PHILOSOPHY, MATHEMATICS AND FORMAL LOGICS

Kinh Tế - Quản Lý - Báo cáo khoa học, luận văn tiến sĩ, luận văn thạc sĩ, nghiên cứu - Công nghệ thông tin Philosophy, Mathematics and Formal Logics Craig Harrison Department of Philosophy San Francisco State University and William M. Pezzaglia Jr. Department of Physics Santa Clara University May 15, 2004 ii Acknowledgements Draft prepared in LA TEX format. iii iv ACKNOWLEDGEMENTS About the Authors Craig Harrison born 1933, London, England B.A. History (1959), Ph.D. Philosophy (1967), Stanford University Associate Professor of Philosophy San Francisco State University 1600 Holloway Avenue San Francisco, CA 94132 William Marvyn Pezzaglia Jr. born 1953, Sacramento, California B.S. Physics (1975), Ph.D. (1983) Physics, University of California, Davis Adjunct Professor of Physics Santa Clara University 500 El Camino Real Santa Clara, CA 95053 v vi ABOUT THE AUTHORS Preface What we Hope to Accomplish in this Work An introductory course in formal or symbolic logic is a subject widely required for philosophy majors at both the graduate and undergraduate levels, and is often considered essential to the pursuit of philosophy. Why this should be so is not entirely clear, especially to the students who take the course. A typical text first teaches the student to “symbolize” certain artificial state- ments. Here is a random sample quoted from a few highly regarded works which happen to be on my bookshelf: Either Adam is blond and loves Eve, or he is not blond and Eve loves him. –Teller (1989)94. If Holmes has bungled or Watson’s on the job, then Moriarty will make a mistake. –Jeffrey (1991)38. Either Sam will come to the party and Max will not, or Sam will not come to the party and Max will enjoy himself. –Mendelson (1987)59. If Jones is ill or Smith is away, then neither will the Argus deal be con- cluded nor will the directors will declare a dividend unless Robinson comes to his senses and takes matters into his own hands. – Quine (1972)68. These are to be symbolized into formulas in two-valued “propositional” or “sentence” logic, depending on whether the author believes in propositions or not. Having done this, the student is to learn to solve problems, typically to determine whether a given argument containing assumptions and a conclusion consisting of statements of this sort, is valid or not, by determining whether the resulting formal argument is valid by means of truth tables, or a formal proof of the conclusion from the assumptions (or a combination of both). A similar though more general method is used to solve such problems in first order logic or “predicate logic”. Truth trees are also gradually increasing in popularity. Having been drilled by constant repetition in these methods, one of two outcomes in, in or experience, all too common: 1. The student satisfies the other requirements, but forgets or rarely applies the methods learned in the introductory course. vii viii PREFACE 2. The student then continues the study of logic and finds herself unprepared and disoriented by the material next presented, which is more or less a formal study of meta-logic or meta-mathematics. Now some students manage to survive the “just do it and don’t ask ques- tions” approach, which is arguably the most efficient, and is also not uncommon in the teaching of high school algebra. But students at the college level, partic- ularly students of philosophy, want to know and understand what the subject is about, what it has to do with philosophy, and what the motivations are for the ideas being introduced, and are uncomfortable in proceeding further until they have got some answers with which they are satisfied. We believe that they deserve answers, and in this book, we try to provide them. To this end, we discuss the basics in each chapter, and reserve more extended discussion of the motivations and philosophical significance of the ideas introduced, as well as more detailed and exact explanations of the nature of the concepts and the abstract objects which represent them, to optional sections. This gives the reader (or the instructor, as the case may be) some flexibility when it comes to deciding which additional material to study. We also try to be forthright and candid about the scope and applicability of the material introduced, and not to pretend by carefully selected examples, that it is wider than it is. Some Philosophical Presuppositions Concerning the Nature of Formal Logic Formal systems of logic, especially two-valued propositional or Boolean logics, have a close affinity to algebra, in particular to Boolean algebras. Experience has shown that Boolean logics, and their extension to first and higher order logics, and other related