Description
Book SynopsisA comprehensive and user-friendly guide to the use of logic in mathematical reasoning Mathematical Logic presents a comprehensive introduction to formal methods of logic and their use as a reliable tool for deductive reasoning.
Trade Review“Overall, he presents the material as if he were holding a dialogue with the reader. An advanced independent reader with a very strong background in mathematics would find the book helpful in learning this area of mathematics. Summing Up: Recommended.” (
Choice, April 2009)
"The book would be ideas as an introduction to classical logic for students of mathematics, computer science or philosophy. Due to the author's clear and approachable style, it can be recommended to a large circle of readers interested in mathematical logic as well." (Mathematical Review, Issue 2009e)
"I give this outstanding book my highest recommendation, whilst being grateful that excellence in the logic-book 'business' is the very opposite of a zero-sum game: there's plenty of room at the top." (Computing Reviews, November 5, 2008)
Table of ContentsPreface.
Acknowledgments.
PART I: BOOLEAN LOGIC.
1. The Beginning.
1.1 Boolean Formulae.
1.2 Induction on the Complexity of WFF: Some Easy Properties of WFF.
1.3 Inductive definitions on formulae.
1.4 Proofs and Theorems.
1.5 Additional Exercises.
2. Theorems and Metatheorems.
2.1 More Hilbertstyle Proofs.
2.2 Equational-style Proofs.
2.3 Equational Proof Layout.
2.4 More Proofs: Enriching our Toolbox.
2.5 Using Special Axioms in Equational Proofs.
2.6 The Deduction Theorem.
2.7 Additional Exercises.
3. The Interplay between Syntax and Semantics.
3.1 Soundness.
3.2 Post’s Theorem.
3.3 Full Circle.
3.4 Single-Formula Leibniz.
3.5 Appendix: Resolution in Boolean Logic.
3.6 Additional Exercises.
PART II: PREDICATE LOGIC.
4. Extending Boolean Logic.
4.1 The First Order Language of Predicate Logic.
4.2 Axioms and Rules of First Order Logic.
4.3 Additional Exercises.
5. Two Equivalent Logics.
6. Generalization and Additional Leibniz Rules.
6.1 Inserting and Removing "(∀x)".
6.2 Leibniz Rules that Affect Quantifier Scopes.
6.3 The Leibniz Rules "8.12".
6.4 More Useful Tools.
6.5 Inserting and Removing "(∃x)".
6.6 Additional Exercises.
7. Properties of Equality.
8. First Order Semantics -- Very Naïvely.
8.1 Interpretations.
8.2 Soundness in Predicate Logic.
8.3 Additional Exercises.
Appendix A: Gödel's Theorems and Computability.
A.1 Revisiting Tarski Semantics.
A.2 Completeness.
A.3 A Brief Theory of Computability.
A.3.1 A Programming Framework for Computable Functions.
A.3.2 Primitive Recursive Functions.
A.3.3 URM Computations.
A.3.4 Semi-Computable Relations; Unsolvability.
A.4 Godel's First Incompleteness Theorem.
A.4.1 Supplement: øx(x) " is first order definable in N.
References.
Index.
