Logic, Mathematics, and Computer Science

Transcription

1 Logic, Mathematics, and Computer Science

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3 Yves Nievergelt Logic, Mathematics, and Computer Science Modern Foundations with Practical Applications Second Edition 123

4 Yves Nievergelt Department of Mathematics Eastern Washington University Cheney, WA, USA Cover art excerpted from triadic truth tables from Charles Sanders Peirce s Logic Notebook. ISBN ISBN (ebook) DOI / Library of Congress Control Number: Mathematics Subject Classification (2010): Primary 03-01; Secondary: 68-01, Springer New York Heidelberg Dordrecht London Springer Science+Business Media New York 2002, 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer Science+Business Media LLC New York is part of Springer Science+Business Media (www. springer.com)

5 Preface This second edition, entitled Logic, Mathematics, and Computer Science: Modern Foundations with Practical Applications, has been adapted from Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography, 2002 by Birkhäuser, from which Chapters 1 5 have been retained but extensively revised. Chapters 6 and 7 have been added. This text discusses the foundations where logic, mathematics, and computer science begin. The intended readership consists of undergraduate students majoring in mathematics or computer science who must learn such foundations either for their own interest or for further studies. For a motivated reader, there are no technical prerequisites: you need not know any technical subject to start reading this text. Although the text does not focus on the history and philosophy of the foundations, the material cites copious references to the literature, where the reader may find additional historical context. Consulting such references is neither suggested nor necessary to study the theory or to work on the exercises, but individual citations document the material by original sources, and all the citations together provide a guide to the variations and chronological developments of logic, mathematics, and computer science. For example, Chapter 1 traces the origin of Truth tables to Charles Sanders Peirce s unpublished 1909 Logic Notebook on philosophy and points out their applications over one half of a century later to the design of computers for use on Earth and on board the Apollo lunar spacecraft. Along informal arguments, this text also shows the corresponding purely symbolic manipulations of formulae, because they clarify the reasoning [11] and can reveal hitherto hidden logical properties, such as the mutual independence of different patterns of reasoning, or the impossibility of some proofs within some logics: As for algebra [of logic], the very idea of the art is that it presents formulae which can be manipulated, and that by observing the effects of such manipulation we find properties not to be otherwise discerned (Charles Sanders Peirce [104, p. 182]). If professionals are unable to learn some topics by any means other than the pure manipulation of symbols, then it would seem unfair to claim that all learning must be intuitive and hide from students such purely manipulative but successful methods. v

6 vi Preface The selection of topics also reflects major accomplishments from the twentieth century: the foundation of all of mathematics, and later computer science, as well as computer-assisted proofs of mathematical theorems, on a formal logic based on only a few axioms, transformation rules, and postulates for set theory [47, 50, 54, 105, 139]. Also, while not written in formal logic, Nobel-Prize winning applications to the social sciences rely on the same foundations, as shown in Chapter 7. To introduce the foundations of logic, the provability theorem in Chapter 1 provides an algorithm to design proofs in propositional logic. Chapter 1 also explains the concept of undecidability with multi-valued ( fuzzy ) logic and presents a proof of unprovability. Chapter 2 introduces logical quantifiers. A working knowledge of logical quantifiers is crucial for the study of basic concepts in modern mathematical analysis and topology, such as the uniform convergence of a sequence of equicontinuous functions. Continuing with the foundations of mathematics, Chapter 3 presents a version of the Zermelo Fraenkel set theory. At the juncture of mathematics and computer science, Chapter 4 develops the concepts of definition and proof by induction. Chapter 4 then uses induction with set theory to define the integers and rational numbers and derive the associative, commutative, and distributive laws, as well as algorithms, for their arithmetics. To give readers some idea of topics at an intermediate level, Chapter 5 shows that in a well-formed theory some paradoxes do not occur, while Chapter 6 completes the foundations of set theory with the axiom of choice. No extragalactic asteroid has yet been found with the universal laws of logic engraved in it. Consequently, not just one logic but many different logics have been invented. Different logics lead to different mathematics and different computer sciences. However, the acid test for adopting a particular logic is its ability to make predictions that are born by subsequent experiments. Formal logic is thus a mathematical model of rational thought processes. In this aspect, logic, mathematics, and computer science are experimental sciences. Only one logic has passed all such tests, which is the one used throughout this text. Other logics are outlined in Chapter 1 as a pedagogical device and to show some of their shortcomings. Acknowledgments I thank Dr. Stephen P. Keeler at the Boeing Company for having corrected many errors in the first edition. New and remaining errors are mine. I also thank Dr. Mary Keeler for guiding me to Charles Sanders Peirce s unpublished work on logic and philosophy. I commend the editors at Birkhäuser and Springer, in particular, Ann Kostant and Elizabeth Loew, for their hard work, patience, encouragements, and positive attitude through the development and production of both texts. Cheney, WA, USA 30 June 2015 Yves Nievergelt

