LOGIC. Mathematics. Computer Science. Stanley N. Burris

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1 LOGIC for Mathematics and Computer Science Stanley N. Burris Department of Pure Mathematics University of Waterloo Prentice Hall Upper Saddle River, New Jersey 07458

Contents Preface The Flow of Topics xi xvii Part I Quantifier-Free Logics 1 Chapter 1 From Aristotle to Boole 3 1.1 Sophistry 3 1.2 The Contributions of Aristotle 5 1.3 The Algebra of Logic 10 1.

2 Contents Preface The Flow of Topics xi xvii Part I Quantifier-Free Logics 1 Chapter 1 From Aristotle to Boole Sophistry The Contributions of Aristotle The Algebra of Logic The Method of Boole, and Venn Diagrams Checking for Validity Finding the Most General Conclusion Historical Remarks 29 Chapter 2 Propositional Logic Propositional Connectives, Propositional Formulas, and Truth Tables Defining Propositional Formulas Truth Tables Equivalent Formulas, Tautologies, and Contradictions Equivalent Formulas Tautologies Contradictions Substitution Replacement Induction Proofs on Formulas The Main Result on Replacement Simplification of Formulas Adequate Connectives 52 V

vi Contents 2.5.1 The Adequacy of Standard Connectives 52 2.5.2 Proving Adequacy 53 2.5.3 Proving Inadequacy 55 2.6 Disjunctive and Conjunctive Forms 59 2.6.1 Rewrite Rules to Obtain Normal Forms 60 2.

3 vi Contents The Adequacy of Standard Connectives Proving Adequacy Proving Inadequacy Disjunctive and Conjunctive Forms Rewrite Rules to Obtain Normal Forms Using Truth Tables to Find Normal Forms Uniqueness of Normal Forms Valid Arguments, Tautologies, and Satisfiability Compactness The Compactness Theorem for Propositional Logic Applications of Compactness The Propositional Proof System PC Simple Equivalences The Proof System Soundness and Completeness Derivations with Premisses Proving Theorems about h Generalized Soundness and Completeness Resolution A Motivation Clauses Resolution The Davis-Putnam Procedure Soundness and Completeness for the DPP Applications of the DPP Soundness and Completeness for Resolution Generalized Soundness and Completeness for Resolution Horn Clauses Graph Clauses Pigeonhole Clauses Historical Remarks The Beginnings Statement Logic and the Algebra of Logic Frege's Work Ignored Bertrand Russell Rescues Frege's Logic The Influence of Principia The Emergence of Truth Tables, Completeness The Hilbert School of Logic The Polish School of Logic 127

Contents VII 2.14.9 Other Propositional Proof Systems 2.14.10 Problems with Algorithms 2.14.11 Reduction to Propositional Logic 2.14.12 Testing for Satisfiability Chapter 3 Equational Logic 3.1 3.2 3.

4 Contents VII Other Propositional Proof Systems Problems with Algorithms Reduction to Propositional Logic Testing for Satisfiability Chapter 3 Equational Logic Interpretations and Algebras Terms Term Functions Evaluation Tables 3.4 Equations The Semantics of Equations Classes of Algebras Defined by Equations Three Very Basic Properties of Equations Valid Arguments Substitution Replacement A Proof System for Equational Logic Birkhoff's Rules Is There a Strategy for Finding Equational Derivations? Soundness Completeness The Construction of Z n The Proof of the Completeness Theorem Valid Arguments Revisited Chain Derivations Unification Unifiers A Unification Algorithm Properties of Prefix Notation for Terms Notation for Substitutions Verification of the Unification Algorithm Unification of Finitely Many Terms 3.13 Term Rewrite Systems (TRSs) Definition of a TRS Terminating TRSs Normal Form TRSs Critical Pairs Critical Pairs Lemma Terms as Strings Confluence

viii Contents 3.14 Reduction Orderings 240 3.14.1 Definition of a Reduction Ordering 241 3.14.2 The Knuth-Bendix Orderings 242 3.14.3 Polynomial Orderings 247 3.15

5 viii Contents 3.14 Reduction Orderings Definition of a Reduction Ordering The Knuth-Bendix Orderings Polynomial Orderings The Knuth-Bendix Procedure Finding a Normal Form TRS for Groups A Formalization of the Knuth-Bendix Procedure Historical Remarks 256 Chapter 4 Predicate Clause Logic First-Order Languages without Equality Interpretations and Structures Clauses Semantics Reduction to Propositional Logic via Ground Clauses, and the Compactness Theorem for Clause Logic Ground Instances Satisfiable over an Algebra The Herbrand Universe Growth of the Herbrand Universe Satisfiability over the Herbrand Universe Compactness for Predicate Clause Logic without Equality Resolution Substitution Opp-Unification Resolution Soundness and Completeness of Resolution The Unification of Literals Unifying Pairs of Literals The Unification Algorithm for Pairs of Literals Most General Unifiers of Finitely Many Literals Resolution with most General Opp-Unifiers Most General Opp-Unifiers An Opp-Unification Algorithm Resolution and Most General Opp-Unifiers Soundness and Completeness with Most General Opp-Unifiers Adding Equality to the Language 307

Contents IX 4.10 Reduction to Propositional Logic 308 4.10.1 Axiomatizing Equality 308 4.10.2 The Reduction 309 4.10.3 Compactness for Clause Logic with Equality 312 4.10.4 Soundness and Completeness 312 4.

6 Contents IX 4.10 Reduction to Propositional Logic Axiomatizing Equality The Reduction Compactness for Clause Logic with Equality Soundness and Completeness Historical Remarks 313 Part II Logic with Quantifiers 315 Chapter 5 First Order Logic: Introduction, and Fundamental Results on Semantics The Syntax of First-Order Logic First-Order Syntax for the Natural Numbers The Semantics of First-Order Sentences in N Other Number Systems First-Order Syntax for (Directed) graphs The Semantics of First-Order Sentences in (Directed) Graphs Semantics for First-Order Logic Equivalent Formulas Replacement and Substitution Prenex Form Valid Arguments Skolemizing The Reduction of First-Order Logic to Predicate Clause Logic The Compactness Theorem Historical Remarks 365 Chapter 6 A Proof System for First Order Logic, and Gödel's Completeness Theorem A Proof System First Facts about Derivations Herbrand's Deduction Lemma Consistent Sets of Formulas Maximal Consistent Sets of Formulas Adding Witness Formulas to a Consistent Sentence 376

X Contents 6.7 Constructing a Model Using a Maximal Consistent Set of Formulas with Witness Formulas 377 6.8 Consistent Sets of Sentences Are Satisfiable 380 6.9 Gödel's Completeness Theorem 380 6.

7 X Contents 6.7 Constructing a Model Using a Maximal Consistent Set of Formulas with Witness Formulas Consistent Sets of Sentences Are Satisfiable Gödel's Completeness Theorem Compactness Historical Remarks 381 Appendix A Appendix В A Simple Timetable of Mathematical Logic and Computing 385 The Dedekind-Peano Number System 391 Appendix С Writing Up an Inductive Definition or Proof 397 C.l Inductive definitions 397 C.2 Inductive proofs 398 Appendix D The FL Propositional Logic 401 Bibliography 409 Index 413

07 Equational Logic and Algebraic Reasoning

CAS 701 Fall 2004 07 Equational Logic and Algebraic Reasoning Instructor: W. M. Farmer Revised: 17 November 2004 1 What is Equational Logic? Equational logic is first-order logic restricted to languages