Three Views of Logic

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