Synopses & Reviews
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
Review
"Overall, this is a well-written text with challenging exercises, proofs of important theorems, and a modern integrated approach."--Choice
Review
"The book can serve as material for a course that teaches the role of logic in several disciplines. It can also be used as a supplementary text for a logic course that emphasizes the more traditional topics of logic but wishes to include a few special topics. Moreover, it can be a valuable resource for researchers and academics."--Roman Murawski, Zentralblatt MATH
Review
"It's always interesting to find a text that reimagines, and offers a novel approach to, a fairly standard subject. This book does that for logic. . . . There is a lot of interesting and well-presented material found here that cannot be easily found elsewhere in a book at this level."--Mark Hunacek, Mathematical Association of America blog
Review
"An instructor of a logic course offered by a mathematics department who is interested in some experimentation will undoubtedly find this book quite rewarding. . . . Even an instructor who is not planning to teach a course along these lines, but who is interested in the subject, will want to look at this text; there is a lot of interesting and well-presented material found here that cannot be easily found elsewhere in a book at this level."--Mark Hunacek, MAA blog
Synopsis
"Formal logic should no longer be taught as a course within a single subject area, but should be taught from an interdisciplinary perspective.
Three Views of Logic has many fine features and combines materials not found together elsewhere. We have needed an accessible textbook like this one for quite some time."
--Hans Halvorson, Princeton University"This concise, precise, and clear textbook is unique in the range of material covered and the level at which it is written, which is intended for undergraduates. The exercises are a considerable help to the student and the examples are useful and interesting."--David Plaisted, University of North Carolina, Chapel Hill
"Loveland, Hodel, and Sterrett are all internationally recognized and leading researchers in their field. Their new textbook gives an excellent introduction to the resolution of propositional and first-order predicate logic, and an outstanding overview of computability theory. The examples and exercises are well chosen, and the material is accessible to students without a logic background."--Frank Wolter, University of Liverpool
About the Author
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.
Table of Contents
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 T
_{n}(e, a
_{1}, . . . , a
_{n}, 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