systems, are helpful when it comes to formalizing and clarifying mathematical concepts and proofs. We also try, especially in the concluding chapter, to explain how all these ideas came about, as well as their range of application, and their limitations. Besides that, we try to show how these ideas developed historically, and their connection with philosophy, which since Plato have been deep and numerous. Introduction Logic is a branch of mathematics, and the mathematical structures that arise from it are extensively studied, within the framework of such mathematical disciplines as set theory, topology, category theory and lattice theory. It is also a traditional part of the philosophy curriculum, especially at the graduate level, and it receives some attention in mathematical studies and in computer science as well. It was long the province of metaphysics and of general philosophy, until as a result of the work of George Boole, it became inseparable from mathematics. The study of mathematics has in turn been important in Western philosophy since its inception, at the time of Pythagoras and Plato, and has remained so to this day. Yet the way in which formal logic is introduced to the beginner is all to often devoid of context and entirely unmotivated. The emphasis is exclusively on the development of “skills”, such as symbolizing statements nobody in their right mind would ever make, or on endless drills in performing algorithms. And this does a disservice to the student and to the subject, and makes further progress all the more difficult. Philosophers do not take things on faith. When they are pursuing a subject, they want to understand what they are doing and why they are doing it. We believe that a basic knowledge of science and of mathematics in particular, is important to philosophy, and to the understanding of reality. You can’t philosophize about science without knowing what you’re talking about. We have tried to introduce the subject of logic with candor and without evasion, to explain the underlying motivations and the context which made them important, and hopefully, to give some idea of the manifold possibilities and the diversity of the subject, which like many other fields, has increased dramatically in the last century or so. In the first introductory chapter, we provide an account of the concepts on which formal logic is based. In the second chapter, we introduce the simple arithmetical operations which define what is meant in symbolic logic by ‘not’, ‘and’, and ‘or’. In the next chapter, we take a closer look at what this all means than is customary in regular algebra, particularly when it concerns the substitution of actual numbers for the variables in algebraic formulas. In Chapter 4, we discuss various methods of determining whether a sentence follows from given assumptions, and we also show that these methods really ix x INTRODUCTION work. In the next chapter, we explore various alternatives to the regular two- valued logic which was heretofore our main concern. In the next three chapters, we extend the discussion to cover generalities and statements of existence, which cannot be done in the logic of “not’s, and’s and or’s” alone. In the final chapter, we provide an account of the philosophical issues which lend importance to the most dramatic discoveries in the past century. We may add that we take formulas to denote the Boolean truth values 0 and 1, much as in ordinary algebra, formulas denote numbers. We also define formu- las involving quantifiers so as not to allow colliding quantifiers, an unnecessary and avoidable subtlety. For similar reasons, we permit only closed formulas in proofs or proof trees. Moreover, we take every element in the domain of a struc- ture to have a name, even if there are uncountably many of them, instead of taking free variables to have variable denotations. If this precludes a physicalist interpretation of proper names, so be it. We have added optional sections to supplement the core material, hopefully to provide further insight into it, or else alternative approaches. Chapters 5 and 8 are also largely optional. This will, we hope, provide the reader (or instructor) with more flexibility in what subjects to address. Contents Acknowledgements iii About the Authors v Preface vii Introduction ix I Propositional Logics 1 1 Introductory Concepts: Sets, Functions,. . . 3 1.1 Logic: An Overview . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1 What is Logic For? . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Naming Sets . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.2 The Principle of Extensionality: When are Two Sets Equal? 6 1.2.3 Unit Sets, the Empty Set and Finite and Infinite Sets . . 