Here are 100 chapter titles for a book on Mathematical Logic, progressing from beginner to advanced:
I. Propositional Logic (20 Chapters)
- Introduction to Logic: What and Why?
- Propositional Variables and Connectives
- Well-Formed Formulas (WFFs)
- Truth Tables and Logical Equivalence
- Tautologies, Contradictions, and Contingencies
- Logical Implication and Equivalence
- Normal Forms: Conjunctive Normal Form (CNF)
- Normal Forms: Disjunctive Normal Form (DNF)
- Propositional Logic and Digital Circuits
- Axiomatic Systems for Propositional Logic
- Natural Deduction for Propositional Logic
- Resolution for Propositional Logic
- Compactness Theorem for Propositional Logic
- Soundness and Completeness of Propositional Logic
- Applications of Propositional Logic: Puzzle Solving
- Applications of Propositional Logic: Circuit Design
- Applications of Propositional Logic: AI
- Many-Valued Logic: Introduction
- Fuzzy Logic: Basic Concepts
- Propositional Logic: Exercises and Review
II. Predicate Logic (First-Order Logic) (30 Chapters)
- Introduction to Predicate Logic: Quantifiers
- Predicates, Functions, and Constants
- First-Order Languages and Structures
- Well-Formed Formulas in Predicate Logic
- Free and Bound Variables
- Interpretations and Models
- Truth and Validity in Predicate Logic
- Logical Equivalence in Predicate Logic
- Prenex Normal Form
- Skolemization
- Axiomatic Systems for Predicate Logic
- Natural Deduction for Predicate Logic
- Resolution for Predicate Logic
- Herbrand's Theorem
- Gödel's Completeness Theorem
- Compactness Theorem for Predicate Logic
- Löwenheim-Skolem Theorem
- Non-Standard Models
- First-Order Theories: Examples
- Arithmetic and Peano Arithmetic
- Set Theory: Axiomatic Approaches
- Zermelo-Fraenkel Set Theory (ZFC)
- Model Theory: Basic Concepts
- Elementary Embeddings
- Isomorphisms and Automorphisms
- Quantifier Elimination
- Decidability and Undecidability
- Church's Theorem
- Applications of Predicate Logic: Database Theory
- Applications of Predicate Logic: Program Verification
III. Advanced Topics in Logic (30 Chapters)
- Incompleteness Theorems: Introduction
- Gödel's First Incompleteness Theorem: Proof Sketch
- Gödel's Second Incompleteness Theorem: Implications
- Computability Theory: Turing Machines
- Recursive Functions
- Lambda Calculus
- Formal Systems and Computability
- Modal Logic: Introduction
- Kripke Semantics
- Temporal Logic
- Dynamic Logic
- Intuitionistic Logic
- Linear Logic
- Higher-Order Logic
- Type Theory
- Category Theory and Logic
- Topos Theory
- Proof Theory: Cut Elimination
- Proof Complexity
- Model Theory: Advanced Topics
- Saturated Models
- Ultraproducts
- Stability Theory
- Set Theory: Advanced Topics
- Axiom of Choice and its Implications
- Continuum Hypothesis
- Large Cardinals
- Philosophical Implications of Logic
- Logic and Philosophy of Mathematics
- Logic and Computer Science
IV. Further Explorations and Specialized Areas (20 Chapters)
- History of Mathematical Logic
- Foundational Issues in Mathematics
- Logic and Artificial Intelligence: Knowledge Representation
- Logic Programming: Prolog
- Automated Theorem Proving
- Model Checking
- Logic and Linguistics
- Logic and Cognitive Science
- Logic and Law
- Logic and Game Theory
- Logic and Quantum Mechanics
- Logic and Music
- Logic and Art
- Computational Logic
- Finite Model Theory
- Descriptive Complexity
- Applications of Logic in other fields
- Open Problems in Logic
- Future Directions in Logic
- Appendix: Foundational Material and References