Introducing Logic (Introducing)

Introducing Logic makes this important discipline accessible to everyone using the series' popular illustrated format. This easy-to-follow guide maps the historical development of logic, explains its symbols and methodology, traces its influence in a variety of disciplines, and shows logic in action in our digital age. What is Logic?......Page 5 Studying Sentences......Page 6 The Square of Oppositions......Page 7 The Syllogism......Page 8 Connective Logic......Page 10 Leibniz’s Law......Page 12 The Reductio ad Absurdum......Page 14 A “New Organon”......Page 16 Frege’s Quantifiers......Page 18 The Context Principle......Page 20 Propositional Calculus......Page 21 Cantor’s Set Theory......Page 22 The Usefulness of Connectives......Page 24 The Russell Paradox......Page 25 The Fatal Flaw......Page 26 The Problem of Surface Grammar......Page 27 Russell’s System......Page 28 Wittgenstein’s Logical Pictures......Page 30 Carnap and the Vienna Circle......Page 32 The Tolerance Principle......Page 33 Hilbert’s Proof Theory......Page 34 The Arrival of Gödel......Page 35 Gödel’s Incompleteness Theorem......Page 36 The Connections to Proof Theory......Page 37 Wittgenstein’s Table of Logical Connectives......Page 39 Wittgenstein’s Truth Tables......Page 40 Discovering Tautologies......Page 41 The Logic Gates of Digital Electronics......Page 42 A Vending Machine......Page 43 Turing and the “Enigma Code”......Page 44 Euclid’s Axiomatic Method......Page 45 Leibniz’s Proof Method......Page 47 Abuse of Contradiction......Page 48 Rules for Connectives......Page 49 Sensitivity to Grammar......Page 50 Predicate Calculus......Page 51 Model Theoretic Semantics......Page 52 Hilbert’s Recursion Model......Page 53 Finite Rules for Infinite Production......Page 56 Simple Instructions......Page 57 Proof Theory and Formal Language......Page 58 Tarski’s Truth Conditions......Page 60 Formal Semantics in Practice......Page 62 Constructing a Soap Opera......Page 63 Prolog to an AI Soap Opera......Page 64 Turing’s recipe for AI......Page 66 The Problem of Paradoxes......Page 68 Can Paradoxes be Avoided?......Page 70 Theory of Types......Page 71 Tarski’s Solution to the Liar......Page 73 The Unexorcised Paradox......Page 74 Gödel’s Incompleteness Theorem......Page 76 The Consequences of Gödel’s Theorem......Page 78 The “Halting Problem”......Page 80 The Limit of Gödel’s Proof......Page 81 Zeno’s Movement Paradox......Page 82 An Infinite Sum......Page 84 A Convergence on Limits......Page 85 How Much is a “Heap”?......Page 86 The Challenge to Sets......Page 87 Undermining Logic......Page 88 The Fiction of Vague Words......Page 89 What Do Words “Mean”?......Page 90 Fuzzy Logic......Page 91 Fuzzy Heaps......Page 92 Can Logic Escape Paradox?......Page 93 Non-Classical Logics: Intuitionism......Page 94 The Devil’s Argument......Page 95 Intuitionistic Logic......Page 96 Intuitionism versus the Reductio Method......Page 97 The Intuitionistic Fad......Page 98 Addressing Some Old Problems......Page 99 The Value of Possible......Page 100 Truth Values as Numbers......Page 101 The Possible and Non-Contradiction......Page 102 From Classical to Fuzzy Logic......Page 104 Electronic “Possible” States......Page 105 The Fuzzy Logic Search Engine......Page 106 The Fuzzy Logical Machine......Page 107 Logic in the Quantum World......Page 108 The Distributive Law of Quantum Logic......Page 109 How Quantum Logic Works......Page 110 Logic by Experiment......Page 111 Logic and Science......Page 113 The Copernican Revolution......Page 114 Galileo’s Revolution......Page 115 Methods of Deduction and Induction......Page 116 Problems with Induction......Page 118 Hume’s Fork......Page 119 Nomological Deduction......Page 120 Induction by Generalization......Page 122 Laws or Empirical Predictions......Page 126 The Raven Paradox......Page 128 A Problem of Cause and Effect......Page 130 Popper’s Answer to Hempel......Page 131 Popper’s Disconfirmation Theory......Page 132 The Probability of Viable Theory......Page 134 Quine’s “Web of Belief”......Page 136 Alterations to the “Web”......Page 138 Insufficient Evidence......Page 140 Quine’s Relativism......Page 142 Davidson’s Reply to Quine......Page 144 The Presentation of Truth......Page 145 Hard-edged Truth versus Relativism......Page 146 Cognitive Science and Logic......Page 147 