Intermediate Logic Richard Zach Philosophy 310 Winter Term 2015 McGill University Intermediate Logic by Richard Zach is licensed under a Creative Commons Attribution 4.0 International License. It is based on The Open Logic Text by the Open Logic Project, used under a Creative Commons Attribution 4.0 In- ternational License. Contents Prefacev I Sets, Relations, Functions1 1 Sets2 1.1 Extensionality ............................ 2 1.2 Subsets and Power Sets....................... 3 1.3 Some Important Sets......................... 5 1.4 Unions and Intersections...................... 5 1.5 Pairs, Tuples, Cartesian Products.................. 8 1.6 Russell’s Paradox .......................... 10 2 Relations 12 2.1 Relations as Sets ........................... 12 2.2 Special Properties of Relations................... 14 2.3 Equivalence Relations........................ 15 2.4 Orders................................. 16 2.5 Graphs................................. 18 2.6 Operations on Relations....................... 19 3 Functions 20 3.1 Basics ................................. 20 3.2 Kinds of Functions.......................... 22 3.3 Functions as Relations........................ 24 3.4 Inverses of Functions ........................ 25 3.5 Composition of Functions...................... 27 3.6 Partial Functions........................... 28 4 The Size of Sets 29 4.1 Introduction ............................. 29 4.2 Enumerations and Enumerable Sets................ 29 4.3 Cantor’s Zig-Zag Method...................... 33 i CONTENTS 4.4 Pairing Functions and Codes.................... 34 4.5 An Alternative Pairing Function.................. 35 4.6 Non-enumerable Sets ........................ 37 4.7 Reduction............................... 39 4.8 Equinumerosity ........................... 41 4.9 Sets of Different Sizes, and Cantor’s Theorem.......... 42 4.10 The Notion of Size, and Schroder-Bernstein¨ ........... 43 II First-Order Logic 44 5 Syntax and Semantics 45 5.1 Introduction ............................. 45 5.2 First-Order Languages........................ 46 5.3 Terms and Formulas......................... 48 5.4 Unique Readability ......................... 50 5.5 Main operator of a Formula..................... 52 5.6 Subformulas ............................. 53 5.7 Free Variables and Sentences.................... 54 5.8 Substitution.............................. 55 5.9 Structures for First-order Languages ............... 57 5.10 Covered Structures for First-order Languages.......... 58 5.11 Satisfaction of a Formula in a Structure.............. 59 5.12 Variable Assignments........................ 64 5.13 Extensionality ............................ 66 5.14 Semantic Notions .......................... 68 6 Theories and Their Models 70 6.1 Introduction ............................. 70 6.2 Expressing Properties of Structures ................ 72 6.3 Examples of First-Order Theories ................. 72 6.4 Expressing Relations in a Structure ................ 75 6.5 The Theory of Sets.......................... 76 6.6 Expressing the Size of Structures.................. 78 III Proofs and Completeness 80 7 The Sequent Calculus 81 7.1 Rules and Derivations........................ 81 7.2 Propositional Rules ......................... 82 7.3 Quantifier Rules ........................... 83 7.4 Structural Rules ........................... 83 7.5 Derivations.............................. 84 ii Contents 7.6 Examples of Derivations ...................... 86 7.7 Derivations with Quantifiers.................... 90 7.8 Proof-Theoretic Notions....................... 91 7.9 Derivability and Consistency.................... 93 7.10 Derivability and the Propositional Connectives......... 94 7.11 Derivability and the Quantifiers.................. 95 7.12 Soundness............................... 96 7.13 Derivations with Identity predicate . 101 7.14 Soundness with Identity predicate . 102 8 The Completeness Theorem 103 8.1 Introduction .............................103 8.2 Outline of the Proof .........................104 8.3 Complete Consistent Sets of Sentences . 106 8.4 Henkin Expansion..........................107 8.5 Lindenbaum’s Lemma........................109 8.6 Construction of a Model.......................110 8.7 Identity ................................112 8.8 The Completeness Theorem ....................114 8.9 The Compactness Theorem.....................115 8.10 A Direct Proof of the Compactness Theorem . 117 8.11 The Lowenheim-Skolem¨ Theorem . 118 8.12 Overspill ...............................119 IV Computability and Incompleteness 120 9 Recursive Functions 121 9.1 Introduction .............................121 9.2 Primitive Recursion .........................122 9.3 Composition .............................124 9.4 Primitive Recursion Functions...................125 9.5 Primitive Recursion Notations...................128 9.6 Primitive Recursive Functions are Computable . 