7 Contents 1 Propositional Logic: Proofs from Axioms and Inference Rules Introduction An Example Demonstrating the Use of Logic in Real Life The Pure Propositional Calculus Formulae, Axioms, Inference Rules, and Proofs The Pure Positive Implicational Propositional Calculus Examples of Proofs in the Implicational Calculus Derived Rules: Implications Subject to Hypotheses A Guide for Proofs: an Implicational Deduction Theorem Example: Law of Assertion from the Deduction Theorem More Examples to Design Proofs of Implicational Theorems Another Guide for Proofs: Substitutivity of Equivalences More Derived Rules of Inference The Laws of Commutation and of Assertion Exercises on the Classical Implicational Calculus Equivalent Implicational Axiom Systems Exercises on Kleene s Axioms Exercises on Tarski s Axioms Proofs by the Converse Law of Contraposition Examples of Proofs in the Full Propositional Calculus Guides for Proofs in the Propositional Calculus Proofs by Reductio ad Absurdum Proofs by Cases Exercises on Frege s and Church s Axioms Other Connectives Definitions of Other Connectives Examples of Proofs of Theorems with Conjunctions Examples of Proofs of Theorems with Equivalences Examples of Proofs of Theorems with Disjunctions vii

8 viii Contents Examples of Proofs with Conjunctions and Disjunctions Exercises on Other Connectives Patterns of Deduction with Other Connectives Conjunctions of Implications Proofs by Cases or by Contradiction Exercises on Patterns of Deduction Equivalent Classical Axiom Systems Exercises on Kleene s, Rosser s, and Tarski s Axioms Completeness, Decidability, Independence, Provability, and Soundness Multi-Valued Fuzzy Logics Sound Multi-Valued Fuzzy Logics Independence and Unprovability Complete Multi-Valued Fuzzy Logics Peirce s Law as a Denial of the Antecedent Exercises on Church s and Łukasiewicz s Triadic Systems Boolean Logic The Truth Table of the Logical Implication Boolean Logic on Earth and in Space Automated Theorem Proving The Provability Theorem The Completeness Theorem Example: Peirce s Law from the Completeness Theorem Exercises on the Deduction Theorem First-Order Logic: Proofs with Quantifiers Introduction The Pure Predicate Calculus of First Order Logical Predicates Variables, Quantifiers, and Formulae Proper Substitutions of Free or Bound Variables Axioms and Rules for the Pure Predicate Calculus Exercises on Quantifiers Examples with Implicational and Predicate Calculi Examples with Pure Propositional and Predicate Calculi Other Axiomatic Systems for the Pure Predicate Calculus Exercises on Kleene s, Margaris s, and Rosser s Axioms Methods of Proof for the Pure Predicate Calculus Substituting Equivalent Formulae Discharging Hypotheses Prenex Normal Form Proofs with More than One Quantifier Exercises on the Substitutivity of Equivalence... 97

