A Short Introduction to Intuitionistic Logic (University Series in Mathematics) (University Series in Mathematics)

intuitionistic Logic Is Presented Here As Part Of Familiar Classical Logic Which Allows Mechanical Extraction Of Programs From Proofs. To Make The Material More Accessible, Basic Techniques Are Presented First For Propositional Logic; Part Ii Contains Extensions To Predicate Logic. This Material Provides An Introduction And A Safe Background For Reading Research Literature In Logic And Computer Science As Well As Advanced Monographs. Readers Are Assumed To Be Familiar With Basic Notions Of First Order Logic. One Device For Making This Book Short Was Inventing New Proofs Of Several Theorems. The Presentation Is Based On Natural Deduction. The Topics Include Programming Interpretation Of Intuitionistic Logic By Simply Typed Lambda-calculus (curry-howard Isomorphism), Negative Translation Of Classical Into Intuitionistic Logic, Normalization Of Natural Deductions, Applications To Category Theory, Kripke Models, Algebraic And Topological Semantics, Proof-search Methods, Interpolation Theorem. The Text Developed From Materal For Several Courses Taught At Stanford University In 1992-1999. Cover......Page 1 Half-Title......Page 3 Title Page......Page 5 Preface......Page 7 Contents......Page 9 Introduction......Page 13 Part I: Intuitionistic Propositional Logic......Page 17 1 Preliminaries......Page 19 2.1. Syntax......Page 21 2.3. Classical Propositional System NKp......Page 22 2.4. Abbreviated Notation for Natural Deductions......Page 23 2.5. Derivable Rules......Page 25 2.6. Direct Chaining and Analysis into Subgoals......Page 27 2.7. Heuristics for Natural Deduction......Page 28 2.8. Replacement of Equivalents......Page 30 2.9.1. Semantics: Truth Tables......Page 31 2.9.2. Logical Computations......Page 32 3 Negative Translation: Glivenko’s Theorem......Page 35 4.1. BHK-Interpretation......Page 37 4.2. Assignment of Deductive Terms......Page 38 4.2.1. Assignment Rules......Page 39 4.3. Properties of Term Assignment ......Page 41 5.1. Conversions and Reductions of Deductive Terms......Page 43 5.2. Conversions and Reductions of Natural Deductions......Page 44 5.3. Normalization......Page 49 5.4. Consequences of Normalization......Page 50 6.1. Structure of Normal Deduction......Page 53 6.3. Coherence Theorem......Page 54 7 Kripke Models......Page 59 7.1. Soundness of the System NJp......Page 62 7.2. Pointed Frames, Partial Orders......Page 63 7.3. Frame Conditions......Page 64 8 Gentzen-type Propositional System LJpm......Page 65 8.2. Completeness and Admissibility of Cut......Page 69 8.3. Translation into the Predicate Logic......Page 73 8.4. Algebraic Models......Page 74 8.5.1. Filtration......Page 77 8.5.2. Lindenbaum Algebra......Page 78 8.5.3. Finite Truth Tables......Page 79 9 Topological Completeness......Page 81 10.1. Tableaux: System LJpm*......Page 87 10.2. Proof-Search Procedure......Page 89 10.3. Complete Proof-Search Strategy......Page 91 11.2. A Disjunctive translation......Page 95 11.3. Pruning, Permutability of Rules......Page 96 12 Interpolation Theorem......Page 101 12.1. Beth Definability Theorem......Page 102 Part II: Intuitionistic Predicate Logic......Page 105 13 Natural Deduction System NJ......Page 107 13.1. Derivable Rules......Page 108 13.2. Gödel’s Negative Translation......Page 109 13.3. Program Interpretation of NJ......Page 111 14 Kripke Models for Predicate Logic......Page 117 14.1. Pointed Models, Frame Conditions......Page 119 15.0.1. Canonical Model, Admissibility of Cut......Page 121 15.1. Translation into the Classical Logic......Page 125 15.2. System LJ......Page 126 15.2.1. Translating LJpm into LJp......Page 127 15.3. Interpolation Theorem......Page 128 16 Proof-Search in Predicate Logic......Page 131 References......Page 137 Index......Page 141 "Intuitionistic logic is presented here as part of familiar classical logic which allows mechanical extraction of programs from proofs. To make the material more accessible, basic techniques are presented first for propositional logic; Part II contains extensions to predicate logic. This material provides an introduction and a safe background for reading research literature in logic and computer science as well as advanced monographs. Readers are assumed to be familiar with basic notions of first order logic.". "One device for making this book short was inventing new proofs of several theorems. The presentation is based on natural deduction. The topics include programming interpretation of intutionistic logic by simply typed lambda-calculus (Curry-Howard isomorphism), negative translation of classical into intuitionistic logic, normalization of natural deductions, applications to category theory, Kripke models, algebraic and topological semantics, proof-search methods, and interpolation theorem. The text developed from material for several courses taught at Stanford University in 1992-1999."--BOOK JACKET. Intuitionistic logic is presented here as part of familiar classical logic which allows mechanical extraction of programs from proofs to make the material more accessible. The presentation is based on natural deduction and readers are assumed to be familiar with basic notions of first order logic Intuitionistic logic is studied here as part of familiar classical logic which allows an effective interpretation and mechanical extraction of programs from proofs.