Abstract
A comprehensive introduction to mathematical logic covering propositional logic, predicate logic, completeness, model theory, and intuitionistic logic. The book provides a balanced treatment of classical and intuitionistic systems, making it particularly valuable for understanding constructive approaches to logic. Van Dalen presents the material with mathematical rigor while maintaining accessibility, including detailed proofs of soundness, completeness, and cut-elimination. The text is widely used for its clear exposition of both semantic and proof-theoretic methods.
Outline
Chapter 1: Propositional Logic
- Syntax and semantics of propositional logic
- Truth tables and valuations
- Tautologies and logical consequence
- Natural deduction for propositional logic
- Completeness and soundness
Chapter 2: Predicate Logic
- Predicate calculus syntax
- Structures and interpretations
- Quantifiers and substitution
- Free and bound variables
- Semantic consequence
Chapter 3: Completeness and Applications
- Gödel’s completeness theorem
- Henkin’s proof using canonical models
- Compactness theorem and applications
- Löwenheim-Skolem theorems
- Elementary equivalence and elementary extensions
Chapter 4: Second-Order Logic
- Syntax and semantics of second-order logic
- Incompleteness of second-order logic
- Standard versus Henkin semantics
- Categoricity results
Chapter 5: Intuitionistic Logic
- Intuitionistic propositional logic
- Kripke semantics for intuitionistic logic
- Natural deduction for intuitionistic logic
- Heyting algebras
- Relation to constructive mathematics
Chapter 6: Normalization
- Cut-elimination for sequent calculus
- Normalization for natural deduction
- Gentzen’s Hauptsatz
- Applications to consistency proofs
Chapter 7: Gödel’s Theorems
- Arithmetization of syntax
- Representability in Peano arithmetic
- Gödel’s first incompleteness theorem
- Gödel’s second incompleteness theorem
- Undecidability results