本书是一部难度适中的本科生数学教材。主要讲述了什么是数学证明,这些证明怎么能够被验证以及电脑在多大程度上能够执行这些数学证明。本书从一阶逻辑以及它在数学基础中的作用的讲述开始，深入阐述了Trachtenbrot的不可判定性理论，基础等值理论的Fraisse特性，一阶逻辑最大化的Lindstrom定理以及逻辑程序设计的基础这些经典的知识点。
本书适用于数学、计算机科学、人工智能和决策学等各个领域的本科生、研究生以及相关专业的研究人员。

Preface
PART A
I Introduction
1.An Example from Group Theory
2.An Example from the Theory of Equivalence Relations
3.A Preliminary Analysis
4.Preview
II Syntax of First-Order Languages
1. Alphabets
2.The Alphabet of a First-Order Language
3.Terms and Formulas in First-Order Languages
4.Induction in the Calculus of Terms and in the Calculus of Formulas
5.Free Variables and Sentences
III Semantics of First-Order Languages
1.Structures and Interpretations
2.Standardization of Connectives
3.The Satisfaction Relation
4.The Consequence Relation
5.Two Lemmas on the Satisfaction Relation
6.Some Simple Formalizations
7.Some Remarks on Formalizability
8.Substitution
IV A Sequent Calculus
1.Sequent Rules
2.Structural Rules and Connective Rules
3.Derivable Connective Rules
4.Quantifier and Equality Rules
5.Further Derivable Rules and Sequents
6.Summary and Example
7.Consistency
V The Completeness Theorem
1.Henkin's Theorem
2.Satisfiability of Consistent Sets of Formulas (the Countable Case)
3.Satisfiability of Consistent Sets of Formulas (the General Case)
4.The Completeness Theorem
VI The Lowenheim-Skolem and the Compactness Theorem
1.The Lowenheim-Skolem Theorem
2.The Compactness Theorem
3.Elementary Classes
4.Elementarily Equivalent Structures
VII The Scope of First-Order Logic
1.The Notion of Formal Proof
2.Mathematics Within the Framework of First-Order Logic
3.The Zermelo-Fraenkel Axioms for Set Theory
4.Set Theory as a Basis for Mathematics
VIII Syntactic Interpretations and Normal Forms
1.Term-Reduced Formulas and Relational Symbol Sets
2.Syntactic Interpretations
3.Extensions by Definitions
4.Normal Forms
PART B
IX Extensions of First-Order Logic
1.Second-Order Logic
2.The System LW1W
3.The System LQ
X Limitations of the Formal Method
1.Decidability and Enumerability
2.Register Machines
3.The Halting Problem for Register Machines
4.The Undecidability of First-Order Logic
5.Trahtenbrot's Theorem and the Incompleteness of Second-Order Logic
6.Theories and Decidability
7.Self-Referential Statements and Godel's Incompleteness Theorems
XI Free Models and Logic Programming
1.Herbrand's Theorem
2.Free Models and Universal Horn Formulas
3.Herbrand Structures
4.Propositional Logic
5.Propositional Resolution
6.First-Order Resolution (without Unification)
7.Logic Programming
XII An Algebraic Characterization of Elementary Equiva-lence
1.Finite and Partial Isomorphisms
2.Fraisse's Theorem
3.Proof of Fraisse's Theorem
4.Ehrenfeucht Games
XIII Lindstrom's Theorems
1.Logical Systems
2.Compact Regular Logical Systems
3.Lindstrom's First Theorem
4.Lindstrom's Second Theorem
References
Symbol Index
Subject Index