A Philosophy Student’s Introduction to Metalogic for Advanced Undergraduate and Beginning Graduate Students John N. Martin Department of Philosophy University of Cincinnati [email protected] © J. N. Martin, December 31, 2003 Revised March 9, 2012 An Introduction to Metalogic Table of Contents Introduction ............................................................................................................ i A. Topics ..................................................................................................... i B. The History of Logic ................................................................................ i C. Method ................................................................................................... ii D. Exercises............................................................................................... iii The Beginning of First Order Logic ....................................................................... 1 I. 19th Century Axiomatics and Logicism ....................................................... 1 A. A Priori Knowledge and Axiom Systems Prior to 1800 ........................... 1 B. Non-Euclidean Geometry ....................................................................... 6 C. Logistical (Axiom) Systems .................................................................... 8 D. Symbolic Logic ..................................................................................... 16 E. Application of Symbolic Logic to Arithmetic .......................................... 23 F. Application of Symbolic Logic to Set Theory ........................................ 25 G. Reduction of Arithmetic to Logic and Set Theory ................................. 30 H. Logicism, Russell’s Paradox, and Principia .......................................... 35 II. Gödel’s Incompleteness Proof ................................................................. 40 A. Kurt Gödel ............................................................................................ 40 B. Strategy Part 1. Arithmetical Calculations and Recursive Functions. .. 46 C. Strategy Part 2. Gödel Numbering: Arithmetization of Syntax ............. 52 D. Proof: Part 1. Tarski’s Theorem .......................................................... 57 E. Proof: Part 2. Expressibility of Theoremhood ...................................... 63 F. Proof of Incompleteness ...................................................................... 66 G. Supplementary Material: Gödel Original Proof Strategy ....................... 71 III. Exercises .............................................................................................. 77 A. Skills: Formal Proof in the System F. ................................................... 77 B. Skills: Informal Proofs in Naïve Set Theory .......................................... 77 i. Naïve Set Theory: The Notion of Implicit Definition. .......................... 77 ii. The Axioms and Definitions of Naive Set Theory .............................. 78 iii. Relations ........................................................................................... 80 C. Logistic Systems .................................................................................. 81 D. Gödel’s Proof: Technical Details .......................................................... 81 E. Gödel’s Proof: Theoretical Implications ................................................ 82 First-Order Logic Soundness and Completeness ............................................... 83 I. Standard Logical Theory .......................................................................... 83 A. Grammar .............................................................................................. 84 i. Parts of Speech ................................................................................ 84 ii. Syntactic Conventions and Abbreviations ......................................... 89 iii. Substitution ....................................................................................... 91 B. Semantics ............................................................................................ 95 i. Sentential Semantics ........................................................................ 95 ii. First-Order Semantics ....................................................................... 96 iii. Inductive Definition of the Interpretation of Terms. ............................ 98 iv. Inductive Definition of the Interpretation of Formulas. ..................... 100 v. Formal Statement of the Primary Semantic Definitions ................... 111 Page 2 An Introduction to Metalogic C. Proof Theory ...................................................................................... 115 i. Axiom Systems ............................................................................... 115 ii. Natural Deduction ........................................................................... 119 iii. Natural Deduction for Sentential Logic ............................................ 124 II. Completeness of First-Order Logic ........................................................ 129 A. Introduction ........................................................................................ 129 i. Strategy ........................................................................................... 129 ii. Background Metatheorems ............................................................. 130 B. Soundness ......................................................................................... 136 C. Completeness .................................................................................... 138 D. Further Metatheorems ........................................................................ 144 III. Exercises ............................................................................................ 146 A. Skills ................................................................................................... 146 i. Syntactic Construction Trees. ......................................................... 146 ii. Proof Theory. .................................................................................. 148 iii. Semantic Entailment and Validity .................................................... 148 iv. Inductive Proofs .............................................................................. 149 v. Metatheorems ................................................................................. 150 B. Theory and Ideas ............................................................................... 150 i. Existence ........................................................................................ 150 ii. Truths of Logic and Valid Arguments .............................................. 150 iii. Intuitionistic Proof Theory as Semantics: Meaning as Use ............. 151 iv. Henkin’s Proof of Soundness and Completeness ........................... 151 Effective Process and Undecidability ................................................................ 152 I. Calculation, Algorithms and Decidable Sets ........................................... 152 A. Introduction ........................................................................................ 152 B. The Concept of Calculation ................................................................ 153 C. Logic and Artificial Intelligence ........................................................... 163 D. Systems in the Language Prolog ....................................................... 168 i. Programs as Axioms ....................................................................... 169 ii. The Resolution Inference Rule ........................................................ 170 iii. Running a Prolog Program: Deduction within an Axiom System ..... 172 iv. Expert Systems in Prolog. ............................................................... 176 II. Decidability and Expert Systems ............................................................ 185 A. Conceptual Introduction ..................................................................... 185 B. Normal Forms and Skolimization. ...................................................... 186 C. Herbrand Models ................................................................................ 189 III. Exercises ............................................................................................ 197 A. Skills ................................................................................................... 197 B. Ideas .................................................................................................. 197 i. Effective Processes ........................................................................ 197 ii. Herbrand Models and Soklemizations............................................. 198 C. Theory ................................................................................................ 198 Many-Valued and Intensional Logic .................................................................. 199 I. Abstract Structures ................................................................................. 199 A. Structure............................................................................................. 199 Page 3 An Introduction to Metalogic B. Sentential Syntax ............................................................................... 200 C. Partial Orderings ................................................................................ 201 D. Standard Abstract Structures ............................................................