ) (C ) D)) ^ (:A)) _ C) 1.16 Restore parentheses to the following forms. (a) C _ :A ^ B (b) B ) :::A ^ C.
(37) 12. 1.17. 1.18. 1.19. 1.20. Introduction to Mathematical Logic. (c) C ) :(A ^ B ) C) ^ A , B (d) C ) A ) A , :A _ B Determine whether the following expressions are abbreviations of statement forms and, if so, restore all parentheses. (a) ::A , A , B _ C (b) :(:A , A) , B _ C (c) :(A ) B) _ C _ D ) B (d) A , (:A _ B) ) (A ^ (B _ C))) (e) :A _ B _ C ^ D , A ^ :A (f) ((A ) B ^ (C _ D) ^ (A _ D)) If we write :b instead of (:b), )bc instead of (b ) c), ^bc instead of (b ^ c), _bc instead of (b _ c), and ,bc instead of (b , c), then there is no need for parentheses. For example, ((:A) ^ (B ) (:D))), which is ordinarily abbreviated as :A ^ (B ) :D), becomes ^:A ) B:D. This way of writing forms is called Polish notation. (a) Write ((C ) (:A)) _ B) and (C _ ((B ^ (:D)) ) C)) in this notation. (b) If we count ), ^, _, and , each as þ1, each statement letter as 1 and : as 0, prove that an expression b in this parenthesis-free notation is a statement form if and only if (i) the sum of the symbols of b is 1 and (ii) the sum of the symbols in any proper initial segment of b is nonnegative. (If an expression b can be written in the form cd, where c 6¼ b, then c is called a proper initial segment of b.) (c) Write the statement forms of Exercise 1.15 in Polish notation. (d) Determine whether the following expressions are statement forms in Polish notation. If so, write the statement forms in the standard way. (i) : ) ABC _ AB:C (ii) )) AB )) BC ) :AC (iii) _ ^ _:A:BC ^ _AC _ :C:A (iv) _ ^ B ^ BBB Determine whether each of the following is a tautology, is contradictory, or neither. (a) B , (B _ B) (b) ((A ) B) ^ B) ) A (c) (:A) ) (A ^ B) (d) (A ) B) ) ((B ) C) ) (A ) C)) (e) (A , :B) ) A _ B (f) A ^ (:(A _ B)) (g) (A ) B) , ((:A) _ B) (h) (A ) B) , :(A ^ (:B)) (i) (B , (B , A)) ) A (j) A ^ :A ) B If A and B are true and C is false, what are the truth values of the following statement forms?.
(38) 13. The Propositional Calculus. 1.21. 1.22. 1.23. 1.24 1.25. 1.26. 1.27. (a) A _ C (b) A ^ C (c) :A ^ :C (d) A , :B _ C (e) B _ :C ) A (f) (B _ A) ) (B ) :C) (g) (B ) :A) , (A , C) (h) (B ) A) ) ((A ) :C) ) (:C ) B)) If A ) B is T, what can be deduced about the truth values of the following? (a) A _ C ) B _ C (b) A ^ C ) B ^ C (c) :A ^ B , A _ B What further truth values can be deduced from those shown? (a) :A _ (A ) B) F (b) :(A ^ B) , :A ) :B T (c) (:A _ B) ) (A ) :C) F (d) (A , B) , (C ) :A) F T If A , B is F, what can be deduced about the truth values of the following? (a) A ^ B (b) A _ B (c) A ) B (d) A ^ C , B ^ C Repeat Exercise 1.23, but assume that A , B is T. What further truth values can be deduced from those given? (a) (A ^ B) , (A _ B) F F (b) (A ) :B) ) (C ) B) F (a) Apply Proposition 1.3 when t is A1 ) A1 _ A2, s1 is B ^ D, and s2 is :B. (b) Apply Proposition 1.4 when b1 is (B ) C) ^ D, b is B ) C, and c is :B _ C. Show that each statement form in column I is logically equivalent to the form next to it in column II.. I (a) A ) (B ) C) (b) A ^ (B _ C) (c) A _ (B ^ C). II (A ^ B) ) C (A ^ B) _ (A ^ C) (A _ B) ^ (A _ C). (Distributive law) (Distributive law).
(39) 14. (d) (A ^ B) _ :B (e) (A _ B) ^ :B (f) A ) B (g) A , B (h) (A , B) , C (i) A , B (j) :(A , B) (k) :(A _ B) (l) :(A ^ B) (m) A _ (A ^ B) (n) A ^ (A _ B) (o) A ^ B (p) A _ B (q) (A ^ B) ^ C (r) (A _ B) _ C (s) A B (t) (A B) C (u) A ^ (B C). Introduction to Mathematical Logic. A _ :B A ^ :B :B ) :A B,A A , (B , C) (A ^ B) _ (:A ^ :B) A , :B (:A) ^ (:B) (:A) _ (:B) A A B^A B_A A ^ (B ^ C) A _ (B _ C) BA A (B C) (A ^ B) (A ^ C). (Law of the contrapositive) (Biconditional commutativity) (Biconditional associativity). (De Morgan’s law) (De Morgan’s law). (Commutativity of conjunction) (Commutativity of disjunction) (Associativity of conjunction) (Associativity of disjunction) (Commutativity of exclusive ‘‘or’’) (Associativity of exclusive ‘‘or’’) (Distributive law). 1.28 Show the logical equivalence of the following pairs. (a) t ^ b and b, where t is a tautology. (b) t _ b and t, where t is a tautology. (c) f ^ b and f, where f is contradictory. (d) f _ b and b, where f is contradictory. 1.29 (a) Show the logical equivalence of :(A ) B) and A ^ :B. (b) Show the logical equivalence of :(A , B) and (A ^ :B) _ (:A ^ B). (c) For each of the following statement forms, find a statement form that is logically equivalent to its negation and in which negation signs apply only to statement letters. (i) A ) (B , :C) (ii) :A _ (B ) C) (iii) A ^ (B _ :C) 1.30 (Duality) (a) If b is a statement form involving only :, ^, and _, and b0 results from b by replacing each ^ by _ and each _ by ^, show that b is a tautology if and only if :b0 is a tautology. Then prove that, if b ) c is a tautology, then so is c0 ) b0 , and if b , c is a tautology, then so is b0 , c0 . (Here c is also assumed to involve only :, ^, and _.) (b) Among the logical equivalences in Exercise 1.27, derive (c) from (b), (e) from (d), (l) from (k), (p) from (o), and (r) from (q). (c) If b is a statement form involving only :, ^, and _, and b* results from b by interchanging ^ and _ and replacing every statement letter by its negation, show that b* is logically equivalent to :b..
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