7 1.2.4 Subsets, and Extensionality Again . . . . . . . . . . . . . 7 1.2.5 Basic Operations on Sets . . . . . . . . . . . . . . . . . . 8 1.2.6 Ordered Pairs, Triples, Quadruples, etc. . . . . . . . . . . 12 1.2.7 What Is an Ordered Pair . . . . . . . . . . . . . . . . . . 13 1.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3.1 On Functions . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3.2 Functions with Several Inputs . . . . . . . . . . . . . . . . 20 1.4 Truth Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.4.1 Truth Valued Functions and Truth Functions . . . . . . . 21 2 Binary Boolean Arithmetic 25 2.1 Reasoning by Calculation from Leibniz to Boole . . . . . . . . . . 25 2.1.1 Boolean Algebra and Modern Logic: Introductory Remarks 25 2.2 Boole’s Arithmetic. . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2.1 Symbolizing Boolean Statements and Calculating their Truth Values . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2.2 Calculating Truth Values: Some Examples . . . . . . . . . 33 xi xii CONTENTS 2.3 Material Implication, . . . Exclusive ‘Or’ . . . . . . . . . . . . . . . 36 2.3.1 Truth Functional or Boolean Implication . . . . . . . . . . 36 2.3.2 Material Equivalence and Exclusive Disjunction . . . . . . 39 2.4 About The Truth Functional Conditional . . . . . . . . . . . . . 40 2.4.1 Why the conditional of Boolean . . . . . . . . . . . . . . . 40 2.4.2 The Material Conditional in Ordinary Reasoning. . . . . . 42 2.5 Boolean Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.5.1 What is a Boolean Language? . . . . . . . . . . . . . . . . 45 2.5.2 Boolean Formation Rules . . . . . . . . . . . . . . . . . . 47 2.5.3 Trees and their Uses . . . . . . . . . ...

Philosophy, Mathematics and Formal Logics Craig Harrison Department of Philosophy San Francisco State University and William M Pezzaglia Jr Department of Physics Santa Clara University May 15, 2004 ii Acknowledgements Draft prepared in LATEX format iii iv ACKNOWLEDGEMENTS About the Authors Craig Harrison born 1933, London, England B.A History (1959), Ph.D Philosophy (1967), Stanford University Associate Professor of Philosophy San Francisco State University 1600 Holloway Avenue San Francisco, CA 94132 William Marvyn Pezzaglia Jr born 1953, Sacramento, California B.S Physics (1975), Ph.D (1983) Physics, University of California, Davis Adjunct Professor of Physics Santa Clara University 500 El Camino Real Santa Clara, CA 95053 v vi ABOUT THE AUTHORS Preface What we Hope to Accomplish in this Work An introductory course in formal or symbolic logic is a subject widely required for philosophy majors at both the graduate and undergraduate levels, and is often considered essential to the pursuit of philosophy Why this should be so is not entirely clear, especially to the students who take the course A typical text first teaches the student to “symbolize” certain artificial state- ments Here is a random sample quoted from a few highly regarded works which happen to be on my bookshelf: • Either Adam is blond and loves Eve, or he is not blond and Eve loves him –Teller (1989)[94] • If Holmes has bungled or Watson’s on the job, then Moriarty will make a mistake –Jeffrey (1991)[38] • Either Sam will come to the party and Max will not, or Sam will not come to the party and Max will enjoy himself –Mendelson (1987)[59] • If Jones is ill or Smith is away, then neither will the Argus deal be con- cluded nor will the directors will declare a dividend unless Robinson comes to his senses and takes matters into his own hands – Quine (1972)[68] These are to be symbolized into formulas in two-valued “propositional” or “sentence” logic, depending on whether the author believes in propositions or not Having done this, the student is to learn to solve problems, typically to determine whether a given argument containing assumptions and a conclusion consisting of statements of this sort, is valid or not, by determining whether the resulting formal argument is valid by means of truth tables, or a formal proof of the conclusion from the assumptions (or a combination of both) A similar though more general method is used to solve such problems in first order logic or “predicate logic” Truth trees are also gradually increasing in popularity Having been drilled by constant repetition in these methods, one of two outcomes in, in or experience, all too common: 1 The student satisfies the other requirements, but forgets or rarely applies the methods learned in the introductory course vii viii PREFACE 2 The student then continues the study of logic and finds herself unprepared and disoriented by the material next presented, which is more or less a formal study of meta-logic or meta-mathematics Now some students