Abstraction Logic
Abstraction Logic
A New Foundation for Reasoning, Computing, and Understanding
About the Book
Abstraction logic is a new logic combining exceptional simplicity with astonishing generality. It combines the best features of first-order logic and higher-order logic, while avoiding their respective drawbacks. It manages to do so because it is based on a simple understanding of the mathematical universe, its operations and, in particular, its operators. Abstraction algebra encodes this understanding as a formal language, generalising abstract algebra. It is the right setting for the treatment of alpha equivalence. Abstraction logic then turns abstraction algebra into a logic by considering truth values as a partially ordered substructure of the mathematical universe. A key property of this logic is that formulas are merely terms. Among the presented proof systems are natural deduction, which is sound if truth values form a complete lattice, and sequent calculus, which is sound if truth values form a complete bi-Heyting algebra. By constructing the Rasiowa model, we prove that natural deduction is a complete proof system for abstraction logic.
This is the first book on abstraction logic. It presents abstraction logic in its most recent and comprehensive form, and supersedes all previous publications on abstraction logic.
With the purchase of this Founder's Edition of the book, you also gain access to any future chapters and updates that are released.
Table of Contents
- Preface
- Chapter 1. The Mathematical Universe
- Operations
- Cantor's Theorem
- Are Operations Mathematical Objects?
- Operators
- Chapter 2. Abstraction Algebra
- Syntax
- Semantics
- Alpha Equivalence
- De Bruijn Terms and Alpha Conversion
- Evaluating De Bruijn Terms
- Alpha Inversion
- Defining Substitution
- Composition of Substitutions
- Evaluation and Substitution
- De Bruijn Abstraction Algebra
- Concluding Remarks
- Chapter 3. Abstraction Logic
- Truth Values and Logical Order
- Complete Bi-Heyting Algebras
- Formulas, Rules, and Sequents
- Logic, Models, and Valuation Spaces
- Proofs and Proof Systems
- Line Notation
- Soundness
- Deduction Logic with Equality
- Extensions
- Deduction Theorem
- Consistency
- Completeness
- Rasiowa Model
- Refutations and Refutation-Completeness
- Is Sequent Calculus Complete?
- Concluding Remarks
- Bibliography
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