LOGIC: Lecture Notes For Philosophy, Mathematics, And Computer Science

Description

This textbook is a logic manual which includes an elementary course and an advanced course. It covers more than most introductory logic textbooks, while maintaining a comfortable pace that students can follow. The technical exposition is clear, precise and follows a paced increase in complexity, allowing the reader to get comfortable with previous definitions and procedures before facing more difficult material. The book also presents an interesting overall balance between formal and philosophical discussion, making it suitable for both philosophy and more formal/science oriented students. This textbook is of great use to undergraduate philosophy students, graduate philosophy students, logic teachers, undergraduates and graduates in mathematics, computer science or related fields in which logic is required. Preface......Page 6 Contents......Page 8 1.1 What Is Logic?......Page 12 1.2 Arguments and Their Formulation......Page 14 1.3 Complex Reasoning......Page 15 1.4 Truth and Falsity......Page 17 1.5 Bivalence......Page 19 Exercises......Page 20 2.1 Some Set-Theoretical Notions......Page 22 2.2 True Premises......Page 23 2.3 Validity as Necessary Truth Preservation......Page 24 2.4 Other Logical Properties and Relations......Page 26 2.5 Important Facts About Validity......Page 27 2.6 Validity Is Not Everything......Page 31 Exercises......Page 33 3.1 Formal Validity......Page 35 3.2 Formal Invalidity......Page 38 3.3 Formal Language......Page 39 3.4 Formal System......Page 40 3.5 Object Language and Metalanguage......Page 41 3.6 Further Set-Theoretical Notions......Page 42 Exercises......Page 43 4.1 Sentence Letters......Page 45 4.2 Sentential Connectives......Page 46 4.3 Brackets......Page 48 4.4 Expressive Completeness......Page 49 4.6 Formalization in a Propositional Language......Page 51 Exercises......Page 52 5.1 Formation Rules......Page 54 5.2 Syntactic Trees......Page 55 5.3 Scope......Page 56 5.4 Interpretation......Page 57 5.5 Truth Tables......Page 58 Exercises......Page 59 6.1 Definition of Logical Consequence......Page 61 6.2 Other Logical Properties and Relations......Page 62 6.3 Important Facts About Logical Consequence......Page 63 6.4 Logical Consequence as a Test for Validity......Page 64 6.5 Effective Computability......Page 65 Exercises......Page 67 7.1 Derivation......Page 68 7.2 Rules for......Page 69 7.3 Rules for......Page 71 7.4 Rules for......Page 72 7.5 Rules for......Page 75 Exercises......Page 77 8.1 Derivability and Related Notions......Page 78 8.2 Important Facts About Derivability......Page 79 8.3 Some Tips......Page 80 8.4 Derived Rules......Page 82 8.5 Other Natural Deduction Systems......Page 83 Exercises......Page 84 9.1 Axioms and Inference Rule......Page 86 9.2 Deduction Theorem......Page 88 9.3 Explosion, Double Negation, Contraposition......Page 90 9.4 Substitution of Equivalents......Page 92 9.5 Reductio Ad Absurdum......Page 94 9.6 Deductive Equivalence Between G- and L......Page 95 9.7 Systems and Theories......Page 96 Exercises......Page 97 10.1 Consistency of L......Page 98 10.2 Definitions of Soundness and Completeness......Page 99 10.4 Completeness of L......Page 100 10.5 Extension to G-......Page 103 Exercises......Page 104 11.1 Quantified Sentences......Page 105 11.2 A Brief Historical Survey......Page 107 11.3 Existential Import......Page 109 11.4 Multiple Generality......Page 110 11.5 Definite Descriptions......Page 112 Exercises......Page 113 12.1 Non-logical Expressions......Page 115 12.2 Logical Constants and Auxiliary Symbols......Page 116 12.3 Other Symbols......Page 117 12.4 Numerical Expressions......Page 119 12.5 Multiple Generality and Scope Ambiguity......Page 120 12.6 Existence......Page 121 Exercises......Page 122 13.1 Syntax......Page 124 13.2 Basic Semantic Notions......Page 126 13.3 Satisfaction......Page 127 13.4 Truth......Page 128 13.5 Logical Consequence......Page 131 13.6 Undecidability......Page 132 Exercises......Page 134 14.1 Axioms and Inference Rule......Page 136 14.2 Derivability in Q......Page 137 14.4 Validity and Derivability......Page 138 14.5 Deduction Theorem and Other Syntactic Results......Page 139 14.6 Alphabetic Variants......Page 140 Exercises......Page 142 15.1 Consistency of Q......Page 144 15.2 Soundness of Q......Page 145 15.3 Completeness of Q......Page 146 15.4 Compactness Theorem......Page 148 15.5 Final Remarks......Page 149 Exercises......Page 150 16.1 Undecidability of Q......Page 151 16.2 Gödel Numbering......Page 152 16.4 A Further Corollary......Page 153 16.5 Recursive Axiomatization and Decidability......Page 154 Exercises......Page 155 17.1 First-Order Languages and Systems......Page 157 17.2 First-Order Logic with Identity......Page 158 17.3 First-Order Theory......Page 159 17.4 The Language of Basic Arithmetic......Page 160 17.5 Peano Arithmetic......Page 162 Exercises......Page 164 18.1 Cardinality......Page 165 18.2 Löwenheim-Skolem Theorems......Page 166 18.3 Isomorphism......Page 168 18.4 Isomorphic Models of a Theory......Page 170 18.5 Categoricity......Page 171 Exercises......Page 173 19.1 Overview......Page 175 19.2 The Arithmetization of Syntax......Page 176 19.3 The Gödel Sentence......Page 178 19.4 First Incompleteness Theorem: Semantic Version......Page 179 19.5 First Incompleteness Theorem: Syntactic Version......Page 180 19.6 Second Incompleteness Theorem......Page 182 Exercises......Page 183 20.1 Modal Operators......Page 184 20.2 A Modal Propositional Language......Page 185 20.3 The System K......Page 187 20.4 The Systems T,B,S4,S5......Page 190 20.5 A Modal Predicate Language......Page 194 20.6 Systems of Modal Predicate Logic......Page 196 20.7 Soundness and Completeness......Page 198 Exercises......Page 199 Chapter 1......Page 201 Chapter 3......Page 202 Chapter 5......Page 203 Chapter 6......Page 204 Chapter 7......Page 206 Chapter 8......Page 209 Chapter 9......Page 211 Chapter 11......Page 213 Chapter 13......Page 215 Chapter 14......Page 216 Chapter 15......Page 218 Chapter 17......Page 219 Chapter 18......Page 220 Chapter 20......Page 221 Bibliography......Page 223 Index......Page 226