Three Views of Logic

Donald W. Loveland is professor emeritus of computer science at Duke University and the author of Automated Theorem Proving: A Logical Basis. Richard E. Hodel is associate professor emeritus of mathematics at Duke University and the author of An Introduction to Mathematical Logic. S. G. Sterrett is the Curtis D. Gridley Distinguished Professor of History and Philosophy of Science at Wichita State University and the author of Wittgenstein Flies a Kite: A Story of Models of Wings and Models of the World.

Preface ix Acknowledgments xiii PART 1. Proof Theory 1 Donald Loveland 1Propositional Logic 3 1.1 Propositional Logic Semantics 5 1.2 Syntax: Deductive Logics 13 1.3 The Resolution Formal Logic 14 1.4 Handling Arbitrary Propositional Wffs 26 2Predicate Logic 31 2.1 First-Order Semantics 32 2.2 Resolution for the Predicate Calculus 40 2.2.1 Substitution 41 2.2.2 The Formal System for Predicate Logic 45 2.2.3 Handling Arbitrary Predicate Wffs 54 3An Application: Linear Resolution and Prolog 61 3.1 OSL-Resolution 62 3.2 Horn Logic 69 3.3 Input Resolution and Prolog 77 Appendix A: The Induction Principle 81 Appendix B: First-Order Valuation 82 Appendix C: A Commentary on Prolog 84 References 91 PART 2. Computability Theory 93 Richard E. Hodel 4Overview of Computability 95 4.1 Decision Problems and Algorithms 95 4.2 Three Informal Concepts 107 5A Machine Model of Computability 123 5.1 RegisterMachines and RM-Computable Functions 123 5.2 Operations with RM-Computable Functions; Church-Turing Thesis; LRM-Computable Functions 136 5.3 RM-Decidable and RM-Semi-Decidable Relations; the Halting Problem 144 5.4 Unsolvability of Hilbert's Decision Problem and Thue'sWord Problem 154 6A Mathematical Model of Computability 165 6.1 Recursive Functions and the Church-Turing Thesis 165 6.2 Recursive Relations and RE Relations 175 6.3 Primitive Recursive Functions and Relations; Coding 187 6.4 Kleene Computation Relation Tn(e, a1, ... , an, c) 197 6.5 Partial Recursive Functions; Enumeration Theorems 203 6.6 Computability and the Incompleteness Theorem 216 List of Symbols 219 References 220 PART 3. Philosophical Logic 221 S. G. Sterrett 7Non-Classical Logics 223 7.1 Alternatives to Classical Logic vs. Extensions of Classical Logic 223 7.2 From Classical Logic to Relevance Logic 228 7.2.1 The (So-Called) "Paradoxes of Implication" 228 7.2.2 Material Implication and Truth Functional Connectives 234 7.2.3 Implication and Relevance 238 7.2.4 Revisiting Classical Propositional Calculus: What to Save,What to Change, What to Add? 240 8Natural Deduction: Classical and Non-Classical 243 8.1 Fitch's Natural Deduction System for Classical Propositional Logic 243 8.2 Revisiting Fitch's Rules of Natural Deduction to Better Formalize the Notion of Entailment-Necessity 251 8.3 Revisiting Fitch's Rules of Natural Deduction to Better Formalize the Notion of Entailment-Relevance 253 8.4 The Rules of System FE (Fitch-Style Formulation ofthe Logic of Entailment) 261 8.5 The Connective "Or," Material Implication,and the Disjunctive Syllogism 281 9Semantics for Relevance Logic: A Useful Four-Valued Logic 288 9.1 Interpretations, Valuations, and Many Valued Logics 288 9.2 Contexts in Which This Four-Valued Logic Is Useful 290 9.3 The Artificial Reasoner's (Computer's) "State of Knowledge" 291 9.4 Negation in This Four-Valued Logic 295 9.5 Lattices: A Brief Tutorial 297 9.6 Finite Approximation Lattices and Scott's Thesis 302 9.7 Applying Scott's Thesis to Negation, Conjunction, and Disjunction 304 9.8 The Logical Lattice L4 307 9.9 Intuitive Descriptions of the Four-Valued Logic Semantics 309 9.10 Inferences and Valid Entailments 312 10Some Concluding Remarks on the Logic of Entailment 315 References 316 Index 319