manage to survive the “just do it and don’t ask ques- tions” approach, which is arguably the most efficient, and is also not uncommon in the teaching of high school algebra But students at the college level, partic- ularly students of philosophy, want to know and understand what the subject is about, what it has to do with philosophy, and what the motivations are for the ideas being introduced, and are uncomfortable in proceeding further until they have got some answers with which they are satisfied We believe that they deserve answers, and in this book, we try to provide them To this end, we discuss the basics in each chapter, and reserve more extended discussion of the motivations and philosophical significance of the ideas introduced, as well as more detailed and exact explanations of the nature of the concepts and the abstract objects which represent them, to optional sections This gives the reader (or the instructor, as the case may be) some flexibility when it comes to deciding which additional material to study We also try to be forthright and candid about the scope and applicability of the material introduced, and not to pretend by carefully selected examples, that it is wider than it is Some Philosophical Presuppositions Concerning the Nature of Formal Logic Formal systems of logic, especially two-valued propositional or Boolean logics, have a close affinity to algebra, in particular to Boolean algebras Experience has shown that Boolean logics, and their extension to first and higher order logics, and other related systems, are helpful when it comes to formalizing and clarifying mathematical concepts and proofs We also try, especially in the concluding chapter, to explain how all these ideas came about, as well as their range of application, and their limitations Besides that, we try to show how these ideas developed historically, and their connection with philosophy, which since Plato have been deep and numerous Introduction Logic is a branch of mathematics, and the mathematical structures that arise from it are extensively studied, within the framework of such mathematical disciplines as set theory, topology, category theory and lattice theory It is also a traditional part of the philosophy curriculum, especially at the graduate level, and it receives some attention in mathematical studies and in computer science as well It was long the province of metaphysics and of general philosophy, until as a result of the work of George Boole, it became inseparable from mathematics The study of mathematics has in turn been important in Western philosophy since its inception, at the time of Pythagoras and Plato, and has remained so to this day Yet the way in which formal logic is introduced to the beginner is all to often devoid of context and entirely unmotivated The emphasis is exclusively on the development of “skills”, such as symbolizing statements nobody in their right mind would ever make, or on endless drills in performing algorithms And this does a disservice to the student and to the subject, and makes further progress all the more difficult Philosophers do not take things on faith When they are pursuing a subject, they want to understand what they are doing and why they are doing it We believe that a basic knowledge of science and of mathematics in particular, is important to philosophy, and to the understanding of reality You can’t philosophize about science without knowing what you’re talking about We have tried to introduce the subject of logic with candor and without evasion, to explain the underlying motivations and the context which made them important, and hopefully, to give some idea of the manifold possibilities and the diversity of the subject, which like many other fields, has increased dramatically in the last century or so In the first introductory chapter, we provide an account of the concepts on which formal logic is based In the second chapter, we introduce the simple arithmetical operations which define what is meant in symbolic logic by ‘not’, ‘and’, and ‘or’ In the next chapter, we take a closer look at what this all means than is customary in regular algebra, particularly when it concerns the substitution of actual numbers for the variables in algebraic formulas In Chapter 4, we discuss various methods of determining whether a sentence follows from given assumptions, and we also show that these methods really ix x INTRODUCTION work In the next chapter, we explore various alternatives to the regular two- valued logic which was heretofore our main concern In the next three chapters, we extend the discussion to cover generalities and statements of existence, which cannot be done in the logic of “not’s, and’s and or’s” alone In the final