First-order logic 一阶逻辑

This completely self-contained study, widely considered the best book in the field, is intended to serve both as an introduction to quantification theory and as an exposition of new results and techniques in "analytic" or "cut-free" methods. Impressed by the simplicity and mathematical elegance of the tableau point of view, the author focuses on it here.
After preliminary material on trees (necessary for the tableau method), Part I deals with propositional logic from the viewpoint of analytic tableaux, covering such topics as formulas of propositional logic, Boolean valuations and truth sets, the method of tableaux and compactness.
　　Part II covers first-order logic, offering detailed treatment of such matters as first-order analytic tableaux, analytic consistency, quantification theory, magic sets and analytic versus synthetic consistency properties.
　　Part III continues coverage of first-order logic. Among the topics discussed are Gentzen systems, elimination theorems, prenex tableaux, symmetric completeness theorems and systems of linear reasoning.

Part Ⅰ. Propositional Logic from the Viewpoint of Analytic Tableaux
　Chapter Ⅰ. Preliminaries
　　0. Foreword on Trees
　　1. Formulas of Propositional Logic
　　2. Boolean Valuations and Truth Sets
　Chapter Ⅱ. Analytic Tableaux
　　1. The Method of Tableaux
　　2. Consistency and Completeness of the System
　Chapter Ⅲ. Compactness
　　1. Analytic Proofs of the Compactness Theorem
　　2. Maximal Consistency: Lindenbaum's Construction
　　3. An Analytic Modification of Lindenbaum's Proof
　　4. The Compactness Theorem for Deducibility
Part Ⅱ. First-Order Logic
　Chapter Ⅳ. First-Order Logic. Preliminaries
　　1. Formulas of Quantification Theory
　　2. First-Order Valuations and Models
　　3. Boolean Valuations vs. First-Order Valuations
　Chapter Ⅴ. First-Order Analytic Tableaux
　　1. Extension of Our Unified Notation
　　2. Analytic Tableaux for Quantification Theory
　　3. The Completeness Theorem
　　4. The Skolem-L6wenheim and Compactness Theorems for First-Order Logic
　Chapter Ⅵ. A Unifying Principle
　　1. Analytic Consistency
　　2. Further Discussion of Analytic Consistency.
　　3. Analytic Consistency Properties for Finite Sets
　Chapter Ⅶ. The Fundamental Theorem of Quantification Theory
　　1. Regular Sets
　　2. The Fundamental Theorem
　　3. Analytic Tableaux and Regular Sets
　　4. The Liberalized Rule D
　Chapter Ⅶ. Axiom Systems for Quantification Theory
　　0. Foreword on Axiom Systems
　　1. The System Q1
　　2. The Systems 02, Q2
　Chapter Ⅸ. Magic Sets
　　1. Magic Sets
　　2. Applications of Magic Sets
　Chapter Ⅹ. Analytic versus Synthetic Consistency Properties
　　1. Synthetic Consistency Properties
　　2. A More Direct Construction
PartⅢ Further Topics in First-Order Logic
　Chapter Ⅺ. Gentzen Systems
　　1. Gentzen Systems for Propositional Logic
　　2. Block Tableaux and Gentzen Systems for First-Order Logic
　Chapter Ⅻ. Elimination Theorems
　　1. Gentzen's Hauptsatz
　　2. An Abstract Form of the Hauptsatz
　　3. Some Applications of the Hauptsatz
　Chapter XIII. Prenex Tableaux
　　1. Prenex Formulas
　　2. Prenex Tableaux
　Chapter XIV. More on Gentzen Systems
　　1. Gentzen's Extended Hauptsatz
　　2. A New Form of the Extended Hauptsatz
　　3. Symmetric Gentzen Systems
　Chapter XV. Craig's Interpolation Lemma and Beth's Definability Theorem
　　1. Craig's Interpolation Lemma
　　2. Beth's Definability Theorem
　Chapter XVI. Symmetric Completeness Theorems
　　1. Clashing Tableaux
　　2. Clashing Prenex Tableaux
　　3. A Symmetric Form of the Fundamental Theorem
　Chapter XVII. Systems of Linear Reasoning
　　1. Configurations
　　2. Linear Reasoning
　　3. Linear Reasoning for Prenex Formulas
　　4. A System Based on the Strong Symmetric Form of the Fundamental
　　Theorem
References