9 Contents ix 2.4 Predicate Calculus with Other Connectives Universal Quantifiers and Conjunctions or Disjunctions Existential Quantifiers and Conjunctions or Disjunctions Exercises on Quantifiers with Other Connectives Equality-Predicates First-Order Predicate Calculi with an Equality-Predicate Simple Applied Predicate Calculi with an Equality-Predicate Other Axiom Systems for the Equality-Predicate Defined Ranking-Predicates Exercises on Equality-Predicates Set Theory: Proofs by Detachment, Contraposition, and Contradiction Introduction Sets and Subsets Equality and Extensionality The Empty Set Subsets and Supersets Exercises on Sets and Subsets Pairing, Power, and Separation Pairing Power Sets Separation of Sets Exercises on Pairing, Power, and Separation of Sets Unions and Intersections of Sets Unions of Sets Intersections of Sets Unions and Intersections of Sets Exercises on Unions and Intersections of Sets Cartesian Products and Relations Cartesian Products of Sets Cartesian Products of Unions and Intersections Mathematical Relations and Directed Graphs Exercises on Cartesian Products of Sets Mathematical Functions Mathematical Functions Images and Inverse Images of Sets by Functions Exercises on Mathematical Functions Composite and Inverse Functions Compositions of Functions Injective, Surjective, Bijective, and Inverse Functions The Set of all Functions from a Set to a Set Exercises on Injective, Surjective, and Inverse Functions

10 x Contents 3.8 Equivalence Relations Reflexive, Symmetric, Transitive, or Anti-Symmetric Relations Partitions and Equivalence Relations Exercises on Equivalence Relations Ordering Relations Preorders and Partial Orders Total Orders and Well-Orderings Exercises on Ordering Relations Mathematical Induction: Definitions and Proofs by Induction Introduction Mathematical Induction The Axiom of Infinity The Principle of Mathematical Induction Definitions by Mathematical Induction Exercises on Mathematical Induction Arithmetic with Natural Numbers Addition with Natural Numbers Multiplication with Natural Numbers Exercises on Arithmetic by Induction Orders and Cancellations Orders on the Natural Numbers Laws of Arithmetic Cancellations Exercises on Orders and Cancellations Integers Negative Integers Arithmetic with Integers Order on the Integers Nonnegative Integral Powers of Integers Exercises on Integers with Induction Rational Numbers Definition of Rational Numbers Arithmetic with Rational Numbers Notation for Sums and Products Order on the Rational Numbers Exercises on Rational Numbers Finite Cardinality Equal Cardinalities Finite Sets Exercises on Finite Sets Infinite Cardinality Infinite Sets Denumerable Sets The Bernstein Cantor Schröder Theorem

11 Contents xi Denumerability of all Finite Sequences of Natural Numbers Other Infinite Sets Further Issues in Cardinality Exercises on Infinite Sets Well-Formed Sets: Proofs by Transfinite Induction with Already Well-Ordered Sets Introduction Transfinite Methods Transfinite Induction Transfinite Construction Exercises on Transfinite Methods Transfinite Sets and Ordinals Transitive Sets Ordinals Well-Ordered Sets of Ordinals Unions and Intersections of Sets of Ordinals Exercises on Ordinals Regularity of Well-Formed Sets Well-Formed Sets Regularity Exercises on Well-Formed Sets The Axiom of Choice: Proofs by Transfinite Induction Introduction The Choice Principle The Choice-Function Principle The Choice-Set Principle Exercises on Choice Principles Maximality and Well-Ordering Principles Zermelo s Well-Ordering Principle Zorn s Maximal-Element Principle Exercises on Maximality and Well-Orderings Unions, Intersections, and Products of Families of Sets The Multiplicative Principle The Distributive Principle Exercises on the Distributive and Multiplicative Principles Equivalence of the Choice, Zorn s, and Zermelo s Principles Towers of Sets Zorn s Maximality from the Choice Principle Exercises on Towers of Sets Yet Other Principles Related to the Axiom of Choice Yet Other Principles Equivalent to the Axiom of Choice

12 xii Contents Consequences of the Axiom of Choice Exercises on Related Principles Applications: Nobel-Prize Winning Applications of Sets, Functions, and Relations Introduction Game Theory Introduction Mathematical Models for The Prisoner s Dilemma Dominant Strategies Mixed Strategies Existence of Nash Equilibria for Two Players with Two Mixed Strategies Exercises on Mathematical Games Match Making Introduction A Mathematical Model for Optimal Match Making An Algorithm for Optimal Match Making with a Match Maker An Algorithm for Optimal Match Making Without a Match Maker Exercises on Gale & Shapley s Algorithms Projects Arrow s Impossibility Theorem Introduction A Mathematical Model for Arrow s Impossibility Theorem Statement and Proof of Arrow s Impossibility Theorem Exercises on Arrow s Impossibility Theorem Solutions to Some Odd-Numbered Exercises References Index