Logic and Proof 的中文译本。
英文原文地址：https://leanprover.github.io/logic_and_proof/index.html
授权于 Apache 2.0 协议
- 1. 介绍/Introduction
- 1.1. 数学证明/Mathematical Proof
- 1.2. 符号逻辑/Symbolic Logic
- 1.3. Interactive Theorem Proving
- 1.4. The Semantic Point of View
- 1.5. Goals Summarized
- 1.6. 关于这本教科书/About this Textbook
- 2. Propositional Logic
- 2.1. A Puzzle
- 2.2. A Solution
- 2.3. Rules of Inference
- 2.4. The Language of Propositional Logic
- 2.5. Exercises
- 3. Natural Deduction for Propositional Logic
- 3.1. Derivations in Natural Deduction
- 3.2. Examples
- 3.3. Forward and Backward Reasoning
- 3.4. Reasoning by Cases
- 3.5. Some Logical Identities
- 3.6. Exercises
- 4. Propositional Logic in Lean
- 4.1. Expressions for Propositions and Proofs
- 4.2. More commands
- 4.3. Building Natural Deduction Proofs
- 4.4. Forward Reasoning
- 4.5. Definitions and Theorems
- 4.6. Additional Syntax
- 4.7. Exercises
- 5. Classical Reasoning
- 5.1. Proof by Contradiction
- 5.2. Some Classical Principles
- 5.3. Exercises
- 6. Semantics of Propositional Logic
- 6.1. Truth Values and Assignments
- 6.2. Truth Tables
- 6.3. Soundness and Completeness
- 6.4. Exercises
- 7. First Order Logic
- 7.1. Functions, Predicates, and Relations
- 7.2. The Universal Quantifier
- 7.3. The Existential Quantifier
- 7.4. Relativization and Sorts
- 7.5. Equality
- 7.6. Exercises
- 8. Natural Deduction for First Order Logic
- 8.1. Rules of Inference
- 8.2. The Universal Quantifier
- 8.3. The Existential Quantifier
- 8.4. Equality
- 8.5. Counterexamples and Relativized Quantifiers
- 8.6. Exercises
- 9. First Order Logic in Lean
- 9.1. Functions, Predicates, and Relations
- 9.2. Using the Universal Quantifier
- 9.3. Using the Existential Quantifier
- 9.4. Equality and calculational proofs
- 9.5. Exercises
- 10. Semantics of First Order Logic
- 10.1. Interpretations
- 10.2. Truth in a Model
- 10.3. Examples
- 10.4. Validity and Logical Consequence
- 10.5. Soundness and Completeness
- 10.6. Exercises
- 11. Sets
- 11.1. Elementary Set Theory
- 11.2. Calculations with Sets
- 11.3. Indexed Families of Sets
- 11.4. Cartesian Product and Power Set
- 11.5. Exercises
- 12. Sets in Lean
- 12.1. Basics
- 12.2. Some Identities
- 12.3. Indexed Families
- 12.4. Power Sets
- 12.5. Exercises
- 13. Relations
- 13.1. Order Relations
- 13.2. More on Orderings
- 13.3. Equivalence Relations and Equality
- 13.4. Exercises
- 14. Relations in Lean
- 14.1. Order Relations
- 14.2. Orderings on Numbers
- 14.3. Equivalence Relations
- 14.4. Exercises
- 15. Functions
- 15.1. The Function Concept
- 15.2. Injective, Surjective, and Bijective Functions
- 15.3. Functions and Subsets of the Domain
- 15.4. Functions and Relations
- 15.5. Exercises
- 16. Functions in Lean
- 16.1. Functions and Symbolic Logic
- 16.2. Second- and Higher-Order Logic
- 16.3. Functions in Lean
- 16.4. Defining the Inverse Classically
- 16.5. Functions and Sets in Lean
- 16.6. Exercises
- 17. The Natural Numbers and Induction
- 17.1. The Principle of Induction
- 17.2. Variants of Induction
- 17.3. Recursive Definitions
- 17.4. Defining Arithmetic Operations
- 17.5. Arithmetic on the Natural Numbers
- 17.6. The Integers
- 17.7. Exercises
- 18. The Natural Numbers and Induction in Lean
- 18.1. Induction and Recursion in Lean
- 18.2. Defining the Arithmetic Operations in Lean
- 18.3. Exercises
- 19. Elementary Number Theory
- 19.1. The Quotient-Remainder Theorem
- 19.2. Divisibility
- 19.3. Prime Numbers
- 19.4. Modular Arithmetic
- 19.5. Properties of Squares
- 19.6. Exercises
- 20. Combinatorics
- 20.1. Finite Sets and Cardinality
- 20.2. Counting Principles
- 20.3. Ordered Selections
- 20.4. Combinations and Binomial Coefficients
- 20.5. The Inclusion-Exclusion Principle
- 20.6. Exercises
- 21. The Real Numbers
- 21.1. The Number Systems
- 21.2. Quotient Constructions
- 21.3. Constructing the Real Numbers
- 21.4. The Completeness of the Real Numbers
- 21.5. An Alternative Construction
- 21.6. Exercises
- 22. The Infinite
- 22.1. Equinumerosity
- 22.2. Countably Infinite Sets
- 22.3. Cantor’s Theorem
- 22.4. An Alternative Definition of Finiteness
- 22.5. The Cantor-Bernstein Theorem
- 22.6. Exercises
- 23. Axiomatic Foundations
- 23.1. Basic Axioms for Sets
- 23.2. The Axiom of Infinity
- 23.3. The Remaining Axioms
- 23.4. Type Theory
- 23.5. Exercises
- 24. Appendix: Natural Deduction Rules