Loveland / Hodel / Sterrett Three Views of Logic

Mathematics, Philosophy, and Computer Science
Course Book
ISBN: 978-1-4008-4875-1
Verlag: De Gruyter
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 344 Seiten

The first interdisciplinary textbook to introduce students to three critical areas in applied logic

Demonstrating the different roles that logic plays in the disciplines of computer science, mathematics, and philosophy, this concise undergraduate textbook covers select topics from three different areas of logic: proof theory, computability theory, and nonclassical logic. The book balances accessibility, breadth, and rigor, and is designed so that its materials will fit into a single semester. Its distinctive presentation of traditional logic material will enhance readers' capabilities and mathematical maturity.

The proof theory portion presents classical propositional logic and first-order logic using a computer-oriented (resolution) formal system. Linear resolution and its connection to the programming language Prolog are also treated. The computability component offers a machine model and mathematical model for computation, proves the equivalence of the two approaches, and includes famous decision problems unsolvable by an algorithm. The section on nonclassical logic discusses the shortcomings of classical logic in its treatment of implication and an alternate approach that improves upon it: Anderson and Belnap's relevance logic. Applications are included in each section. The material on a four-valued semantics for relevance logic is presented in textbook form for the first time.

Aimed at upper-level undergraduates of moderate analytical background, Three Views of Logic will be useful in a variety of classroom settings.
- Gives an exceptionally broad view of logic
- Treats traditional logic in a modern format
- Presents relevance logic with applications
- Provides an ideal text for a variety of one-semester upper-level undergraduate courses

Autoren/Hrsg.

Loveland, Donald W.

Hodel, Richard

Sterrett, S. G.

Sterrett, Susan

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein Geschichte der Mathematik
• Mathematik | Informatik EDV | Informatik Informatik
• Geisteswissenschaften Philosophie Philosophische Logik, Argumentationstheorie
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Optimierung
• Mathematik | Informatik Mathematik Mathematik Allgemein Philosophie der Mathematik
• Mathematik | Informatik Mathematik Mathematik Allgemein Mathematische Logik

Inhaltsverzeichnis

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

Über Autor(innen)

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.