chapter, we provide an account of the philosophical issues which lend importance to the most dramatic discoveries in the past century We may add that we take formulas to denote the Boolean truth values 0 and 1, much as in ordinary algebra, formulas denote numbers We also define formu- las involving quantifiers so as not to allow colliding quantifiers, an unnecessary and avoidable subtlety For similar reasons, we permit only closed formulas in proofs or proof trees Moreover, we take every element in the domain of a struc- ture to have a name, even if there are uncountably many of them, instead of taking free variables to have variable denotations If this precludes a physicalist interpretation of proper names, so be it We have added optional sections to supplement the core material, hopefully to provide further insight into it, or else alternative approaches Chapters 5 and 8 are also largely optional This will, we hope, provide the reader (or instructor) with more flexibility in what subjects to address Contents Acknowledgements iii About the Authors v Preface vii Introduction ix I Propositional Logics 1 1 Introductory Concepts: Sets, Functions, 3 1.1 Logic: An Overview 3 1.1.1 What is Logic For? 3 1.2 Sets 4 1.2.1 Naming Sets 5 1.2.2 The Principle of Extensionality: When are Two Sets Equal? 6 1.2.3 Unit Sets, the Empty Set and Finite and Infinite Sets 7 1.2.4 Subsets, and Extensionality Again 7 1.2.5 Basic Operations on Sets 8 1.2.6 Ordered Pairs, Triples, Quadruples, etc 12 1.2.7 What Is an Ordered Pair 13 1.3 Functions 14 1.3.1 On Functions 14 1.3.2 Functions with Several Inputs 20 1.4 Truth Functions 21 1.4.1 Truth Valued Functions and Truth Functions 21 2 Binary Boolean Arithmetic 25 2.1 Reasoning by Calculation from Leibniz to Boole 25 2.1.1 Boolean Algebra and Modern Logic: Introductory Remarks 25 2.2 Boole’s Arithmetic 29 2.2.1 Symbolizing Boolean Statements and Calculating their Truth Values 31 2.2.2 Calculating Truth Values: Some Examples 33 xi xii CONTENTS 2.3 Material Implication, Exclusive ‘Or’ 36 2.3.1 Truth Functional or Boolean Implication 36 2.3.2 Material Equivalence and Exclusive Disjunction 39 2.4 About The Truth Functional Conditional 40 2.4.1 Why the conditional of Boolean 40 2.4.2 The Material Conditional in Ordinary Reasoning 42 2.5 Boolean Syntax 44 2.5.1 What is a Boolean Language? 45 2.5.2 Boolean Formation Rules 47 2.5.3 Trees and their Uses 48 2.5.4 The Unique Readability Theorem 52 2.6 Game-Theoretical Semantics 56 2.6.1 Boolean Games 56 2.6.2 How To Win a Boolean Game 59 2.7 How to Believe that you will Believe a Falsehood 62 2.7.1 On the Island of Knights and Knaves 62 3 Truth Functional Logic: Determining Validity and Satisfiability by Calculation 37 3.1 Boolean Paralogisms as “Errors in Calculation” 37 3.1.1 Boolean Word Problems 37 3.2 TRUTH FUNCTIONAL ENTAILMENT 39 3.2.1 Boolean Formulas and Truth Functional Entailment Be- tween Them 39 3.3 PROPOSITIONAL LANGUAGES 41 3.3.1 What is a Boolean Language? 41 3.4 BOOLEAN SEMANTICS: TRUTH AND SATISFIABILITY 42 3.4.1 Truth Value Assignments and Boolean Valuations 42 3.4.2 Satisfying Formulas and Sets of Formulas 43 3.5 The Basic Problems of Boolean Logic 43 3.5.1 The Basic Problems of Boolean Logic, and How to Solve Them by Calculating with Vectors 43 3.5.2 Solving the Basic Problems by Truth Value Analysis 43 3.6 The Syntax of Boolean Formulas II 43 3.6.1 Subformula Trees and Valuation Trees 43 3.6.2 An Algorithm for Finding the Main Connective and the Immediate Subformulas of a Given Formula 43 3.7 Game-Theoretical Semantics for Boolean Logic: An Alternative to Direct Calculation 43 3.7.1 Boolean Games 43 3.8 About the Truth Functional or Material Conditional 43 3.8.1 Why the Conditional of Boolean Reasoning is the Material Conditional 43 3.8.2 The Material Conditional in Ordinary Reasoning: Some Pitfalls 43 3.9 Spatializing Logical Relations: The Geometry of Entailment 43 CONTENTS xiii 3.9.1 Formal Propositions a Points in Space 43 4 Boolean Reasoning 45 4.1 Methods of Boolean Logic: Boolean Equations 46 4.1.1 Boolean Equations 46 4.1.2 Finding Solutions to Sets of Boolean Equations, or Show- ing that None Exist 46 4.1.3 Solving the Basic Problems of Boolean Logic with Equa- tion Trees 46 4.2 Methods of Boolean Logic: Formal Proofs 46 4.2.1 Boolean Natural Deduction 46 4.2.2 Summarizing the Rules 46 4.3 Methods of Boolean Logic: Truth Trees 46 4.3.1 Truth Trees 46 4.3.2 Solving the Satisfiability, Validity and Tautology Prob- lems using Truth Trees 46 4.4 Reasoning about Boolean Reasoning 46 4.4.1 Basis Vectors 46 4.4.2 Vectors and Duality 46 4.4.3 The Proof of the Pudding: the soundness and Complete- ness Theorems for the Natural Deduction System 46 5 Non-Classical Logics 47 5.1 Some