Abstraction Logic: A New Foundation for Reasoning, Computing, and Understanding (Draft)

Abstraction logic is a new logic combining exceptional simplicity with astonishing generality. It might be adequate to describe it as combining the best features of first-order logic and higher-order logic, while avoiding their respective drawbacks.

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.

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.