128 9.7 Examples of Primitive Recursive Functions . 129 9.8 Primitive Recursive Relations ...................132 9.9 Bounded Minimization .......................134 9.10 Primes.................................135 9.11 Sequences...............................136 9.12 Trees..................................139 9.13 Other Recursions...........................140 9.14 Non-Primitive Recursive Functions . 141 9.15 Partial Recursive Functions.....................142 9.16 The Normal Form Theorem.....................144 iii CONTENTS 9.17 The Halting Problem.........................145 9.18 General Recursive Functions....................146 10 Arithmetization of Syntax 147 10.1 Introduction .............................147 10.2 Coding Symbols ...........................148 10.3 Coding Terms.............................150 10.4 Coding Formulas...........................151 10.5 Substitution..............................152 10.6 Derivations in LK . 153 11 Representability in Q 157 11.1 Introduction .............................157 11.2 Functions Representable in Q are Computable . 159 11.3 The Beta Function Lemma .....................160 11.4 Simulating Primitive Recursion ..................163 11.5 Basic Functions are Representable in Q . 164 11.6 Composition is Representable in Q . 167 11.7 Regular Minimization is Representable in Q . 168 11.8 Computable Functions are Representable in Q . 171 11.9 Representing Relations .......................172 11.10 Undecidability............................173 12 Incompleteness and Provability 174 12.1 Introduction .............................174 12.2 The Fixed-Point Lemma.......................175 12.3 The First Incompleteness Theorem . 177 12.4 Rosser’s Theorem ..........................179 12.5 Comparison with Godel’s¨ Original Paper . 181 12.6 The Derivability Conditions for PA . 181 12.7 The Second Incompleteness Theorem . 182 12.8 Lob’s¨ Theorem............................184 12.9 The Undefinability of Truth.....................187 Problems 189 Bibliography 200 iv Preface Formal logic has many applications both within philosophy and outside (es- pecially in mathematics, computer science, and linguistics). This second course will introduce you to the concepts, results, and methods of formal logic neces- sary to understand and appreciate these applications as well as the limitations of formal logic. It will be mathematical in that you will be required to master abstract formal concepts and to prove theorems about logic (not just in logic the way you did in Phil 210); but it does not presuppose any advanced knowledge of mathematics. We will begin by studying some basic formal concepts: sets, relations, and functions and sizes of infinite sets. We will then consider the language, seman- tics, and proof theory of first-order logic (FOL), and ways in which we can use first-order logic to formalize facts and reasoning abouts some domains ofin- terest to philosophers and logicians. In the second part of the course, we will begin to investigate the meta- theory of first-order logic. We will concentrate on a few central results: the completeness theorem, which relates the proof theory and semantics of first- order logic, and the compactness theorem and Lowenheim-Skolem¨ theorems, which concern the existence and size of first-order interpretations. In the third part of the course, we will discuss a particular way of mak- ing precise what it means for a function to be computable, namely, when it is recursive. This will enable us to prove important results in the metatheory of logic and of formal systems formulated in first-order logic: Godel’s¨ incom- pleteness theorem, the Church-Turing undecidability theorem, and Tarski’s theorem about the undefinability of truth. Week 1 (Jan 5, 7). Introduction. Sets and Relations. Week 2 (Jan 12, 14). Functions. Enumerability. Week 3 (Jan 19, 21). Syntax and Semantics of FOL. Week 4 (Jan 26, 28). Structures and Theories. Week 5 (Feb 2, 5). Sequent Calculus and Proofs in FOL. Week 6 (Feb 9, 12). The Completeness Theorem. v PREFACE Week 7 (Feb 16, 18). Compactness and Lowenheim-Skolem¨ Theorems Week 8 (Mar 23, 25). Recursive Functions Week 9 (Mar 9, 11). Arithmetization of Syntax Week 10 (Mar 16, 18). Theories and Computability Week 11 (Mar 23, 25). Godel’s¨ Incompleteness Theorems Week 12 (Mar 30, Apr 1). The Undefinability of Truth. Week 13, 14 (Apr 8, 13). Applications. vi Part I Sets, Relations, Functions 1 Chapter 1 Sets 1.1 Extensionality A set is a collection of objects, considered as a single object. The objects making up the set are called elements or members of the set. If x is an element of a set a, we write