Alternative Propositional Logics 47 5.1.1 Philosophy and Logic 47 5.1.2 On Modal Logics 48 5.1.3 Intuitionist Logic 50 5.1.4 Many-Values Logics and Fuzzy Logics 51 5.1.5 Quantum Logic 53 5.1.6 Philosophy and Logic Again: The Struggle Between Op- posites 55 5.1.7 Embracing Contradictions: Formal Systems which are Non- Trivial but Inconsistent 57 II Quantifier Logics 63 6 Zero Order Reasoning and 65 6.1 Zero Order Logic 65 6.1.1 Zero Order vs Boolean Languages 66 6.1.2 Doing Arithmetic in a Zero Order Language 69 6.1.3 The Completeness of the Axiom System 72 6.1.4 Structures and Valuations of Zero Order Languages 72 6.2 Interpretations and Validity in 74 6.2.1 Transition from Monadic Zero Order Languages to Uni- form Monadic First Order Languages 75 xiv CONTENTS 6.2.2 Categorical Syllogisms in Uniform First Order Languages 77 7 First Order Reasoning 81 7.1 First Order Semantics 81 7.1.1 First Order Languages and Multiple Quantification: Sen- tences, Interpretations and Truth 82 7.1.2 Elementary and Higher Order Structures 84 7.1.3 Models of Sets of Sentences: Some Examples 86 7.1.4 First Order Games 89 7.2 Methods of First Order Logic I: Truth Trees 90 7.2.1 First Order Truth Trees 91 7.3 Methods of First Order Logic II 97 7.3.1 Natural Deduction in First Order Logic: Formal Proofs 98 7.3.2 The Basic Problems of First Order Logic 103 7.4 First Order Theories 104 7.4.1 First Order Theories 106 7.4.2 Introducing Functions and Equality: First Order Theories with Equality 108 7.4.3 Some First Order Theories with Functions and Equality 111 7.5 Reasoning about First Order Reasoning 118 7.5.1 The First Order Soundness Theorem 118 7.5.2 The Completeness Theorem for First Order Natural De- duction 124 7.5.3 Deriving the First Order Completeness Theorem 127 8 Questions of Consistency, 135 8.1 First Crisis: Arithmetizing Geometry 135 8.1.1 Crises in Mathematical Foundations I: Facing the Irrational136 8.2 Taming the Infinite: The analysis of Change 141 8.2.1 Concepts of the Calculus from Newton and Leibniz Through the Seventeenth Century: the Analysis of Change and its Cumulative Effects in Space and Time 141 8.3 Second Crisis: Paradoxical Series 151 8.3.1 The Problem of the Convergence of Power Series: Cauchy’s Criterion 151 8.3.2 The Problem of the convergence of Fourier Series: Can- tor’s Introduction of Sets and Completed Infinities 155 8.3.3 Reducing the Concept of Real Numbers to the Concept of Natural Numbers, with a Little Help from Sets 156 8.3.4 Defining Natural Numbers as Sets 161 8.4 Third Crisis, Set Theory Paradoxes 161 8.4.1 Trouble in Cantor’s Paradise: Hilbert to the Rescue 161 8.4.2 Which Sets are Admissible? Logicism and the Formaliza- tion of Mathematics within Type Theory 161 8.4.3 The Consistency of Mathematics: Hilbert’s Program 161 8.4.4 Set-Theoretical Foundations: First Order Set Theories 161 CONTENTS xv 8.5 Classical Mathematics and Loss of Certainty 161 8.5.1 Prelude to G¨odel I: Exact Definitions of Decidability 161 8.5.2 Prelude to G¨odel II: G¨odel Numbering: the Code for Talk- ing about Arithmetic in the Language of Arithmetic 161 8.5.3 Prelude to G¨odel III: Putting all the Ducks in a Row and How to Make Sentences Talk about Themselves 161 8.5.4 The Heart of the Matter: the Basic Arguments 161 8.5.5 Formalism and the “Loss of Certainty”: Implications for Analysis 161 8.5.6 The Demise of Metamathematical Certainty: Some Con- sequences 161 Appendix 1: Operator Precedence 63 xvi CONTENTS List of Figures 1.1 Euler Diagram 9 1.2 Intersecting Circles 9 1.3 Euler Diagram with 3 Regions 10 1.4 Tree Diagram 10 1.5 Internal Diagram for Five Functions 15 1.6 Maps 16 1.7 Natural Number Equinumerous with Even Natural Numbers 17 1.8 Composition 19 2.1 The Two Element Boolean Lattice B2 33 2.2 Subformula Tree for U 54 2.3 Valuation Tree for U 55 2.4 Game Tree for 1 ∨ (1 ∧ 0) (Example 1) 57 2.5 Game Tree for (1 ∨ 1) ∧ 0 (Example 2) 58 2.6 Example 3 Valuation and Game Trees 61 2.7 Example 4 Valuation and Game Trees 61 5.1 Deducing Disjunctive Syllogism from the Spread Rule 58 5.2 Proof of What using Minimal Logic 59 6.1 Formalization of Argument 71 7.1 Directed Graph of Strict Partial Order on A = {1, 2, 3, 6} 88 7.2 Game Tree for ‘ ∀x∃y xF y ’ 90 7.3 Valuation Tree for ‘ ∀x∃y xF y ’ 90 7.4 Game Tree for ‘ ∃y∀x xGy ’ 90 7.5 Valuation Tree for ‘ ∃y∀x xGy ’ 91 7.6 Closed tree constructed from sentence 94 8.1 Plot of Falling Speed versus Time 143 8.2 Plot of Falling Distance versus Time 144 8.3 Something About Limits 147 xvii xviii LIST OF FIGURES List of Tables 2.1 Logical Multiplication for B2 30 2.2 Logical Addition for B2 30 2.3 Logical Complementation for B2 30 2.4 Truth Functions 32 2.5 Matrix for Material Conditional 36 6.1 Categorical Statements 77 xix