Preface
Acknowledgments
PART 1. Proof Theory by DONALD W. LOVELAND
1 Propositional Logic
1.1 PropositionalLogicSemantics
1.2 Syntax: Deductive Logics
1.3 The Resolution Formal Logic
1.4 Handling Arbitrary Propositional Wffs
2 Predicate Logic
2.1 First-Order Semantics
2.2 Resolution for the Predicate Calculus
2.2.1 Substitution
2.2.2 The Formal System for Predicate Logic
2.2.3 Handling Arbitrary Predicate Wffs
3 An Application: Linear Resolution and Prolog
3.1 OSL-Resolution
3.2 Horn Logic
3.3 Input Resolution and Prolog
Appendix A: The Induction Principle
Appendix B: First-Order Valuation
Appendix C: A Commentary on Prolog
References 91
PART 2. Computability Theory by RICHARD E. HODEL
4 Overview of Computability
4.1 Decision Problems and Algorithms
4.2 Three Informal Concepts
5 A Machine Model of Computability
5.1 Register Machines and RM-Computable Functions
5.2 Operations with RM-Computable Functions;
Church-Turing Thesis; LRM-Computable Functions
5.3 RM-Decidable and RM-Semi-Decidable Relations; the
Halting Problem
5.4 Unsolvability of Hilbert’s Decision Problem and
Thue’s Word Problem
6 A Mathematical Model of Computability
6.1 Recursive Functions and the Church-Turing Thesis
6.2 Recursive Relations and RE Relations
6.3 Primitive Recursive Functions and Relations; Coding
6.4 Kleene Computation Relation Tn(e, a1, . . . , an, c)
6.5 Partial Recursive Functions; Enumeration Theorems
6.6 Computability and the Incompleteness Theorem
List of Symbols 219 References
PART 3. Philosophical Logic by S. G. STERRETT
7 Non-Classical Logics
7.1 AlternativestoClassicalLogicvs.Extensionsof
Classical Logic
7.2 From Classical Logic to Relevance Logic
7.2.1 The (So-Called) “Paradoxes of Implication”
7.2.2 Material Implication and Truth Functional
Connectives
7.2.3 Implication and Relevance
7.2.4 Revisiting Classical Propositional Calculus: What to
Save, What to Change, What to Add?
8 Natural Deduction: Classical and Non-Classical
8.1 Fitch’s Natural Deduction System for Classical
Propositional Logic
8.2 Revisiting Fitch’s Rules of Natural Deduction to Better
Formalize the Notion of Entailment — Necessity
8.3 Revisiting Fitch’s Rules of Natural Deduction to Better
Formalize the Notion of Entailment — Relevance
8.4 The Rules of System FE (Fitch-Style Formulation of
the Logic of Entailment)
8.5 The Connective “Or,” Material Implication,
and the Disjunctive Syllogism
Semantics for Relevance Logic: A Useful Four-Valued Logic
9.1 Interpretations, Valuations, and Many Valued Logics
9.2 Contexts in Which This Four-Valued Logic Is Useful
9.3 The Artificial Reasoner’s (Computer’s) “State of Knowledge”
9.4 Negation in This Four-Valued Logic
9.5 Lattices: A Brief Tutorial
9.6 Finite Approximation Lattices and Scott’s Thesis
9.7 Applying Scott’s Thesis to Negation, Conjunction, and Disjunction
9.8 The Logical Lattice L4
9.9 Intuitive Descriptions of the Four-Valued Logic Semantics
9.10 InferencesandValidEntailments
Some Concluding Remarks on the Logic of Entailment
References
Index
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