Chomsky’s Universal Grammar......Page 148 Noun and Verb Categories......Page 150 Recursive Rules of Grammar......Page 153 The X-bar Theory......Page 154 A Logical Theory......Page 155 Problems of Syntax and Semantics......Page 156 Complex Grammatical Structures......Page 158 Problems with “Universal” Grammar......Page 160 The Symbolic Brain Model......Page 162 Training a Neural Net......Page 164 Pattern Recognition......Page 166 The Rational Behaviour Model......Page 168 Practical Reason......Page 169 What is Consciousness?......Page 170 The Place of Logic......Page 171 Wittgenstein’s Change of View......Page 172 Further Reading......Page 176 Index......Page 178 Introducing Logic - Online Publication What is Logic? 5 Studying Sentences 6 The Square of Oppositions 7 The Syllogism 8 Connective Logic 10 Leibniz’s Law 12 The Reductio ad Absurdum 14 A “New Organon” 16 Frege’s Quantifiers 18 The Context Principle 20 Propositional Calculus 21 Cantor’s Set Theory 22 The Usefulness of Connectives 24 The Russell Paradox 25 The Fatal Flaw 26 The Problem of Surface Grammar 27 Russell’s System 28 Wittgenstein’s Logical Pictures 30 Carnap and the Vienna Circle 32 The Tolerance Principle 33 Hilbert’s Proof Theory 34 The Arrival of Gödel 35 Gödel’s Incompleteness Theorem 36 The Connections to Proof Theory 37 Wittgenstein’s Table of Logical Connectives 39 Wittgenstein’s Truth Tables 40 Discovering Tautologies 41 The Logic Gates of Digital Electronics 42 A Vending Machine 43 Turing and the “Enigma Code” 44 Euclid’s Axiomatic Method 45 Leibniz’s Proof Method 47 Abuse of Contradiction 48 Rules for Connectives 49 Sensitivity to Grammar 50 Predicate Calculus 51 Model Theoretic Semantics 52 Hilbert’s Recursion Model 53 Finite Rules for Infinite Production 56 Simple Instructions 57 Proof Theory and Formal Language 58 Tarski’s Truth Conditions 60 Formal Semantics in Practice 62 Constructing a Soap Opera 63 Prolog to an AI Soap Opera 64 Turing’s recipe for AI 66 The Problem of Paradoxes 68 Can Paradoxes be Avoided? 70 Theory of Types 71 Tarski’s Solution to the Liar 73 The Unexorcised Paradox 74 Gödel’s Incompleteness Theorem 76 The Consequences of Gödel’s Theorem 78 The “Halting Problem” 80 The Limit of Gödel’s Proof 81 Zeno’s Movement Paradox 82 An Infinite Sum 84 A Convergence on Limits 85 How Much is a “Heap”? 86 The Challenge to Sets 87 Undermining Logic 88 The Fiction of Vague Words 89 What Do Words “Mean”? 90 Fuzzy Logic 91 Fuzzy Heaps 92 Can Logic Escape Paradox? 93 Non-Classical Logics: Intuitionism 94 The Devil’s Argument 95 Intuitionistic Logic 96 Intuitionism versus the Reductio Method 97 The Intuitionistic Fad 98 Addressing Some Old Problems 99 The Value of Possible 100 Truth Values as Numbers 101 The Possible and Non-Contradiction 102 From Classical to Fuzzy Logic 104 Electronic “Possible” States 105 The Fuzzy Logic Search Engine 106 The Fuzzy Logical Machine 107 Logic in the Quantum World 108 The Distributive Law of Quantum Logic 109 How Quantum Logic Works 110 Logic by Experiment 111 Logic and Science 113 The Copernican Revolution 114 Galileo’s Revolution 115 Methods of Deduction and Induction 116 Problems with Induction 118 Hume’s Fork 119 Nomological Deduction 120 Induction by Generalization 122 Laws or Empirical Predictions 126 The Raven Paradox 128 A Problem of Cause and Effect 130 Popper’s Answer to Hempel 131 Popper’s Disconfirmation Theory 132 The Probability of Viable Theory 134 Quine’s “Web of Belief” 136 Alterations to the “Web” 138 Insufficient Evidence 140 Quine’s Relativism 142 Davidson’s Reply to Quine 144 The Presentation of Truth 145 Hard-edged Truth versus Relativism 146 Cognitive Science and Logic 147 Chomsky’s Universal Grammar 148 Noun and Verb Categories 150 Recursive Rules of Grammar 153 The X-bar Theory 154 A Logical Theory 155 Problems of Syntax and Semantics 156 Complex Grammatical Structures 158 Problems with “Universal” Grammar 160 The Symbolic Brain Model 162 Training a Neural Net 164 Pattern Recognition 166 The Rational Behaviour Model 168 Practical Reason 169 What is Consciousness? 170 The Place of Logic 171 Wittgenstein’s Change of View 172 Further Reading 176 Index 178 Introducing,Logic Introducing Logic follows the historical development of this intriguing subject, explaining its symbols and methods, exploring complex philosophical issues and intricate mathematics.